Obstructions to Rational and Integral Points by David Alexander Corwin Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of PhD in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 ○c Massachusetts Institute of Technology 2018. All rights reserved.
Author...... Department of Mathematics May 18, 2018
Certified by...... Bjorn Poonen Claude Shannon Professor of Mathematics Thesis Supervisor
Accepted by...... Davesh Maulik Graduate Co-Chair, Department of Mathematics 2 Obstructions to Rational and Integral Points by David Alexander Corwin
Submitted to the Department of Mathematics on May 18, 2018, in partial fulfillment of the requirements for the degree of PhD in Mathematics
Abstract In this thesis, I study two examples of obstructions to rational and integral points on varieties. The first concerns the 푆-unit equation, which asks for solutions to 푥 + 푦 = 1 with 푥 and 푦 both 푆-units, or units in 푍 = Z[1/푆]. This is equivalent to finding the set of 푍-points 1 of P ∖ {0, 1, ∞}. We follow work of Dan-Cohen–Wewers and Brown in applying a motivic version of the non-abelian Chabauty’s method of Minhyong Kim to find polynomials in 푝-adic polylogarithms that vanish on this set of integral points. More specifically, we extend the computations already done by Dan-Cohen–Wewers to the integer ring Z[1/3], and we provide some significant simplifications to a previous algorithm of Dan-Cohen, especially inthecase of Z[1/푆]. One of the reasons for doing this is to verify cases of Kim’s conjecture, which states that these 푝-adic functions precisely cut out the set of integral points. This is joint work with Ishai Dan-Cohen. The second is about obstructions to the local-global principle. The étale Brauer-Manin obstruction of Skorobogatov can be used to explain the failure of the local-global principle for many algebraic varieties. In 2010, Poonen gave the first example of failure of the local-global principle that cannot be explained by the étale Brauer-Manin obstruction. Further obstructions such as the étale homotopy obstruction and the descent obstruction are unfortunately equivalent to the étale Brauer-Manin obstruction. However, Poonen’s construction was not accompanied by a definition of a new, finer obstruction. Here, we present a possible definition for such an obstruction by applying the Brauer-Manin obstruction to each piece of every stratification of the variety. We prove that this obstruction is necessary and sufficient, over imaginary quadratic fields and totally real fields unconditionally, andover all number fields conditionally on the section conjecture. This is part of a joint project with Tomer Schlank.
Thesis Supervisor: Bjorn Poonen Title: Claude Shannon Professor of Mathematics
3 4 Acknowledgments
To my father Jim Corwin, who inspired my love of mathematics, served as my first teacher of math, found enrichment programs in math for me, and provided the support for me to attend Princeton University. To my mother Pam Shira Fleetman, who gave me the unconditional love, support, and equanimity needed for me to succeed; inspired me to value education and academic achievement; taught me the value of hard work; encouraged my intellectual curiosity; and raised me to think analytically. I would like to thank all of the teachers at Mathcamp for giving me a chance to learn more math and in particular, for introducing me to these ideas. Dave Jensen introduced me to algebraic geometry, Nina White taught me topology, and Mark Krusemeyer taught me complex analysis and some ring and field theory. Julian Rosen introduced me to fundamental groups, in which I first asked whether there was a connection to Galois theory, prompting me to discover étale fundamental groups a bit later. David Roe also taught me a lot about elliptic curves in my final year and helped inspire me to learn algebraic number theory at Princeton. And of course, Mira Bernstein, created the whole environment behind this. I would like to thank my professors at Princeton, especially Chris Skinner, who taught me algebraic number theory and advised my independent work. I would of course like to thank my thesis advisor Bjorn Poonen, especially for the countless hours he put into advising me for multiple years, and in particular for reading over and giving comments on much of my writing. I would like to thank Ishai Dan-Cohen for introducing me to the project in Part I and working on it with me. I would also like to thank the 2015 Algebraic Geometry Summer Institute, which is where we met. I would also like to thank Francis Brown for providing ideas that went into the current work. I would like to thank Tomer Schlank, both for spending lots of time advising me as an advisor would do for a student, as well as working on this project with me. I would like to thank the Hebrew University of Jerusalem for financial support for me to
5 visit Tomer Schlank at HUJI. I have a number of specific acknowledgements for Part II. I would like to thank Borys Kadets for a simplification to the proof of Proposition 8.1.2 and Will Sawin for the version of Hensel’s Lemma in Lemma 8.1.3 ([hs]). He would also like to thank Alexei Skorobogatov for discussing an earlier version of the paper, Gereon Quick for responding to queries about étale homotopy, and Jakob Stix for providing a reference. I would like to thank his host at ENS, Olivier Wittenberg, for going through parts of the paper and help on some of the arguments, especially in Section 6.4 and on a model-theoretic argument in the proof of Theorem 7.1.4. He would finally like to thank his thesis advisor Bjorn Poonen for an enormous amountof help, including multiple careful readings of the paper and numerous discussions about the ideas. I would like to thank the Chateaubriand Fellowship for providing me the opportunity to spend a semester at ENS in Paris, where I worked on both of these projects. I would like to thank MIT and the MIT math department as a whole. Barbara Peskin was especially helpful in helping navigate through all of my degree milestones. I would like to thank the other number theory students at MIT for creating a wonderful environment for doing number theory. I would also like to thank specific people in my year for helping me navigate the administrative work surrounding the thesis and the degree, especially Eva Belmont and Renee Bell. I would like to thank Eva Belmont, Oron Propp, and Nicholas Triantafillou for help submitting the thesis while I was abroad.
6 This doctoral thesis has been examined by a Committee of the Department of Mathematics as follows:
Professor Bjorn Poonen ...... Thesis Supervisor Claude Shannon Professor of Mathematics
Professor Wei Zhang ...... Member, Thesis Committee Professor of Mathematics
Professor Davesh Maulik...... Member, Thesis Committee Professor of Mathematics 8 Contents
1 Combined Introduction 13
1 I Motivic Chabauty-Kim Theory for P ∖ {0, 1, ∞} 15
2 Introduction 17 2.1 Motivic Non-Abelian Chabauty’s Method ...... 17 2.1.1 Classical Chabauty’s Method ...... 17 2.1.2 Minhyong Kim’s Non-abelian Chabauty ...... 18 2.1.3 Motivic Chabauty-Kim Theory ...... 19 2.1.4 Motivic Periods ...... 19 2.2 This Project ...... 20 2.3 Notation ...... 22
3 Main Result 23 3.1 Technical Preliminaries ...... 23 3.1.1 Generalities on Graded Pro-Unipotent Groups ...... 23 3.1.2 The Various Fundamental Groups ...... 31 3.1.3 Cohomology and Cocycles ...... 36 3.1.4 p-adic Realization and Kim’s Cutter ...... 38 3.1.5 Chabauty-Kim Locus ...... 39 3.1.6 The Polylogarithmic Quotient ...... 40
9 3.1.7 Kim’s Conjecture ...... 41 3.2 Coordinates ...... 42 3.2.1 Coordinates on the Fundamental Group ...... 42 3.2.2 Generating 퐴(푍) ...... 44 3.2.3 Coordinates on the Space of Cocycles ...... 46 3.2.4 Kummer and Period Maps in Coordinates ...... 51
3.3 Computations for 푍 = Spec Z[1/푆] ...... 52 3.3.1 Abstract Coordinates for 푍 = Spec Z[1/푆] ...... 52 3.3.2 The Geometric Step for Z[1/ℓ] in Half-Weight 4 ...... 55 3.3.3 Coordinates on the Galois Group for 푍 = Z[1/ℓ] ...... 56 3.3.4 The Chabauty-Kim Locus for Z[1/3] in Half-Weight 4 ...... 61
3.4 Answer to Question 3.1.24 and 푆3-Symmetrization ...... 61 3.4.1 Answer to Question 3.1.24 ...... 61
3.4.2 푆3-Symmetrization ...... 63
II Obstructions to the Local-Global Principle 65
4 Introduction and Setup 67 4.1 Introduction ...... 67 4.1.1 Notation and Conventions ...... 69 4.2 Obstructions to Rational Points ...... 70 4.2.1 Generalized Obstructions ...... 70 4.2.2 Functor Obstructions ...... 72 4.2.3 Descent Obstructions ...... 73 4.2.4 Comparison with Brauer and Homotopy Obstructions ...... 75 4.3 Statement of Our Main Results ...... 76 4.3.1 A New Obstruction ...... 78 4.3.2 Relationship to the Birational Section Conjecture ...... 80
10 5 Main Result via Embeddings 81 5.1 General Setup ...... 81 5.2 Embeddings of Varieties into Tori ...... 83 5.2.1 The Result When 푘 is Imaginary Quadratic ...... 85 5.2.2 The Result When 푘 is Totally Real ...... 85 5.3 Finite Descent and the Section Conjecture ...... 86 5.3.1 Grothendieck’s Section Conjecture ...... 87 5.3.2 VSA via the Section Conjecture ...... 93 5.4 Finite Abelian Descent for all 푘 ...... 94
6 The Fibration Method 97 6.1 The Homotopy Section Conjecture ...... 97 6.1.1 Basepoints and Homotopy Groups ...... 100 6.1.2 Obstructions to the Hasse Principle ...... 103 6.2 Homotopy Section Conjecture in Fibrations ...... 105 6.3 Elementary Fibrations ...... 112 6.3.1 Good Neighborhoods ...... 114 6.4 Cuspidal Sections in Arbitrary Dimension ...... 116 6.4.1 Cuspidal Sections Associated to a Compactification ...... 117 6.4.2 Cuspidal Sections in Fibrations ...... 118
7 Examples 119 7.1 Poonen’s Counterexample ...... 119 7.1.1 Conic Bundles ...... 119 7.1.2 Poonen’s Variety ...... 120 7.1.3 VSA Stratifications and Open Covers ...... 122 7.2 The Brauer-Manin obstruction applied to ramified covers ...... 124 7.2.1 Quasi-Torsors and the Ramified Etale-Brauer Obstruction . . . . 124 7.2.2 Quasi-torsors in Poonen’s Example ...... 126
11 8 Appendices to Part II 135 8.1 Appendix A: Obstructions Without Functors ...... 135 8.2 Appendix B: Reformulation in Terms of Cosheaves ...... 138
12 Chapter 1
Combined Introduction
A common question in number theory and algebraic geometry is to ask whether a system of polynomial equations has a solution in a ring 푅, usually a field or a ring of 푆-integers. This can be rephrased as asking whether the set 푋(푅) of 푅-points of a certain scheme 푋 over 푅 is nonempty. To answer questions like these, arithmetic geometers make two important observations:
∙ It is easier to understand the set of 푅-points when 푅 is a completion (or collection of completions).
∙ There are topological invariants of schemes that contain arithmetic information, and we can understand points by their effect on these invariants.
In line with these two philosophies, one can map 푋(푅) both to a “local” version 푋(푅)loc
and a “cohomological” or “homotopical” version 푋(푅)hom, respectively. In this general framework, one obtains a diagram
푋(푅) / 푋(푅)loc .
휅 휅loc 푋(푅) / 푋(푅) hom loc hom,loc
−1 One then considers the set 휅loc(Im(loc)) in 푋(푅)loc and uses it to obtain information
13 about 푋(푅). This thesis considers two examples of this framework.
∙ The first is that of Chabauty’s method and its non-abelian variant due to Minhyong Kim. In this case, the local version is the set of points at a single completion of 푅, and the cohomological version is the non-abelian Galois (or motivic) cohomology of the unipotent fundamental group of 푋.
∙ The second is that of obstructions to the local-global principle such as the Brauer-Manin obstruction (resp., étale homotopy obstruction). In this case, the local version is the set of points with values in the adeles of 푅, and the cohomological version is the Brauer group of 푅 (resp., the set of homotopy fixed points 푋(ℎ푅)).
The first example forms Part I, is joint work with Ishai Dan-Cohen, and will soonappear as a preprint. The second is Part II, is joint work with Tomer Schlank, and largely overlaps with the preprint [CS17].
14 Part I
Motivic Chabauty-Kim Theory for 1 P ∖ {0, 1, ∞}
15 16 Chapter 2
Introduction
2.1 Motivic Non-Abelian Chabauty’s Method
2.1.1 Classical Chabauty’s Method
The Chabauty-Skolem method proves that certain equations have finitely many integral solutions, and, by innovations of Coleman, allows one to find all solutions. The idea isto embed the scheme 푋 defined by the equations (which, geometrically, should be a smooth
irreducible curve of negative Euler characteristic) over an open integer scheme 푍 = 풪퐾 [1/푆] (for 퐾 a number field and 푆 a finite set of finite places of 퐾) into a scheme 퐽 with more structure, specifically that of a semiabelian variety (to do this, one takes a generalized Jacobian of 푋). One then choose a closed point p of 푍 and looks at the intersection of the
푝-adic closure of 퐽(푍) with 푋(푍p) (with 푍p the completed local ring of 푍 at p) inside the
푝-adic analytic space 퐽(푍p) (where p is a prime of good reduction for 푋 lying above 푝). The method of Coleman then sometimes allows one to find explicit 푝-adic analytic functions on
푋(푍p) that vanish on 푋(푍) and have finitely many zeroes (i.e., Coleman functions).
17 2.1.2 Minhyong Kim’s Non-abelian Chabauty
This method often does not work, because 퐽(푍) can be too large. Philosophically, the reason why 퐽(푍) can be large while 푋(푍) remains finite is because the geometric fundamental group of 퐽 is abelian (as opposed to hyperbolic curves like 푋, whose fundamental groups are non-abelian, even center-free)1. The groundbreaking work of Minhyong Kim ([Kim05]) gets around this fact. First, it reinterprets the Chabauty-Skolem method by replacing 퐽(푍) (or its 푝-adic closure) by a corresponding 푝-adic Selmer group. By the work of Bloch-Kato ([BK90]), this can be expressed intrinsically in terms of the 푝-adic Tate module of 퐽, which
is the abelianization of the pro-푝 étale fundamental group of 푋퐾 . Then, one replaces this abelianization by a mildly non-abelian version, namely, the 푛th quotient of the descending
central series of the pro-푝 étale fundamental group of 푋퐾 for a natural number 푛.
Thus the 푝-adic Selmer group is replaced by the Selmer variety Sel(푋/푍)푛 of [Kim09]. In particular, for a place p of 퐾 over 푝, Kim constructs a commutative diagram ([Kim05],[Kim09])
푋(푍) / 푋(푍p)
휅 휅p Sel(푋/푍)푛 / Sel(푋/푍p)푛 loc푛
1 The goal, which was realized when 푋 = P ∖ {0, 1, ∞} in [Kim05], is to prove that the
morphism of schemes loc푛 is non-dominant for sufficiently large 푛. This means that there 푍 is a nonzero ideal ℐ푛 of functions on Sel(푋/푍p)푛 that vanish on the image of loc푛. These
functions pull back via 휅p to Coleman functions on 푋(푍p), which, by the commutativity of
this diagram, vanish on 푋(푍). We let 푋(푍p)푛 be the set of common zeroes of all pullbacks of 푍 elements of ℐ푛 , and we note that this sequence of sets is decreasing in 푛 and contains 푋(푍).
The map 휅푝 can be expressed in terms of 푝-adic iterated integrals, which makes the method more concrete.
1 The original work of Kim ([Kim05]) dealt with the case of 푋 = P ∖ {0, 1, ∞}. That work
1This is not to say that all varieties with abelian fundamental group have infinitely many integral points, but that in the context of the distinction between a curve and its Jacobian, this principle applies.
18 푍 shows that ℐ푛 is nonzero for sufficiently large 푛 and hence that 푋(푍p)푛 is finite for such 푛, thereby reproving a 1929 theorem of Siegel that 푋(푍) is finite. Part I of this thesis is
part of the effort to develop algorithms for computing 푋(푍p)푛 and compute examples for specific 푛 and 푍, in hopes of giving evidence for the conjecture of Kim ([BDCKW]) that
푋(푍) = 푋(푍p)푛 for sufficiently large 푛.
2.1.3 Motivic Chabauty-Kim Theory
In order to compute this set concretely (i.e., in terms that can be applied to computing
푍 elements of ℐ푛 ), it is actually better to upgrade this Galois cohomology to a motivic version of the Selmer variety, as developed in [Had11] and [DCW16]. This motivic Selmer variety is
MT just the group cohomology of the mixed Tate Galois group 휋1 (푍) of 푍, as developed in [DG05], with coefficients in (a quotient depending on 푛 of) the unipotent fundamental
un group 휋1 (푋) of 푋. This group (as well as all of the quotients we take) has the structure MT of a mixed Tate motive, i.e., an action of 휋1 (푍). We in fact work only with a certain PL un quotient 휋1 (푋) of 휋1 (푍), known as the polylogarithmic quotient, as well as its finite- PL dimensional quotients 휋1 (푋)≥푛. In this case, we refer to the Chabauty-Kim ideal and loci 푍 by ℐPL,푛 and 푋(푍p)PL,푛, respectively. The reason why this can be made explicit is that MT 휋1 (푍) can be described as an abstract (pro-algebraic) group over Q, which ultimately results from Borel’s computation of the algebraic K-theory of integer rings. More specifically,
MT un un 휋1 (푍) = 휋1 (푍) o G푚, where 휋1 (푍) is isomorphic to a free pro-unipotent group on the (︃ ∞ )︃∨ graded vector space 퐾2푛−1(푍)Q . This is in contrast with classical Galois groups of 푛=1 number fields, which cannot be easily described as abstract profinite groups.
2.1.4 Motivic Periods
MT While we have an abstract description of the group 휋1 (푍) ultimately coming from algebraic K-theory and Voevodsky’s work on motives, we need to understand this group in more
concrete terms. More specifically, we need to compute the image of the map loc푛 in the diagram, which is a 푝-adic regulator map. This leads to motivic polylogarithms of the
19 u u form Li푛(푧) for 푛 ∈ Z≥1 and 푧 ∈ Q, as well as motivic logarithms log (푧) and motivic u zeta values 휁 (푛) for 푛 ∈ Z≥1 (Section 3.2.2). These are all elements of the coordinate ring un un 푝 풪(휋1 (푍)) of 휋1 (푍) whose 푝-adic realizations are the Coleman 푝-adic polylogarithms Li푛(푧). There is an explicit formula for the (reduced) coproduct of these elements in the Hopf algebra
un 풪(휋1 (푍)), due to Goncharov ([Gon05a]):
푛−1 ∑︁ (logu(푧))푖 Δ′ Liu (푧) := Δ Liu (푧) − 1 ⊗ Liu (푧) − Liu (푧) ⊗ 1 = Liu (푧) ⊗ . 푛 푛 푛 푛 푛−푖 푖! 푖=1
Our current work, building on ideas in [DCW16] and [DC15], shows how this can be used to
un algorithmically compute bases of 풪(휋1 (푍)) in low degrees.
2.2 This Project
Before this work, the only cases that had been computed were those for which 푍 = Z, Z[1/2], both in weights 푛 = 2, 4 (in [DCW16]). In addition, an algorithm had been proposed, and this algorithm was shown to compute the set of points assuming various conjectures ([DC15]). In this work, we make the algorithm more efficient and explicit and use it to findan
푍 element of ℐPL,4 for 푍 = Z[1/3]. More specifically, we have:
Theorem 2.2.1 (Theorem 3.3.6). The element
(︂ )︂ u u u 12 u −1 u u u 휁 (3) log (3) Li4 − − Li4(3) + (2푤2(9)) Li4(9) log Li3 푤2(9) u 3 u (︂ (︂ )︂)︂ (log ) Li1 u u 12 u −1 u − 휁 (3) log (3) − 4 − Li4(3) + (2푤2(9)) Li4(9) 24 푤2(9)
un 푍 of 풪(ΠPL,4 × 휋1 (푍)) (defined in Section 3.1.6) is in ℐPL,4 for 푍 = Spec Z[1/3], where 푤2(9) 26 is a number 푝-adically close to − for 푝 = 5, 7. 3
By computing values of this function at elements of 푋(푍p)PL,2 already found in [BDCKW], we get:
20 Theorem 2.2.2 (Theorem 3.3.7). For 푍 = Spec Z[1/3], we have 푋(푍p)PL,4 ⊆ {−1} for 푝 = 5, 7.
Part of the reason for computing information about these Chabauty-Kim loci is that they help verify cases of the following conjecture of Kim:
Conjecture 2.2.3 (Conjecture 3.1 of [BDCKW]). 푋(푍) = 푋(푍p)푛 for sufficiently large 푛.
In order to do computations, it is easier to work only with polylogarithms, i.e., only
푍 with the polylogarithmic quotient. Luckily, not only is ℐ푛 nonzero for sufficiently large 푛, 푍 but ℐPL,푛 is as well, and hence 푋(푍p)PL,푛 is finite for such 푛. To make the method truly computable, one would like an analogue of Conjecture 2.2.3 for the polylogarithmic quotient, which leads one to ask the question:
Question 2.2.4 (Question 3.1.24). Does 푋(푍) = 푋(푍p)PL,푛 for sufficiently large 푛?
A positive answer would be a strengthening of Conjecture 2.2.3. However, we prove:
Theorem 2.2.5 (Theorem 3.4.2). For any prime ℓ and positive integer 푛, we have
−1 ∈ 푋(푍p)PL,푛, where 푍 = Spec Z[1/ℓ]. In particular, for ℓ odd, Question 2.2.4 has a negative answer.
In particular, it does not make sense to make a version of Kim’s conjecture with 푋(푍p)PL,푛
in place of 푋(푍p)푛. However, there is an action of 푆3 on the scheme 푋, and we may use it to propose a strengthening of Kim’s conjecture. We write
푆3 ⋂︁ 푋(푍p)PL,푛 := 휎(푋(푍p)PL,푛). 휎∈푆3
Our strengthened Kim’s conjecture is the following:
푆3 Conjecture 2.2.6 (Conjecture 3.1.25). 푋(푍) = 푋(푍p)PL,푛 for sufficiently large 푛.
21 As explained in Section 3.1.7, Conjecture 2.2.6 implies Conjecture 2.2.3.
We then use our computation from Theorem 2.2.2 to verify Conjecture 2.2.6 for 푍 = Z[1/3] and 푝 = 5, 7:
Theorem 2.2.7 (Theorem 3.4.4). For 푍 = Spec Z[1/3] and 푝 = 5, 7, Conjecture 2.2.6 (and hence Conjecture 3.1.23) holds (with 푛 = 4).
2.3 Notation
For a scheme 푌 , we let 풪(푌 ) denote its coordinate ring. If 푅 is a ring, we let 푌 ⊗ 푅 or 푌푅 denote the product (or ‘base-change’) 푌 × Spec 푅. If 푌 and Spec 푅 are over an implicit base
scheme 푆 (often Spec Q), we take the product over 푆. Similarly, if 푀 is a linear object (such
as a module, an algebra, a Lie algebra, or a Hopf algebra), then 푀푅 denotes 푀 ⊗ 푅 (again, with the tensor product taken over an implicit base ring). If 푓 : 푌 → 푍 is a morphism of schemes, we denote by 푓 # : 풪(푍) → 풪(푌 ) the corresponding homomorphism of rings. Similarly, if 훼 ∈ 푌 (푅), we have a homomorphism 훼# : 풪(푌 ) → 푅.
22 Chapter 3
Main Result
3.1 Technical Preliminaries
This work builds on the work of [DCW16], [DC15], and [Bro17]. We recall some of the important objects in the theory.
3.1.1 Generalities on Graded Pro-Unipotent Groups
Conventions for Graded Vector Spaces
Our definition of graded vector space is the following:
Definition 3.1.1. A graded vector space is a collection of vector spaces 푉푖 indexed by 푖 ∈ Z.
Definition 3.1.2. A graded vector space is positive (respectively negative, strictly positive,
strictly negative) if 푉푖 = 0 for 푖 < 0 (respectively, for 푖 > 0, for 푖 ≤ 0, for 푖 ≥ 0).
In general, we will only consider graded vector spaces satisfying one of these four conditions.
Furthermore, unless otherwise stated, we will only consider graded vector spaces {푉푖} such that each 푉푖 is finite-dimensional, which ensures that the double dual is the identity. One must then be careful when taking tensor constructions, as follows. Specifically, we only consider the tensor product between two graded vector spaces if they are either both positive
23 or both negative, and we consider the tensor algebra only of a strictly positive or strictly negative graded vector space. We now make a technical remark. For a collection
푉 = {푉푖}푖∈Z
of (finite-dimensional) vector spaces indexed by the integers, we may either take thedirect sum 푉 := 푉푖 푖 or the direct product ∏︀ ∏︁ 푉 := 푉푖 푖 as our notion of a graded vector space. In general, we use the former for coordinate rings and Lie coalgebras and the latter for universal enveloping algebras and Lie algebras. As all of our pro-unipotent groups will be negatively graded, we use the notion for positively ∏︁ graded vector spaces and the notion for negatively graded vector spaces. ∏︁ When considering the notion, we take completed tensor product instead of tensor product (and similarly for tensor algebras and universal enveloping algebras), and a coproduct
is a complete coproduct, i.e., a homomorphism 푉 → 푉 ⊗̂︀푉 . In addition, a set of homogeneous elements is considered a basis if it generates 푉 in each degree, and a similar remark applies to bases of algebras and Lie algebras. When taking the dual of a negatively graded vector space, we take the graded dual and then view the resulting (positively) graded vector space via the notion. In particular, the double dual is always the original vector space. Nonetheless, we have the following relation between graded and ordinary (non-graded) duals:
︀∏ (푉 )∨ = (푉 ∨) .
Graded Pro-unipotent Groups
Let 푈 be a pro-unipotent group over Q.
24 Then 푈 is a group scheme over Q, so its coordinate ring 풪(푈) is a Hopf algebra over Q. We recall that if 풪(푈) is a Hopf algebra over Q, then it is equipped with a product 풪(푈)⊗풪(푈) → 풪(푈), a coproduct Δ: 풪(푈) → 풪(푈) ⊗ 풪(푈), and a counit 휖: 풪(푈) → Q. The kernel 퐼(푈) of 휖 is known as the augmentation ideal. We also write Δ′(푥) := Δ(푥) − 푥 ⊗ 1 − 1 ⊗ 푥 for the reduced coproduct. Finally, we say that an element is primitive if it is in the kernel of Δ′. We say that a Hopf algebra 퐴 is a graded Hopf algebra if the multiplication 퐴 ⊗ 퐴 → 퐴,
the coproduct 퐴 → 퐴 ⊗ 퐴, unit Q → 퐴, and counit 퐴 → Q, are morphisms of graded vector spaces, where 퐴 ⊗ 퐴 has the standard grading on a tensor product, and Q is in degree zero.
Definition 3.1.3. By a grading on 푈, we mean a positive grading on 풪(푈) as a Q-vector space such that the degree zero part of 풪(푈) is one-dimensional over Q.
Definition 3.1.4. If 퐴 is a Hopf algebra graded in the sense of Definition 3.1.3, we let Δ푛 ′ ′ and Δ푛 denote the restrictions of Δ and Δ , respectively, to 퐴푛, the 푛th graded piece.
′ Furthermore, Δ푛 and Δ푛 map 퐴푛 into the 푛th graded piece of 퐴 ⊗ 퐴, which is
퐴푖 ⊗ 퐴푗. 푖+푗=푛
′ Definition 3.1.5. For 푖 + 푗 = 푛 and 푖, 푗 ≥ 0, we let Δ푖,푗 and Δ푖,푗 denote the projections of ′ Δ푛 and Δ푛, respectively, to 퐴푖 ⊗ 퐴푗. One may check via the axioms defining a Hopf algebra ′ that Δ푖,푗 = 0 when either of 푖 or 푗 is zero.
The reduced coproduct Δ′ induces a (graded) Lie coalgebra structure on 퐼(푈)/퐼(푈)2, and the dual Lie algebra (퐼(푈)/퐼(푈)2)∨ is the Lie algebra n of 푈. It is a strictly negatively graded pro-nilpotent Lie algebra. We let 풰푈 = 풰n denote the dual Hopf algebra of 풪(푈), which is the (completed) universal enveloping algebra of n. The composition
′ ∨ ∨ ker(Δ ) ˓→ 풰푈 = 풪(푈) 퐼(푈)
induces an isomorphism between the set of primitive elements of 풰푈 and the Lie algebra n = (퐼(푈)/퐼(푈)2)∨ ⊆ 퐼(푈)∨.
25 Furthermore, for a Q-algebra 푅, we may identify 푈(푅) with the group of grouplike elements in (풰푈)푅, i.e., 푥 such that
Δ푥 = 푥 ⊗ 푥.
Evaluation of an an element of 풪(푈) on an element of 푈(푅) is given by evaluation on the corresponding grouplike element of (풰푈)푅. The functor sending 푈 to n is known to be an equivalence of categories between graded pro-unipotent groups and strictly negatively graded pro-nilpotent Lie algebras. For each positive integer 푛, the set of elements n<−푛 is a Lie ideal, and we denote by n≥−푛 the quotient n/n<−푛. We denote the corresponding quotient pro-unipotent group by
푈≥−푛, and it is a unipotent algebraic group. In fact, 푈 is the inverse limit
lim 푈 . ←− ≥−푛 푛
Example 3.1.6. If n is a one-dimensional Lie algebra generated by an element 푥, then n is nilpotent. We have 풰n = Q[[푥]], and 풪(푈) = Q[푓푥], with both 푥 and 푓푥 primitive. Then
푈 is the group G푎, and the set of grouplike elements of 풰n is the set of elements of the form exp(푟푥) for 푟 ∈ Q. In particular, this demonstrates the usefulness of taking completed universal enveloping algebras.
Free Pro-unipotent Groups
Definition 3.1.7. If 푉 is a strictly negative graded vector space, we may form the free pro-nilpotent Lie algebra on 푉 as follows. We take the graded tensor algebra 푇 푉 on 푉 and put the unique coproduct on it such that all elements of 푉 are primitive. The subspace of primitive elements of 푇 푉 forms a graded pro-nilpotent Lie algebra, denoted n(푉 ), with corresponding
26 pro-unipotent group 푈(푉 ). Then n(푉 ), 푈(푉 ) are known as the free pro-nilpotent Lie algebra and free pro-unipotent group, respectively, on the graded vector space 푉 .
The construction 푉 ↦→ n(푉 ) is left adjoint to the forgetful functor from graded pro- nilpotent Lie algebras to graded vector spaces.
Finally, if 퐼 is an index set with a degree function 푑: 퐼 → Z<0 with finite fibers, then the free pro-unipotent group on the set 퐼 is just the free pro-unipotent group on the free graded vector space on the set 퐼. In particular, the free pro-unipotent group on a graded vector space is isomorphic to the free pro-unipotent group on a (graded) basis of that vector space. The Lie algebra is the pro-nilpotent completion1 of the free Lie algebra on the set 퐼 and as such is generated by the elements of 퐼. The graded dual Hopf algebra of 푇 푉 = 풰n(푉 ) is the coordinate ring 풪(푈(푉 )). Let
{푥푖} be a graded basis of 푉 , so that words 푤 in the {푥푖} form a basis of 풰n(푉 ), and let
{푓푤}푤 denote the basis of 풪(푈(푉 )) dual to {푤}. Then 풪(푈(푉 )) is isomorphic to the free shuffle algebra on the graded vector space 푉 ∨. Its coproduct is known as the deconcatenation coproduct and is given by ∑︁ Δ푓푤 := 푓푤1 ⊗ 푓푤2 , 푤1푤2=푤 and the (commutative) product X on 풪(푈(푉 )), known as the shuffle product, is given by:
∑︁ 푓푤1 X푓푤2 := 휎(푓푤1푤2 ),
휎∈X(ℓ(푤1),ℓ(푤2))
where ℓ denotes the length of a word, X(ℓ(푤1), ℓ(푤2)) ⊆ 푆ℓ(푤1)+ℓ(푤2) denotes the group of
shuffle permutations of type (ℓ(푤1), ℓ(푤2)), and 푤1푤2 denotes concatenation of words.
Remark 3.1.8. It follows from the definition of the deconcatenation coproduct that aword consisting of a single letter is a primitive element of the free shuffle algebra.
∏︁ 1In fact, if one takes a free Lie algebra in the graded sense and views it via the notion, then it is already pro-nilpotent.
27 Conventions for Products
Let 훼 and 훽 be two paths in a space 푋. Then in the literature, the sybmol 훼훽 can have two different meanings. It can either denote:
(i) The path given by going along 훼 and then 훽
(ii) The path given by going along 훽 and then 훼
The first is known as the ‘lexical’ order and the second as the ‘functional’ order.We will use the lexical order, in contrast to the convention of [DCW16] (but consistent with [Bro17]). However, we would like to take a moment to explain how these two conventions help clarify differing conventions in the literature for iterated integrals, multiple zeta values, polylogarithms, and more. We will refer to these differing conventions as the lexical and functional conventions, respectively. In fact, either convention necessitates a particular convention for iterated integrals. In general, one wants a “coproduct” formula to hold for iterated integrals (c.f. Section 2.1 of [Bro12b], 5.1(ii) of [Hai94], or Proposition 5 of [Hai05]), by which we mean
푛 ∫︁ ∑︁ ∫︁ ∫︁ 휔1 ··· 휔푛 = 휔1 ··· 휔푖 휔푖+1 ··· 휔푛 (3.1) 훼훽 푖=0 훼 훽
In order for the coproduct to take this nice form (3.1), our convention for path composition determines our convention for iterated integrals. More specifically, those that use the lexical order for path composition use the formula
∫︁ ∫︁ 퐼(훾(0); 휔1, ··· , 휔푛; 훾(1)) := 휔1 ··· 휔푛 = 푓1(푡1) ··· 푓푛(푡푛)푑푡1 ··· 푑푡푛, 훾 0≤푡1≤···≤푡푛≤1
* where 훾 (휔푖) = 푓푖(푡)푑푡, and those that use the function order for path composition use the formula
28 ∫︁ ∫︁ 퐼(훾(0); 휔1, ··· , 휔푛; 훾(1)) := 휔1 ··· 휔푛 = 푓1(푡1) ··· 푓푛(푡푛)푑푡1 ··· 푑푡푛. 훾 0≤푡푛≤···≤푡1≤1
Given that these conventions are opposite, the corresponding conventions for the iterated integral expression for polylogarithms must be opposite. More precisely, the iterated integral 푑푧 defining a multiple polylogarithm must always begin with in the lexical convention, 1 − 푧 푑푧 while it must always end with in the functional convention. More precisely, let us 1 − 푧 define ∑︁ 푧푘푟 Li푠 ,··· ,푠 (푧) := . (3.2) 1 푟 푠1 푠푟 푘1 ··· 푘푟 0<푘1<···<푘푟 푑푧 푑푧 Set 푒0 = and 푒1 = . Then using the lexical convention for iterated integration, we 푧 1 − 푧 have:
1 0 0 1 0 0 1 0 0 Li푠1,··· ,푠푟 (푧) = 퐼(0; 푒 , 푒 , ··· , 푒 , 푒 , 푒 , ··· , 푒 , ··· , 푒 , 푒 , ··· , 푒 ; 푧). ⏟ ⏞ ⏟ ⏞ ⏟ ⏞ 푠1−1 푠2−1 푠푟−1
In fact, the definition itself of Li푠1,··· ,푠푟 (푧) depends on the convention. More specifically, if we were to use the functional convention for iterated integrals in tandem with (3.2), we would get:
0 0 1 0 0 1 0 0 1 Li푠1,··· ,푠푟 (푧) = 퐼(0; 푒 , ··· , 푒 , 푒 , 푒 , ··· , 푒 , ··· , 푒 , 푒 , ··· , 푒 , 푒 ; 푧). ⏟ ⏞ ⏟ ⏞ ⏟ ⏞ 푠푟−1 푠푟−1−1 푠1−1
This is precisely the formula that appears in [BL11] (1.4). However, most authors prefer
the 푠푖’s to appear in the iterated integral in the same order as they do in the argument of the function. Therefore, almost all articles that use the functional convention for iterated integration will write:
∑︁ 푧푘푟 Li푠 ,··· ,푠 (푧) := . 1 푟 푠1 푠푟 푘1 ··· 푘푟 푘1>···>푘푟>0
29 As a result, one then writes:
0 0 1 0 0 1 0 0 1 Li푠1,··· ,푠푟 (푧) = 퐼(0; 푒 , ··· , 푒 , 푒 , 푒 , ··· , 푒 , ··· , 푒 , 푒 , ··· , 푒 , 푒 ; 푧). ⏟ ⏞ ⏟ ⏞ ⏟ ⏞ 푠1−1 푠2−1 푠푟−1
Thus, the convention one uses for path composition determines the convention one uses
for Li푠1,··· ,푠푟 (푧) (except in [BL11]). Similarly, the two conventions for multiple zeta values follow this paradigm. Specifically, those who use the lexical convention write
∑︁ 1 휁(푠1, ··· , 푠푟) = , 푠1 푠푟 푘1 ··· 푘푟 푘1>···>푘푟>0 and those who use the functional convention write
∑︁ 1 휁(푠1, ··· , 푠푟) = . 푠1 푠푟 푘1 ··· 푘푟 푘1>···>푘푟>0
Note, however, that Li푛 always denotes the same function (both as a multi-valued complex analytic function, a Coleman function, and an abstract function on the de Rham fundamental group), no matter which convention one uses. In fact, this brings us back to the two u conventions for path composition. The fact that some write Li푛 (c.f. Section 3.2.1) as 푒1 푒0 ··· 푒0 and others write it as 푒0 ··· 푒0 푒1, yet both denote the exact same regular function ⏟ ⏞ ⏟ ⏞ 푛−1 푛−1 1 on the unipotent de Rham fundamental group of P ∖ {0, 1, ∞}, is due to the differing conventions for path composition. As we use the lexical convention, one will find the coproduct formula
푛−1 ∑︁ (logu)X푖 Δ′ Liu = Liu ⊗ (3.3) 푛 푛−푖 푖! 푖=1
푛−1 ∑︁ (logu)X푖 in this thesis (3.2.1). With the other convention, one must write Δ′ Liu = ⊗Liu . 푛 푖! 푛−푖 푖=1 More subtly, these conventions affect the convention one uses for the motivic coproduct.
30 More specifically, if we use the lexical convention, we want to also be able to write
푛−1 ∑︁ (logu(푧))X푖 Δ′ Liu (푧) = Liu (푧) ⊗ (3.4) 푛 푛−푖 푖! 푖=1
rather than its opposite. This formula is correct as long as we use the lexical order for
un composition in 휋1 (푍) (which, in particular, comes out in how one writes the Goncharov coproduct of [Gon05a]; Goncharov uses the lexical order himself). Theoretically, one could
un un use one convention for composition in 휋1 (푋) and another for composition in 휋1 (푍), but that would cause formulas (3.3) and (3.4) not to appear identical. One important implication of the difference in formulas for the motivic coproduct is:
Remark 3.1.9. Our 푓휎휏 is actually the 휑1.3 of 7.6.1 of [DCW16], even though the notation would suggest it is 휑3.1. In terms of authors and sources, one may find the lexical convention and/or the other con- ventions that go along with it in [Bro14] (Section 1.2), [Hai94] (Section 5),[Hai05] (Definition 2), [Bro17] (9.1), [Hai87] (Definition 1.1),Bro12b [ ] (Definition 2.1),Gon05a [ ], [Che77] (1.1.1), [Fur04] (0.1), [Gon05b] (3.4.3), [Gon] (Formula (4) and Definition 1.2),Gon95 [ ] (Section 12), [Zag93], [Zag94], [Ter02], and [Rab96]. One may also find conventions for multiple zeta values consistent with this convention in other articles by Francis Brown, e.g., [Bro12a], [Bro], and [Bro13]. The sources [DCW16] (1.16), [DG05] (5.16.1),[Car02] (Formula (79), [Del10] (Formula 0.1 and 5.1A), [Del13], [Hof97], [Rac02], [Sou10], [DC15] (2.2.4), and [Bro04] use the functional conventions.
3.1.2 The Various Fundamental Groups
The Mixed Tate Fundamental Group of 푍
Definition 3.1.10. An open integer scheme is an open subscheme of Spec 풪퐾 , where 퐾 is a
number field and 풪퐾 its ring of integers.
Let 푍 ⊆ Spec 풪퐾 be an open integer scheme, MT(푍, Q) its Tannakian category of mixed
31 2 Tate motives with Q-coefficients, which existsDG05 by[ ] . This is a category with realization functors 휎 real : MT(푍, Q) → MHSQ
to mixed Hodge structures with Q-coefficients for each embedding 휎 : 퐾 ˓→ C and
realℓ : MT(푍, ) → Rep (퐺 ) Q Qℓ 퐾
to ℓ-adic representations of 퐺퐾 for each prime ℓ. The image of each realization functor consists of mixed Tate objects, i.e., objects with a composition series consisting of tensor
1 powers of the image of the Tate object Q(1) := ℎ2(P ; Q).
Definition 3.1.11. A continuous Qℓ-representation of 퐺퐾 for a number field 퐾 is said to have good reduction at a non-archimedean place 푣 of 퐾 if either 푣 - ℓ, and the representation is unramified at 푣, or if 푣 | ℓ, and the representation is crystalline at 푣.
The ℓ-adic realizations of an object of MT(푍, Q) form a compatible system of Qℓ-Galois representations with good reduction at closed points of 푍 (in particular, crystalline at primes dividing ℓ). If (푋, 퐷) is a pair of a scheme and codimension 1 subscheme, both smooth
* and proper over 푍 and rationally connected, then the relative cohomology ℎ (푋, 퐷; Q) is an object of this category such that
휎 * * an an real (ℎ (푋, 퐷; Q)) = 퐻Betti(푋휎 (C), 퐷휎 (C); Q),
ℓ * * real (ℎ (푋, 퐷; Q)) = 퐻´et(푋퐾 , 퐷퐾 ; Qℓ), with their associated mixed Hodge structure and continuous 퐺퐾 -action, respectively. ⊗푛 The only simple objects of MT(푍, Q) are the objects Q(푛) := Q(1) for 푛 ∈ Z, each 2It is constructed by putting a t-structure on a certain subcategory of Voevodsky’s triangulated category 퐷푀푔푚(퐾) of [Voe00], taking the heart of that t-structure, and finally taking the subcategory of objects with good reduction at closed points of 푍. The category 퐷푀푔푚(퐾) is defined by taking a certain localization of the category of complexes of smooth varieties over a field with correspondences as morphisms, then taking its pseudo-abelian envelope, and finally inverting the Tate object Q(1). However, we will only need the properties of MT(푍, Q), not its construction.
32 object has a finite composition series, and the extensions are determined by the fact that
1 Ext (Q(0), Q(푛)) = 퐾2푛−1(푍)Q
Ext푖 = 0 ∀ 푖 ≥ 2.
The groups 퐾2푛−1(푍)Q are known by the work of Borel ([Bor74]). For 푛 = 1, we have × 퐾1(푍) = 풪(푍) , and for 푛 ≥ 2, 퐾2푛−1(푍)Q = 퐾2푛−1(퐾)Q has dimension 푟2 for 푛 even and 푟1 + 푟2 for 푛 odd, where 푟1 and 푟2 are the numbers of real and complex places of 퐾, respectively.
Definition 3.1.12. Let 푀 be an object of MT(푍, Q). Then 푀 has an increasing filtration
푊푖푀 known as the weight filtration. The quotient 푊푖푀/푊푖−1푀 is trivial when 푖 is odd and is isomorphic to a direct sum of copies of Q(−푖/2) when 푖 is even. We let
Can(푀) := HomMT(푍,Q)(Q(−푖), 푊2푖푀/푊2푖−1푀), 푖∈Z and we call this the canonical fiber functor.
Then the category MT(푍, Q) is a neutral Q-linear Tannakian category with fiber functor MT Can, and we let 휋1 (푍) denote its Tannakian fundamental group, which is therefore a pro-algebraic group over Q. The subcategory of simple objects of MT(푍, Q) consists of direct sums of tensor powers of Q(1) and is therefore equivalent as a Tannakian category to the category of representations MT un of G푚. This inclusion induces a quotient map 휋1 (푍) G푚, and we let 휋1 (푍) denote the kernel of this quotient. The functor sending an object 푀 of MT(푍, Q) to the direct sum 푊푖푀/푊푖−1푀 gives a splitting of this inclusion of categories, which implies that the 푖∈Z MT quotient map 휋1 (푍) G푚 splits. This implies that MT(푍, Q) has fundamental group
MT un 휋1 (푍) = 휋1 (푍) o G푚,
33 un MT un where 휋1 (푍) the maximal pro-unipotent subgroup of 휋1 (푍). The action of G푚 on 휋1 (푍)
by conjugation gives an action of G푚, or equivalently, a grading, on the Hopf algebra of un 휋1 (푍). This associated graded Hopf algebra is denoted
∞ un 퐴(푍)푖 = 퐴(푍) := 풪(휋1 (푍)), 푖=0 where 퐴(푍)푖 denotes the 푛th graded piece. We refer to the degree on 퐴(푍) as the half-weight, as it is half the ordinary motivic weight. In fact, the description of the Ext groups gives us the following information. It gives a canonical embedding of graded vector spaces
∞ 퐾2푛−1(푍)Q ˓→ 퐴(푍), 푛=1
with 퐾2푛−1(푍)Q in degree 푛, and whose image is the set of primitive elements of 퐴(푍). Equivalently, this gives a canonical isomorphism
(︃ ∞ )︃∨ ∽ un ab 휋1 (푍) = 퐾2푛−1(푍)Q . 푛=1
un In fact, this canonical isomorphism extends to an isomorphism between 휋1 (푍) and the free (︃ ∞ )︃∨ pro-unipotent group (Definition 3.1.7) on the graded vector space 퐾2푛−1(푍)Q , with 푛=1 ∨ (퐾2푛−1(푍)) in degree −푛, but this extension is not canonical. This last fact tells us that there is a non-canonical isomorphism between 퐴(푍) and the free shuffle algebra on the graded ∞ vector space 퐾2푛−1(푍)Q, with 퐾2푛−1(푍) in degree 푛. This non-canonicity is the key to a 푛=1 later consideration; see Remark 3.3.2. 1 Furthermore, in Proposition 3.3.3, we will show that Ext (Q(0), Q(푛)) is isomorphic to the space of degree 푛 primitive elements of 퐴(푍).
′ ′ For 푍 ⊆ 푍 an open subscheme, we have an inclusion MT(푍, Q) ⊆ MT(푍 , Q), which MT ′ MT gives rise to a quotient map 휋1 (푍 ) 휋1 (푍) that is an isomorphism on G푚 and hence
34 to an inclusion 퐴(푍) ⊆ 퐴(푍′)
of graded Hopf algebras. There is also a graded Hopf algebra 퐴(Spec 퐾), which is the union
of 퐴(푍) for 푍 ⊆ Spec 풪퐾 , and we may view all such 퐴(푍) as lying inside 퐴(Spec 퐾). If p is
a point of Spec 풪퐾 ∖ 푍 and 훼 ∈ 퐴(푍), we say 훼 is unramified at p if
훼 ∈ 퐴(푍 ∪ {p}) ⊆ 퐴(푍).
The de Rham Unipotent Fundamental Group of 푋
1 For the rest of Part I, we let 푋 = P ∖ {0, 1, ∞} over Z.
Definition 3.1.13. We let
un 휋1 (푋)
denote the unipotent de Rham fundamental group of 푋Q. It is the fundamental group of the
Tannakian category of algebraic vector bundles with connection on 푋Q.
This pro-unipotent group over Q is a free pro-unipotent group the graded vector space dR consisting of 퐻1 (푋Q) in degree −1 (and zero in other degrees). Dually, its coordinate ring is generated by words in holomorphic differential forms on 푋, which are integrands of interated integrals.
un By the construction in Section 3 of [DG05], 휋1 (푋) is in MT(푍, Q) (in the sense that its coordinate ring and Lie algebra are each an Ind-object and pro-object, respectively, of
MT MT(푍, Q)). It therefore carries an action of 휋1 (푍), whose restriction to G푚 induces the un MT grading. While the group 휋1 (푋) itself is independent of basepoint, the action of 휋1 (푍) ⃗ 1 is not, and we always use the tangential basepoint 10, i.e., the element 1 ∈ 푇0(PQ) = Q at 1 0 ∈ P (Q). Remark 3.1.14. By an argument analogous to the proof of Corollary 2.9 of [BDCKW], one may check that all of these constructions (in particular, the Chabauty-Kim locus, c.f., Section
3.1.5) are the same if we replace ⃗10 by any other 푍-integral basepoint of 푋.
35 MT un Remark 3.1.15. We will often consider 휋1 (푍)-equivariant quotients 휋1 (푋) Π, especially when Π is finite-dimensional as a scheme over Q (hence an algebraic group). Unless otherwise stated, it is always understood that Π is such a quotient.
un A standard example is Π푛 := 휋1 (푋)≥−푛. However, in most of our calculations, we will PL be concerned with quotients that factor through 휋1 (푋), a specific quotient to be introduced in 3.1.6.
We write n(푍), n(푋), and nPL(푋) for the corresponding Lie algebras.
3.1.3 Cohomology and Cocycles
1 MT un MT We let 퐻 (휋1 (푍), 휋1 (푋)) denote the (pointed) set of 휋1 (푍)-equivariant torsor schemes un under 휋1 (푋). For 푏 ∈ 푋(푍), there is the torsor of paths 푃 constructed in Section 3 of [DG05] (as 푏 ⃗10 [DG05] uses the functional convention for path composition; this is in fact the torsor of paths from ⃗1 to 푏). It is a 휋MT(푍)-equivariant torsor under 푃 = 휋un(푋). 0 1 ⃗10 ⃗10 1 We therefore have the Kummer map
휅 1 MT un 푋(푍) −→ 퐻 (휋1 (푍), 휋1 (푋)),
un and by composition with the map induced by 휋1 (푋) Π,
휅 1 MT 푋(푍) −→ 퐻 (휋1 (푍), Π).
Definition 3.1.16. For any Q-algebra 푅, we define
1,G푚 1 un G푚 푍Π (푅) := 푍 (휋1 (푍)푅, Π푅) ,
un which is the set of morphisms of schemes from 휋1 (푍)푅 to Π푅 over 푅, equivariant with respect ′ ′ to the G푚-action, and satisfying the cocycle condition on the 푅 -points for any 푅-algebra 푅 .
Proposition 6.4 of [Bro17] ensures that this is representable by a scheme (also see Corollary
36 3.2.10). We thus write
1,G푚 1,G푚 1 un G푚 푍Π = 푍Π (푍) := 푍 (휋1 (푍), Π)
for the scheme of G푚-equivariant cocycles. If Π is finite-dimensional, then this is in facta finite-dimensional variety over Q. By Proposition 5.2.1 of [DCW16], we have
1 MT 1,G푚 퐻 (휋1 (푍), Π) = 푍Π (Q).
We have a universal cocycle evaluation map
1,G푚 un un 풞Π : 푍Π × 휋1 (푍) → Π × 휋1 (푍).
It is defined on the functor of points as follows. Fora Q-algebra 푅 and an element (푐, 훾) ∈ 1,G푚 un 1,G푚 un (푍Π )푅(푅)×휋1 (푍)푅(푅) = (푍Π ×휋1 (푍))(푅), we have 푐(훾) ∈ Π푅(푅) = Π(푅). We define
풞Π(푐, 훾) to be the pair (푐(훾), 훾). un In fact, the morphism 풞Π lies over the identity morphism on 휋1 (푍), so letting 풦 denote un un the function field of 휋1 (푍), we also have the base change from 휋1 (푍) to Spec 풦
풦 1,G푚 풞Π :(푍Π )풦 → Π풦.
Remark 3.1.17. If Π is finite dimensional, it will turn out that this is a morphism ofaffine spaces over the field 풦, and our “geometric step” (see 3.3.2) consists in computing its scheme-theoretic image.
푍 Definition 3.1.18. We let ℐΠ denote the ideal of functions in the coordinate ring of Π풦 풦 vanishing on the image of 풞Π . This is known as the Chabauty-Kim ideal (associated to Π). 푍 For Π = Π푛, we denote it by ℐ푛 .
37 3.1.4 p-adic Realization and Kim’s Cutter
∼ Let p be a closed point of 푍. For simplicity, we suppose that 푍p = Spec Z푝. un If 휔 ∈ 풪(휋1 (푋)), and 푎, 푏 are Z푝-basepoints of 푋 (rational or tangential), then we can ∫︁ 푏 extract an element of Q푝 known as the Coleman iterated integral 휔. The Coleman iterated 푎 integral is originally due to Coleman ([Col82]) and was reformulated by Besser [Bes02] into
the form that is used in [DCW16]. Fixing 푎 = ⃗10 and letting 푏 vary over 푋(푍p), one may define local Kummer map
휅p un 푋(푍p) −→ 휋1 (푋)(Q푝)
un ♯ by sending a regular function 휔 on 휋1 (푋) to the Coleman function 휅p(휔): 푋(푍p) → Q푝 ∫︁ 푏 defined by 푏 ↦→ 휔. 푎 un Its composition with 휋1 (푋)(Q푝) → Π(Q푝) is also denoted by 휅p. This map is Coleman- un analytic, meaning that regular functions on 휋1 (푋) pull back to Coleman functions on 푋(푍p). These are locally analytic functions, and a nonzero such function has finitely many zeroes.
Remark 3.1.19. The local Kummer map in this form was originally referred to in [Kim05] as the 푝-adic unipotent Albanese map. It is the same as the map 훼 of 1.3 of [DCW16]. The latter is defined by sending 푏 to the torsor of paths from 푎 to 푏 (with its structure as a filtered 휑-module)
p-adic Period Map
un In addition, there is a morphism Spec Q푝 → 휋1 (푍), or equivalently, a Q-algebra homomor- phism
perp : 퐴(푍) → Q푝,
known as the p-adic period map. While elements of 퐴(푍) are represented by formal (motivic) iterated integrals, this homomorphism takes the value of the iterated integral in the sense of Coleman integration. The following may be regarded as a 푝-adic analogue of a small piece of the Kontsevich- Zagier period conjecture ([KZ01]). It has been in folklore for some time and appears in the
38 literature for 퐾/Q abelian as Conjecture 4 of [Yam10].
Conjecture 3.1.20 (푝-adic Period Conjecture). For any open integer scheme 푍, the period map perp : 퐴(푍) → Q푝 is injective.
Kim’s Cutter
un Viewing 풞Π as a morphism of schemes over 휋1 (푍) and base-changing along the p-adic period
un 1,G푚 map Spec Q푝 → 휋1 (푍), we get a morphism (푍Π )Q푝 → ΠQ푝 . The induced map on Q푝- 휅 1 MT 1,G푚 points is denoted by locΠ. Denoting the composition 푋(푍) −→ 퐻 (휋1 (푍), Π) = 푍Π (Q) ⊆ 1,G푚 푍Π (Q푝) by 휅 as well, this fits into a diagram:
푋(푍) / 푋(푍p) ,
휅 휅p 1,G푚 푍Π (Q푝) / Π(Q푝) locΠ which we call Kim’s Cutter. This diagram is commutative (c.f. [DCW16], 4.9). In Section 3.2.4, we will describe 휅 and 휅p explicitly in terms of coordinates.
3.1.5 Chabauty-Kim Locus
un un Let 푓 ∈ 풪(Π × 휋1 (푍)). Now perp induces a morphism ΠQ푝 → Π × 휋1 (푍), and 푓 pulls back 푓 via this morphism to an element of the coordinate ring of ΠQ푝 , hence a function Π(Q푝) −→ Q푝. 휅p The composite 푓 ∘ 휅p with 푋(푍p) −→ Π(Q푝) is a Coleman function on 푋(푍p) and is denoted
푓 푋(푍p). We then define
푍 un 푍 푋(푍p)Π = 푋(푍p)Π := {푧 ∈ 푋(푍p): 푓 푋(푍p) (푧) = 0 ∀ 푓 ∈ 풪(Π × 휋1 (푍)) ∩ ℐΠ }.
While this set depends on 푍, we write 푋(푍p)Π when there is no confusion.
un 푍 1,G푚 un un Any 푓 ∈ 풪(Π × 휋1 (푍)) ∩ ℐΠ vanishes on the image of 풞Π : 푍Π × 휋1 (푍) → Π × 휋1 (푍),
39 hence also on the image of locΠ. By the commutativity of Kim’s Cutter, 푓 푋(푍p) vanishes on 푋(푍), hence
푋(푍) ⊂ 푋(푍p)Π.
푝 ′ un ′ # 푍 푍 We note that if Π dominates Π (i.e., we have 휋1 (푋) Π −− Π), then 푝 (ℐΠ ) ⊆ ℐΠ′ ,
hence 푋(푍p)Π′ ⊆ 푋(푍p)Π.
For Π = Π푛, we denote 푋(푍p)Π by 푋(푍p)푛. The importance of Kim’s method lies in
the fact that 푋(푍p)푛 is known to be finite for sufficiently large 푛 when 퐾 is totally real by [Kim05] and [Kim12].
Remark 3.1.21. The locus defined inDCW16 [ ] differs slightly from our own in that it uses
푍 a version of ℐΠ defined as the ideal of functions vanishing on the image of the basechange
of 풞Π along perp, which could in principle be larger than our own. However, as explained in 4.2.6 of [DC15], Conjecture 3.1.20 implies that these two are the same, which is why we see no harm in doing it this way. Furthermore, our version of Kim’s conjecture is a priori stronger than the original version, so our theorems apply to his conjecture either way.
3.1.6 The Polylogarithmic Quotient
un un We let 푁 denote the kernel of the homomorphism 휋1 (푋) → 휋1 (G푚) induced by the inclusion 1 푋 ˓→ G푚, where 휋 un(G푚) refers to the unipotent de Rham fundamental group of G푚Q. Then, following [Del89], we define the polylogarithmic quotient
PL un 휋1 (푋) := 휋1 (푋)/[푁, 푁].
MT un un un un The group 휋1 (푍) acts on 휋1 (G푚) as well as 휋1 (푋), and because 휋1 (푋) → 휋1 (G푚) MT is induced by a map of schemes over Z, it is 휋1 (푍)-equivariant. Therefore, 푁 and hence MT PL MT [푁, 푁] are 휋1 (푍)-stable, so 휋1 (푋) has a structure of a motive, i.e., an action of 휋1 (푍). As a motive, it has the structure
∞ PL ∏︁ 휋1 (푋) = Q(1) n Q(푖), 푖=1
40 MT so in particular the action of 휋1 (푍) factors through G푚.
PL Remark 3.1.22. In Section 3.2.1, we will describe 휋1 (푋) more explicitly in coordinates.
PL 1,G푚 In the specific case Π = ΠPL,푛 := 휋1 (푋)≥−푛 for a positive integer 푛, we write 푍PL,푛 ,
푍 1,G푚 푍 풞PL,푛, locPL,푛, ℐPL,푛, and 푋(푍p)PL,푛 to denote 푍Π , 풞Π, locΠ, ℐΠ , and 푋(푍p)Π, respectively. un PL Since 휋1 (푍) acts trivially on 휋1 (푋), a G푚-equivariant cocycle
un PL 푐: 휋1 (푍) → 휋1 (푋)
is just a G푚-equivariant homomorphism. 푢 If Π is a quotient of ΠPL,푛, then any G푚-equivariant homomorphism 휋1 푛(푍) → Π must un 1,G푚 be zero on 휋1 (푍)<−푛 by the G푚-equivariance, so 푍Π is the same as
1 un G푚 푍 (휋1 (푍)≥−푛, Π) .
It follows that we can in fact view 풞PL,푛 as a morphism
1,G푚 un un 푍PL,푛 × 휋1 (푍)≥−푛 → ΠPL,푛 × 휋1 (푍)≥−푛
un un lying over the identity on 휋1 (푍)≥−푛. Letting 풦푛 denote the function field of 휋1 (푍)≥−푛, it induces a map
풦푛 1,G푚 풞PL,푛 :(푍PL,푛 )풦푛 → (ΠPL,푛)풦푛
푍 of finite-dimensional affine spaces over the field 풦푛, and we view ℐPL,푛 as an ideal in
풪((ΠPL,푛)풦푛 ).
3.1.7 Kim’s Conjecture
We recall Conjecture 2.2.3 from the introduction:
Conjecture 3.1.23. 푋(푍) = 푋(푍p)푛 for sufficiently large 푛.
푍 In fact, dimension counts show that ℐPL,푛 is nonzero and hence 푋(푍p)PL,푛 is finite for
41 sufficiently large 푛. One might wonder the following:
Question 3.1.24. Does 푋(푍) = 푋(푍p)PL,푛 for sufficiently large 푛? This is a strengthening of Conjecture 3.1.23. However, as we will show in Section 3.4, this
question has a negative answer as stated (at least for 푍 = Spec Z[1/ℓ] and odd primes ℓ) and
needs to be modified by an 푆3-symmetrization.
More specifically, there is an action of 푆3 on the scheme 푋. This induces an action of 푆3 un on 휋1 (푋). For a quotient Π and 휎 ∈ 푆3, we may define 휎(Π) by
MT MT ker(휋1 (푍) 휎(Π)) = 휎(ker(휋1 (푍) Π)).
It follows by independence of basepoint (Remark 3.1.14) and functoriality of all the construc-
tions that 휎(푋(푍p)Π) = 푋(푍p)휎(Π). We then write
푆3 ⋂︁ ⋂︁ 푋(푍p)PL,푛 := 휎(푋(푍p)PL,푛) = 푋(푍p)휎(ΠPL,푛). 휎∈푆3 휎∈푆3
Our symmetrized conjecture is that
푆3 Conjecture 3.1.25. 푋(푍) = 푋(푍p)PL,푛 for sufficiently large 푛.
We note that Π푛 dominates 휎(ΠPL,푛) for all 휎 ∈ 푆3; hence 푋(푍p)푛 ⊆ 휎(푋(푍p)PL,푛), and so
푆3 푋(푍p)Π푛 ⊆ 푋(푍p)PL,푛,
so Conjecture 3.1.25 implies Conjecture 3.1.23. In Theorem 3.4.4, we use our computations
to verify Conjecture 3.1.25 for 푍 = Spec Z[1/3] and 푝 = 5, 7.
3.2 Coordinates
3.2.1 Coordinates on the Fundamental Group {︂푑푧 푑푧 }︂ Let {푒 , 푒 } be a basis of 퐻dR(푋 ) dual to the basis , of 퐻1 (푋 ). The algebra 0 1 1 Q 푧 1 − 푧 dR Q un 풰휋1 (푋) is the free (completed) non-commutative algebra on the generators 푒0 and 푒1, with
42 coproduct given by declaring that 푒0 and 푒1 are primitive and grading given by putting both in degree −1. We refer to the words
푒0, 푒1, 푒1푒0, 푒1푒0푒0,...
as the polylogarithmic words. We let the elements
u u u un log , Li1, Li2, · · · ∈ 풪(휋1 (푋))
un be the duals of these words with respect to the standard basis of 풰휋1 (푋). More generally, u un for a word 푤 in 푒0, 푒1, we let Li푤 denote the dual basis element of 풪(휋1 (푋)).
Proposition 3.2.1. We have Δ′ logu = 0
푛−1 ∑︁ (logu)X푖 Δ′ Liu = Liu ⊗ . 푛 푛−푖 푖! 푖=1 Proof. The first equation follows from Remark 3.1.8. By the discussion following Definition 3.1.7, we have the formula
′ u ′ u u u u u u u Δ Li푛 = Δ Li푒1푒0···푒0 = Li푒1 ⊗ Li푒0···푒0 + Li푒1푒0 ⊗ Li푒0···푒0 + ··· + Li푒1푒0···푒0 ⊗ Li푒0 .
By the definition of the shuffle product, we have the formula
u X푖 u X푖 u (log ) = (Li ) = 푖! Li 푖 . 푒0 (푒0)
u The previous two formulas together with the definition of Li푛 then imply the proposition.
PL u u u Letting 풫 be the subalgebra generated by log , Li1, Li2,... , the formula above implies PL un that 풫 is a Hopf subalgebra. It therefore corresponds to a quotient group of 휋1 (푋). It PL follows from Proposition 7.1.3 of [DCW16] that this quotient group is the group 휋1 (푋).
43 Furthermore, we have u u u 풪(ΠPL,푛) = Q[log , Li1,..., Li푛]
PL PL as a Hopf subalgebra of 풫 = 풪(휋1 (푋)).
1,G푚 ♯ PL Given a cocycle 푐 ∈ 푍PL , we write 푐 : 풫 → 퐴(푍) for the associated homomorphism of Q-algebras, and we write logu(푐) := 푐♯ logu,
u ♯ u Li푛(푐) := 푐 Li푛 .
1,G푚 Corollary 3.2.2. For 푐 ∈ 푍PL (Q), we have
Δ′ logu(푐) = 0
푛−1 ∑︁ (logu(푐))X푖 Δ′ Liu (푐) = Liu (푐) ⊗ . 푛 푛−푖 푖! 푖=1 Proof. The cocycle condition reduces to the homomorphism conditions (as we are restricting ourselves to the polylogarithmic quotient), which means, dually, that 푐♯ respects the coproduct. The corollary then follows immediately from Proposition 3.2.1.
3.2.2 Generating 퐴(푍)
un We take a moment to discuss coordinates for the Galois group 휋1 (푍). On an abstract level, this is a free unipotent group, and its structure is governed by the theory of 3.1.1. However, we need to be able to write elements of 퐴(푍) in a way that allows us to compute their 푝-adic periods. This essentially means writing them as explicit combinations of special values of polylogarithms and zeta functions. We introduce the notation
u u log (푧) := log (휅(푧)) ∈ 퐴(푍)1
u u Li푛(푧) := Li푛(휅(푧)) ∈ 퐴(푍)푛
44 u for 푧 ∈ 푋(푍). It is the same as the motivic period Li푛(푧) mentioned in (9.1) of [Bro17] and u 2.2.4 of [DC15], which justifies the notation Li푛(푐) in the previous section. u u Because log is pulled back from G푚, we in fact have log (푧) ∈ 퐴(푍) whenever 푧 ∈ G푚(푍). We also note:
Fact 3.2.3. For 푧, 푤 ∈ 푋(푍), we have
logu(푧푤) = logu(푧) + logu(푤)
and u u Li1(푧) = − log (1 − 푧).
Letting 푐 denote the cocycle corresponding to the class of the path torsor 푃 in 1 −⃗11 ⃗10 1 MT un 퐻 (휋1 (푍), 휋1 (푋)), we write
u u 휁 (푛) := Li푛(푐1) ∈ 퐴(푍)푛.
u It is not clear a priori that the special values Li푛(푧) for 푧 ∈ 푋(푍) span the space 퐴(푍) (whether we leave 푍 fixed or take a union over various 푍, such as all 푍 with a fixed function field 퐾 or all 푍 whose function field is cyclotomic). However, there is the following conjecture of Goncharov:
Conjecture 3.2.4 ([Gon], Conjecture 7.4). The ring 퐴(Q) is spanned by elements of the u 1 form Li푤(푧) for 푧 a rational point or rational tangential basepoint of P ∖ {0, 1, ∞} and 푤 a word in 푒0, 푒1.
Similar to [DC15], we state a version of this conjecture with control on ramification.
Definition 3.2.5. We say that an integral scheme 푍 with function field Q is saturated if ∼ there exists 푁 ∈ Z>0 ∪ {∞} such that 푍 = Spec Z[1/푆] for 푆 = {푝 prime; 푝 ≤ 푁}.
Conjecture 3.2.6. If 푍 is saturated, then the ring 퐴(푍) is spanned by elements of the form
u 1 Li푤(푧) for 푧 a 푍-integral point or tangential basepoint of P ∖ {0, 1, ∞}.
45 This is a strengthening of Conjecture 2.2.6 of [DC15] and is in fact the motivation behind that conjecture. It is known in the cases 푁 = 1 ([Bro12a]) and 푁 = 2 ([Del10]), and the case 푁 = ∞ is Conjecture 3.2.4.
In Section 3.3.3, we will demonstrate this in practice by writing elements of 퐴(Z[1/3]) in 1 terms of values of polylogarithms at elements of P ∖ {0, 1, ∞}(Z[1/6]).
Remark 3.2.7. In doing so, we will find that all of these may be written in terms of single (rather than multiple) polylogarithms. This is in line with a conjecture of Goncharov about the depth filtration, as we now describe. We let
퐺 n (푍) := n(푍)/[n(푍)<−1, [n(푍)<−1, n(푍)<−1]],
퐺 퐺 휋1 (푍) the corresponding quotient group, and 퐴 (푍) its coordinate ring. It follows by the PL definition of 휋1 (푋) that
1,G푚 1 퐺 G푚 푍PL,푛 = 푍 (휋1 (푍)≥푛, ΠPL,푛) .
u 퐺 In particular, each Li푛(푧) is in 퐴 (푍), and Goncharov’s conjecture (Conjecture 7.6 of [Gon]) 퐺 u says that 퐴 (푍) is spanned by elements of the form Li푛(푧), at least when 푍 = Spec Q. We might expect this to hold whenever 푍 is saturated. Whenever 퐾 is totally real, n퐺(푍) agrees
퐺 with n(푍) in degrees ≥ −4; hence 퐴 (푍)≤4 = 퐴(푍)≤4 should be spanned by elements of the u form Li푛(푧).
3.2.3 Coordinates on the Space of Cocycles
Fix an arbitrary family Σ = {휎푛,푖} of homogeneous free generators for n(푍) with 휎푛,푖 in
half-weight −푛, and for each word 푤 of half-weight −푛 in the above generators, let 푓푤 denote
the associated element of 퐴푛 = 퐴(푍)푛. The following proposition was communicated to the authors in an unpublished letter by Francis Brown. It provides equivalent information to the geometric algorithm in [DC15], but it allows one to avoid computing the logarithm in a unipotent group, making the algorithm much more practical.
46 1,G푚 Definition 3.2.8. For an arbitrary cocycle 푐 ∈ 푍PL (푅) for a Q-algebra 푅, a polylogarithmic word 휆 of half-weight −푛, and a word 푤 in the elements of Σ of half-weight −푛, let
푤 휑휆 (푐) ∈ Q
denote the associated matrix entry of 푐♯, so that in the notation above, we have
u ∑︁ 푤 Li휆(푐) = 휑휆 (푐)푓푤. 푤
1,G푚 Proposition 3.2.9. Let 푐 ∈ 푍PL (푅) for a Q-algebra 푅. For 0 ≤ 푟 < 푛, 휏1, . . . , 휏푟 ∈ Σ−1, and 휎 ∈ Σ푟−푛, we have
휑휎휏1···휏푟 (푐) = 휑휏1 (푐) ··· 휑휏푟 (푐)휑휎 (푐), 푒1푒0 ··· 푒0 푒0 푒0 푒1푒0 ··· 푒0
⏟ 푛⏞ ⏟ 푛 −⏞푟
푤 and all other matrix entries 휑휆 (푐) vanish.
Proof. This amounts to a straightforward verification, but we nevertheless give the details.
휏 We begin with a formal calculation, in which Σ−1 may be an arbitrary finite set, and {푎 }휏∈Σ−1 a family of commuting coefficients. In this abstract setting, we claim that
⎛ ⎞X푛
∑︁ 휏 ∑︁ 휏1 휏푛 ⎝ 푎 푓휏 ⎠ = 푛! 푎 ··· 푎 푓휏1···휏푛 . 휏∈Σ−1 휏1,...,휏푛∈Σ−1
47 Indeed, the left side of the equation
∑︁ 휏1 휏푛 = (푎 푓휏1 )X ··· X(푎 푓휏푛 ) 휏1,...,휏푛 ⎛ ⎞ ⎜ ⎟ ∑︁ 휏1 휏푛 ⎜ ∑︁ ⎟ = 푎 ··· 푎 ⎜ 푓휏 푝···휏 푝 ⎟ ⎜ 1 푛 ⎟ 휏1,...,휏푛 ⎝permutations 푝 ⎠
of 휏1,...,휏푛 ∑︁ ∑︁ 휏 휏 = 푎 1 ··· 푎 푛 푓 푝 푝 , 휏1 ···휏푛 푝 휏1,...,휏푛 ⏟ ⏞ independent of 푝
which equals the right side of the equation. Returning to our concrete situation, we apply the reduced coproduct Δ′ to both sides of
∑︁ 푤 u 휑푛+1(푐)푓푤 = Li푛+1(푐), |푤|=−(푛+1)
and compute:
푛 u X푖 ∑︁ 푤′푤′′ ∑︁ u (log (푐)) 휑 (푐)푓 ′ ⊗ 푓 ′′ = Li (푐) ⊗ (3.5) 푛+1 푤 푤 푛+1−푖 푖! 푤′,푤′′ 푖=1 푛 (︃ (∑︀ 휑휏 (푐)푓 )X푖 )︃ ∑︁ 휏∈Σ−1 푒0 휏 = Liu (푐) ⊗ (3.6) 푛+1−푖 푖! 푖=1 푛 ∑︁ ∑︁ ∑︁ 휏1 휏푖 푤 = 휑푒0 (푐) ··· 휑푒0 (푐)휑푛+1−푖(푐)푓푤 ⊗ 푓휏1···휏푖 . (3.7) 푖=1 휏1,...,휏푖∈Σ−1 |푤|=푛+1−푖
Taking the coefficient of 푓푣 ⊗ 푓휏 , with 휏 ∈ Σ−1, and 푣 an arbitrary word of length 푛 ≥ 1, we obtain
푣휏 휏 푣 휑푛+1(푐) = 휑푒0 (푐) · 휑푛(푐), while taking the coefficient of 푓푣 ⊗ 푓휎, with 휎 ∈ Σ−푖 <−1, and 푣 an arbitrary word of length
48 푛 + 1 − 푖 ≥ 1, we obtain
푣휎 휑푛+1(푐) = 0.
We have a morphism
1,G푚 un PL 풞PL : 푍PL × 휋1 (푍) → 휋1 (푋)
given by (푐, 훾) ↦→ 푐(훾).
We refer to 풞PL as the universal polylogarithmic cocycle, and it is just the first component of PL the universal cocycle evaluation morphism 풞Π for Π = 휋1 (푋). We define a morphism
[︁ ]︁ 1,G푚 휌 Ψ: 푍PL → Spec Q {Φ휆}푤푡(휌)=푤푡(휆) , where 휌 ranges over Σ and 휆 ranges over the set of polylogarithmic words, by
휌 푐 ↦→ (휑휆(푐))휌,휆.
We define a homomorphism of rings
♯ u u u 휌 휃 : Q[log , Li1, Li2,... ] → 퐴(푍)[{Φ휆}휌,휆]
by
u ∑︁ 휏 log ↦→ 푓휏 Φ푒0 휏∈Σ−1 and ∑︁ Liu ↦→ 푓 Φ휏1 ··· Φ휏푟 Φ휎 . 푛 휎휏1···휏푟 푒0 푒0 푒1푒0 ··· 푒0
휏1,...,휏푟∈Σ−1 ⏟ 푠⏞ 휎∈Σ−푠 푟+푠=푛 1≤푠≤푛
1,G푚 Corollary 3.2.10. The morphism Ψ is an isomorphism. In particular, 푍PL is canonically
49 an affine space endowed with coordinates. Moreover, the triangle
1,G푚 un Ψ×푖푑 휌 un 푍PL × 휋1 (푍) ∼ / (Spec Q[{Φ휆}휌,휆]) × 휋1 (푍)
풞PL 휃 ' PL u 휋1 (푋) commutes.
Proof. The injectivity of Ψ, as well as the commutativity of the diagram, follow directly from 휌 Proposition 3.2.9. The surjectivity of Ψ amounts to the statement that given any family 푎휆 of elements of a Q-algebra 푅, the 푅-algebra homomorphism
♯ u u u 푐 : 푅[log , Li1, Li2,... ] → 푅 ⊗ 퐴(푍)
given by
u ∑︁ 휏 log ↦→ 푎푒0 푓휏 휏∈Σ−1 and ∑︁ Liu ↦→ 푎휏1 ··· 푎휏푟 푎휎 푓 푛 푒0 푒0 푒1푒0 ··· 푒0 휎휏1···휏푟 휏 ,...,휏 ∈Σ 1 푟 −1 ⏟ 푛 −⏞푟 휎∈Σ푟−푛 0≤푟≤푛−1
u u is compatible with the coproduct. For log , this is because both sides are primitive. For Li푛, this follows by the computation of (3.5)–(3.7) and the fact that the terms for which 푟 = 0 are primitive.
50 Variant in Bounded Weight
1,G푚 Ψ푛 휌 Let 푛 be a positive integer. Then there is a natural isomorphism 푍PL,푛 −→ (Spec Q[{Φ휆}휌,|휆|≤푛]) such that the square
1,G푚 Ψ 휌 푍PL ∼ / Spec Q[{Φ휆}휌,휆]
1,G푚 Ψ푛 휌 푍PL,푛 ∼ / Spec Q[{Φ휆}휌,|휆|≤푛]
commutes, where the vertical arrows are the natural projections.
3.2.4 Kummer and Period Maps in Coordinates
Given 푧 ∈ 푋(푍), we recall that, by definition,
logu(푧) = logu(휅(푧)),
u u Li푛(푧) = Li푛(휅(푧)).
We describe 휅p in these coordinates. More precisely, for 푧 ∈ 푋(푍p), the value 휅p is the u 푝 u 푝 element of ΠPL,푛(Q푝) sending log to log (푧) and Li푛 to Li푛(푧).
Finally, we describe perp in these coordinates. More precisely, we have
u 푝 perp(log (푧)) = log (푧),
u 푝 perp(Li푛(푧)) = Li푛(푧), which in particular expresses the commutativity of Kim’s cutter.
51 3.3 Computations for 푍 = Spec Z[1/푆]
3.3.1 Abstract Coordinates for 푍 = Spec Z[1/푆]
We let ℓ denote a prime number. We want to fix free generators (휏ℓ)ℓ∈푆 and (휎2푛+1)푛≥1 for
n(푍), with 휏ℓ in degree −1 and 휎2푛+1 in degree −2푛 − 1. We first recall some notation from Section 3.2 of [DC15]. ∞ Letting 퐴푛 = 퐴(푍)푛 denote the degree 푛 part of 퐴(푍), and 퐴>0 = 퐴푛, we let 푛=1 ′ 퐸푛 = 퐸푛(푍) denote the kernel of Δ 퐴푛 and 퐷푛 = 퐷푛(푍) the subspace of decomposable elements of degree 푛, i.e., the degree 푛 elements in the image of
mult 퐴>0 ⊗ 퐴>0 −−→ 퐴.
We let 푃푛 = 푃푛(푍) denote a vector subspace of 퐴푛 complementary to 퐸푛 and 퐷푛. The
symbols ℰ푛, 풟푛, and 풫푛 refer to bases of 퐸푛, 퐷푛, and 푃푛, respectively.
Proposition 3.3.1. One may choose free generators (휏ℓ)ℓ∈푆 and (휎2푛+1)푛≥1 for n(푍), with u u 휏ℓ in degree −1 and 휎2푛+1 in degree −2푛 − 1, such that 푓휏ℓ = log (ℓ) and 푓휎2푛+1 = 휁 (2푛 + 1).
Furthermore a choice of 푃2푛+1 uniquely determines 휎2푛+1.
Proof. A computation using Corollary 3.2.2 shows that logu(ℓ) and 휁u(2푛 + 1) are primitive
elements of the Hopf algebra 퐴(푍) (or, in the terminology of [DC15], they lie in the space 퐸푛 of extensions). In fact, by our knowledge of the rational algebraic K-theory of 푍, we know
that 퐸푛 is one-dimensional when 푛 is odd and zero-dimensional otherwise. Therefore, the u u elements log (ℓ) and 휁 (2푛 + 1) must span the spaces 퐸푛, and we take them as our ℰ푛.
By Proposition 3.2.3 of [DC15], for a choice of ℰ푛, 풟푛, and 풫푛, we get a set of generators
for the Lie algebra, which are dual to the elements of ℰ푛. In fact, the part of the condition
of being dual that depends on 풟푛 and 풫푛 is that the element of the Lie algebra pairs to
zero with all of 풟푛 ∪ 풫푛, so it in fact depends only on 푃푛 and 퐷푛. The latter is uniquely
determined, so a choice of 푃푛 determines such a choice of generators.
52 Remark 3.3.2. The choice of logu(ℓ) and 휁u(2푛 + 1) corresponds to choosing generators for
the rational algebraic K-groups of 푍. The arbitrariness in choosing 푃푛 then corresponds precisely to the non-canonicity discussed toward the end of Section 3.1.2.
Give such a set of generators, we get an abstract basis for 퐴(푍). For each word 푤 of half-weight −푛 in the above generators, we have an element 푤 ∈ 풰n(푍), and these form a
basis of 풰n(푍). We let (푓푤)푤 denote the dual basis for 퐴(푍). With the choices above, we have u 푓휏ℓ = log (ℓ)
u 푓휎2푛+1 = 휁 (2푛 + 1).
u In order to find bases of 푃푛 and verify relations between different Li푛(푧)’s, we need to apply the reduced coproduct to reduce the computation in degree 푛 to the computation in degrees 푚 < 푛. For this, we need the exact sequence
Proposition 3.3.3. The sequence
′Δ′⊗id−id⊗Δ ′Δ 0 → 퐸푛 → 퐴푛 −→ 퐴푖 ⊗ 퐴푗 −−−−−−−−→ 퐴푖 ⊗ 퐴푗 ⊗ 퐴푘 푖+푗=푛 푖+푗+푘=푛 푖,푗≥1 푖,푗,푘≥1
1 is exact, and 퐸푛 = ExtMT(푍,Q)(Q(0), Q(푛)) = 퐾2푛−1(푍)Q.
Proof. We have the reduced cobar complex
Δ′ Δ′⊗id−id⊗Δ′ Δ′⊗id⊗id−id⊗Δ′⊗id+id⊗id⊗Δ′ 퐴>0 −→ 퐴>0 ⊗ 퐴>0 −−−−−−−−→ 퐴>0 ⊗ 퐴>0 ⊗ 퐴>0 −−−−−−−−−−−−−−−−−−→· · · , which is a complex of graded vector spaces. The proof of Lemma 3.8 of [Gon] shows that 푘 the 푘th cohomology of this complex is the graded vector space ExtGrComod(퐴)(Q(0), Q(푛)) 푛 (graded by 푛). The result then follows because the category of graded comodules over 퐴 is the
MT same as the category of representations of 휋1 (푍), or equivalently, the category MT(푍, Q), 1 and we know that ExtMT(푍,Q)(Q(0), Q(푛)) = 퐾2푛−1(푍)Q. from our knowledge of the Ext-groups in the category MT(푍, Q).
53 ′ We recall Definitions 3.1.4 and 3.1.5. In those definitions, welet Δ푛 denote the restriction ′ ′ ′ of Δ to 퐴푛. For 푖 + 푗 = 푛, we let Δ푖,푗 denote the component of Δ푛 landing in 퐴푖 ⊗ 퐴푗. The following corollary will be useful for computations in half-weight 4:
′ ′ Corollary 3.3.4. If 퐾 is totally real, then ker(Δ1,2) = ker(Δ3) = 퐸3.
′ Proof. Let 훼 ∈ 퐴3. Since Δ1 is zero (by Remark 3.1.8), we have
′ ′ ′ ′ ′ ′ ′ (Δ2 ⊗ id − id ⊗ Δ2)(Δ3(훼)) = (Δ2 ⊗ id)(Δ2,1(훼)) − (id ⊗ Δ2)(Δ1,2(훼)) = 0.
As 퐾 is totally real, it has no complex places, so we have 퐾3(푍)Q = 0. Then by Proposition ′ ′ 3.3.3, we have that 퐸2 = 0. Therefore, Δ2 is injective, hence Δ2 ⊗ id is injective on 퐴2 ⊗ 퐴1. ′ By the displayed equation and the previous sentence, it follows that if Δ1,2(훼) = 0, then ′ ′ ′ Δ2,1(훼) = 0 as well. Therefore, ker(Δ1,2) = ker(Δ3) = 퐸3.
Coordinates on the Space of Cocycles for 푍 = Spec Z[1/ℓ]
1,G푚 Relative to the chosen coordinates, we name coordinates for 푍PL,푛 when 푆 = {ℓ}. Specifically, we set
: 휏ℓ 푤0 = 휑푒0
: 휏ℓ 푤1 = 휑푒1
푤 := 휑휎2푘+1 2 ≤ 푘 ≤ 푛 푘 푒1푒0 ··· 푒0
⏟ 2 푘+1⏞ in the notation of Proposition 3.2.9. In fact, for 푘 ≥ 2, this makes sense even when |푆| > 1, and we use it in Section 3.3.3.
If 푧 ∈ 푋(푍), we write 푤푘(푧) for 푤푘(휅(푧)). In this notation, 푤0(푧) = ordℓ 푧, and 푤1(푧) =
− ordℓ(1 − 푧), both by Fact 3.2.3.
54 In this case, for 푛 ≥ 푘, we have
# u # u 풞PL,푛(log ) = 풞PL(log ) = 푤0푓휏
푘 ⌈ 2 ⌉ # u # u 푘−1 푘 ∑︁ 푘−2푖+1 푘−2푖+1 풞PL,푛(Li푘) = 풞PL(Li푘) = 푤1푤0 푓휏 /푘! + 푤푖푤0 푓휎2푖−1휏 . (3.8) 푖=2
3.3.2 The Geometric Step for Z[1/ℓ] in Half-Weight 4
Coordinates for the Galois Group
Let 푍 = Z[1/ℓ]. Then n(푍)≥−4 is three-dimensional as a vector space, generated by 휏 = 휏ℓ,
휎 = 휎3, and [휎, 휏]. As 푃3 = 0 when |푆| = 1, the elements 휏, 휎 are already well-defined. We un choose 푓휏 , 푓휎, and 푓휎휏 as a set of affine coordinates on 휋1 (푍)≥−4.
Coordinates for the Selmer Variety
In this case, the only nonzero coordinates are 푤0, 푤1, and 푤2.
The Universal Cocycle Evaluation Morphism
# We now write the morphism 풞PL,4 in these coordinates. We have 풞PL,4(푓푖) = 푓푖 for 푖 = 휏, 휎, 휎휏. # u # u Using Equation (3.8), we find that 풞PL,4(log ) = 푤0푓휏 , 풞PL,4(Li1) = 푤1푓휏 , and
# u 2 풞PL,4(Li2) = 푤0푤1푓휏 /2.
un At this point, we already see that the function on ΠPL,4 × 휋1 (푍)≥−4 given by
1 Liu − logu Liu 2 2 1
푍 vanishes on the image of 풞PL,4, i.e., is in ℐPL,4.
55 Again, by Equation (3.8), we have
# u 2 3 풞PL,4(Li3) = 푤1푤0푓휏 /6 + 푤2푓휎,
# u 3 4 풞PL,4(Li4) = 푤1푤0푓휏 /24 + 푤0푤2푓휎휏 .
푍 To construct a second element of ℐPL,4, we first eliminate 푤2 by considering
# u u u 3 5 3 4 풞PL,4(푓휎푓휏 Li4 −푓휎휏 log Li3) = 푤1푤0푓휎푓휏 /24 − 푓휎휏 푤1푤0푓휏 /6 푤 푤3푓 4 = 1 0 휏 (푓 푓 − 4푓 ) 24 휎 휏 휎휏 풞# ((logu)3 Liu) = PL,4 1 (푓 푓 − 4푓 ) 24 휎 휏 휎휏
It follows that
(︂ (logu)3 Liu )︂ 풞# 푓 푓 Liu −푓 logu Liu − 1 (푓 푓 − 4푓 ) = 0 PL,4 휎 휏 4 휎휏 3 24 휎 휏 휎휏
In other words,
Proposition 3.3.5. The two functions
1 Liu − logu Liu 2 2 1
(logu)3 Liu 푓 푓 Liu −푓 logu Liu − 1 (푓 푓 − 4푓 ) 휎 휏 4 휎휏 3 24 휎 휏 휎휏
un 푍 on ΠPL,4 × 휋1 (푍)≥−4 are in ℐPL,4 for 푍 = Spec Z[1/ℓ].
3.3.3 Coordinates on the Galois Group for 푍 = Z[1/ℓ]
In order to evaluate the functions of Proposition 3.3.5, we need to choose a prime 푝 ̸= ℓ and
interpret 푓휏 , 푓휎, and 푓휎휏 in such a way that we can take (approximate) their 푝-adic periods. Essentially, this means writing them as special values of polylogarithms. As mentioned in
56 Proposition 3.3.1, we have chosen the first two to correspond to logu(ℓ) and 휁u(3), respectively.
It remains to understand 푓휎휏 . This will depend on ℓ. We start with the case ℓ = 2, in an effort to re-derive Theorem 1.16 of [DCW16].
The Case 푍 = Z[1/2]
We let 퐴 = 퐴(푍) as usual. We compute using Corollary 3.2.2 and Fact 3.2.3 that
′ u u u 2 u u 2 2 Δ1,2(Li3(1/2)) = Li1(1/2) ⊗ (log (1/2)) /2 = − log (1/2) ⊗ (log (2)) /2 = 푓휏 ⊗ 푓휏 /2.
′ u 3 u Using the fact that Δ is a ring homomorphism, we note that Δ1,2(log (2) ) = 3 log (2) ⊗ logu(2)2. Therefore, Corollary 3.3.4 implies that
(logu(2))3 Liu(1/2) − 3 6
u is in 퐸3. As 퐸3 = 퐾5(푍)Q is one-dimensional, this is a rational multiple of 휁 (3). The identity in the appendix to [DCW16] says that
(logu(2))3 7 Liu(1/2) = + 휁u(3). (3.9) 3 6 8
By Definition 3.2.8, we have
# u 3 4 풞PL,4(Li4) = 푤1푤0푓휏 /24 + 푤0푤2푓휎휏 .
It is easy to check using the formulas at the end of Section 3.3.1 that 푤0(휅(1/2)) = −1
and 푤1(휅(1/2)) = 1, and (3.9) together with Definition 3.2.8 implies that 푤2(휅(1/2)) = 7/8. We therefore get 7 Liu(1/2) = −(logu(2))4/24 − 푓 . 4 8 휎휏 57 It follows that 8 (︂logu(2)4 )︂ 푓 = − + Liu(1/2) . 휎휏 7 24 4
Choosing 푃3(Spec Z[1/6])
′ To deal with the case 푍 = Spec Z[1/3], we have to consider 푍 = Spec Z[1/6], as 푍 is not ′ saturated. In this case, the Lie algebra n(푍 ) is generated by 휏2, 휏 = 휏3, and 휎 = 휎3. ′ ′ In this case, 푃1(푍 ) is zero. As 푃3(푍 ) is nonzero, there is some choice in the definition of ′ ′ 휎 as an element of n(푍 ). We therefore seek to choose a set 풫3(푍 ). ′ u u ′ ′ While 퐴(푍 )1 has basis {log (2), log (3)}, we must first find a basis of 퐴(푍 )2. As 퐸2(푍 ) = ′ ′ u 2 u u u 2 0, the map Δ2 is injective. We may take 풟2(푍 ) = {log (2) , log (2) log (3), log (3) }. Then
′ u u u u u Δ (Li2(−2)) = Li1(−2) ⊗ log (−2) = − log (3) ⊗ log (2), which is independent from
{Δ′ logu(2)2, Δ′ logu(2) logu(3), Δ′ logu(3)2}
′ ′ ′ ′ in 퐴1(푍 ) ⊗ 퐴1(푍 ) (as seen by checking via the basis of 퐴1(푍 ) ⊗ 퐴1(푍 ) induced by the u u ′ u basis {log (2), log (3)} of 퐴1(푍 )). In fact, Li2(−2) = −푓휏휏2 , as they have the same reduced coproduct.
u 2 u u u 2 u ′ This gives us a basis {log (2) , log (2) log (3), log (3) , Li2(−2)} of 퐴2(푍 ), so by taking ′ ′ all degree 3 products of basis elements of 퐴1(푍 ) and 퐴2(푍 ), we may take
′ u 3 u 3 u 2 u u u 2 u u u u 풟3(푍 ) = {log (2) , log (3) , log (2) log (3), log (2) log (3) , log (2) Li2(−2), log (3) Li2(−2)}.
We would like to show that
u 3 u 3 u 2 u u u 2 u u u u u u ℬ := {log (2) , log (3) , log (2) log (3), log (2) log (3) , log (2) Li2(−2), log (3) Li2(−2), Li3(−2), Li3(3)}
′ ′ u u ′ is a basis of 퐴3(푍 )/퐸3(푍 ), which would imply that {Li3(−2), Li3(3)} can be taken as 풫3(푍 ).
58 ′ To do this, we apply Δ1,2 to each element of ℬ and expand in the basis
u u 2 u u u u u 2 u {log (2) ⊗ log (2) , log (2) ⊗ log (2) log (3), log (2) ⊗ log (3) , log (2) ⊗ Li2(−2),
u u 2 u u u u u 2 u log (3) ⊗ log (2) , log (3) ⊗ log (2) log (3), log (3) ⊗ log (3) , log (3) ⊗ Li2(−2)}
′ ′ of 퐴1(푍 ) ⊗ 퐴2(푍 ). Using Corollary 3.2.2, this produces the matrix
logu(2) ⊗ logu(2)2 3 0 0 0 0 0 0 0 logu(2) ⊗ logu(2) logu(3) 0 0 2 0 0 0 0 0 logu(2) ⊗ logu(3)2 0 0 0 1 0 0 0 −1/2
u log (2) ⊗ Li2(−2) 0 0 0 0 1 0 0 0 , logu(3) ⊗ logu(2)2 0 0 1 0 −1 0 −1/2 0 logu(3) ⊗ logu(2) logu(3) 0 0 0 2 0 −1 0 0 logu(3) ⊗ logu(3)2 0 3 0 0 0 0 0 0
u log (3) ⊗ Li2(−2) 0 0 0 0 0 1 0 0 where the columns correspond to the element of ℬ. This matrix has determinant 9, which
′ ′ by Corollary 3.3.4 and the fact that 퐴3(푍 )/퐸3(푍 ) is eight-dimensional (by our knowledge ′ ′ ′ of the shuffle algebra 퐴(푍 )) implies that ℬ is in fact a basis of 퐴3(푍 )/퐸3(푍 ). We choose ′ u u 푃3(푍 ) to be the space generated Li3(−2) and Li3(3), and 휎 such that it pairs to 0 with this ′ choice of 푃3(푍 ).
The Case 푍 = Z[1/3]
′ Armed with our choice of 휎 ∈ n(푍 ), we seek to write 푓휎휏 ∈ 퐴(푍) as an explicit combination
of motivic polylogarithms. With our choice of 휎, we have 푤2(−2) = 푤2(3) = 0 because u u Li3(−2) and Li3(−3) are elements of 풫3. From the matrix above, we see that
logu(2) ⊗ logu(3)2 Δ′ Liu(3) = − . 1,2 3 2 59 We also compute
logu(9)2 Δ′ Liu(9) = Liu(9) ⊗ = −6 logu(2) ⊗ logu(3)2, 1,2 3 1 2
u u ′ u ′ ′ so Li3(9) = 12 Li3(3) is in 퐸3(푍 ), hence a rational multiple of 휁 (3) (because 퐸3(푍 ) = 퐾5(푍 )Q
is one-dimensional). In fact, since 푤2(3) = 0, this rational number is the 푓휎-coordinate of u ′ u Li3(9) in the basis (푓푤)푤 of 퐴(푍 ), hence it equals 푤2(9) because 푓휎 = 휁 (3). Numerical computation using the code [DCC] in the 5-adic and 7-adic realizations suggests that Liu(9) − 12 Liu(3) 26 푤 (9) = 3 3 = − . 2 휁u(3) 3 We may now compute using Definition 3.2.8 that
u 휏 휏2 휏 3 Li4(3) = 푤2(3)휑푒0 (3)푓휎휏 + 휑푒1 (3)휑푒0 (3) 푓휏2휏휏휏 = −푓휏2휏휏휏
and
u 휏 휏2 휏 3 Li4(9) = 푤2(9)휑푒0 (9)푓휎휏 + 휑푒1 (9)휑푒0 (9) 푓휏2휏휏휏 = 2푤2(9)푓휎휏 − 24푓휏2휏휏휏
This implies that
12 u −1 u − Li4(3) + (2푤2(9)) Li4(9) = 푓휎휏 . 푤2(9)
un un ′ We note that this corresponds precisely to 푓휎휏 in the image of 풪(휋1 (푍)) ˓→ 풪(휋1 (푍 )). Plugging this into the second equation of Proposition 3.3.5, we find
Theorem 3.3.6. The element
(︂ )︂ u u u 12 u −1 u u u 휁 (3) log (3) Li4 − − Li4(3) + (2푤2(9)) Li4(9) log Li3 푤2(9) u 3 u (︂ (︂ )︂)︂ (log ) Li1 u u 12 u −1 u − 휁 (3) log (3) − 4 − Li4(3) + (2푤2(9)) Li4(9) 24 푤2(9)
un 푍 of 풪(ΠPL,4 × 휋1 (푍)) is in ℐPL,4 for 푍 = Spec Z[1/3], where 푤2(9) is a number 푝-adically 26 close to − for 푝 = 5, 7. 3 60 For 푝 ̸= 2, 3, the corresponding Coleman function is
(︂ )︂ 푝 푝 푝 12 푝 −1 푝 푝 푝 휁 (3) log (3)Li4(푧) − − Li4(3) + (2푤2(9)) Li4(9) log (푧)Li3(푧) (3.10) 푤2(9) 푝 3 푝 (︂ (︂ )︂)︂ (log (푧)) Li1(푧) 푝 푝 12 푝 −1 푝 − 휁 (3) log (3) − 4 − Li4(3) + (2푤2(9)) Li4(9) . 24 푤2(9)
3.3.4 The Chabauty-Kim Locus for Z[1/3] in Half-Weight 4 1 As noted in Section 8.2 of [BDCKW], the function Li푝(푧) − log푝(푧) log푝(1 − 푧) has the zero 2 2 1 1 set {2, , −1} for 푝 = 5, 7. By numerical evaluation of (3.10) at 2, , and −1, we conclude 2 2 1 that 2, ∈/ 푋(푍 ) . It follows that: 2 p PL,4
Theorem 3.3.7. For 푍 = Spec Z[1/3], we have 푋(푍p)PL,4 ⊆ {−1} for 푝 = 5, 7.
3.4 Answer to Question 3.1.24 and 푆3-Symmetrization
3.4.1 Answer to Question 3.1.24
We fix a set of generators as in Section 3.3.1. We first need the following lemma:
푝 Lemma 3.4.1. We have Li푘(−1) = 0 for 푘 ≥ 2 even.
−푘 푝 2 푝 푝 Proof. This follows from the identity 2 Li푘(푧 ) = Li푘(푧) + Li푘(−푧), which is Proposition 6.1 푝 −푘 푝 of [Col82] for 푚 = 1. Indeed, setting 푧 = 1 in the identity shows Li푘(−1) = (2 − 1)Li푘(1), 푝 푝 푝 and since Li푘(1) = 휁 (푘), which is 0 for 푘 even, we have Li푘(−1) = 0.
Theorem 3.4.2. For any prime ℓ and positive integer 푛, we have
−1 ∈ 푋(푍p)PL,푛, where 푍 = Spec Z[1/ℓ].
61 In particular, for ℓ odd, Question 3.1.24 has a negative answer.
u Proof. We use the coordinates of Section 3.3.1. We also write 휏 = 휏ℓ. We have 푓휏 = log (ℓ).
1,G푚 To prove the theorem, we produce an element 푐−1 of 푍PL,푛 (풦) whose image 훼−1 under
풞PL,푛 lies in ΠPL,푛(퐴(푍)) ⊆ ΠPL,푛(풦) and satisfies perp(훼−1) = 휅p(−1) ∈ ΠPL,푛(Q푝). Since un 푍 풦 any element of 풪(ΠPL,푛 × 휋1 (푍)) ∩ ℐPL,푛 vanishes on the image of 풞PL,푛, it vanishes on 훼−1
and therefore on 휅p(−1), which proves the theorem.
We define 푐−1 by setting
푤0(푐−1) := 0, Liu(−1) 푤 (푐 ) := 1 , 1 −1 logu(ℓ) Liu (−1) 푤 (푐 ) := 2푘−1 푘 = 2, ··· , 푛. 푘 −1 휁u(2푘 − 1)
u u Setting 훼−1 := 풞PL,푛(푐−1), we now compute log (훼−1) and Li푘(훼−1) for 1 ≤ 푘 ≤ 푛. # u We have 풞PL,푛(log ) = 푤0푓휏 , so
u log (훼−1) = 푤0(푐−1)푓휏 = 0.
# u We have 풞PL,푛(Li1) = 푤1푓휏 , so
Liu(−1) Liu(훼 ) = 푤 (푐 )푓 = 1 logu(ℓ) = Liu(−1). 1 −1 1 −1 휏 logu(ℓ) 1
For 푘 ≥ 2 even, we have
푘 ⌈ 2 ⌉ # u 푘−1 푘 ∑︁ 푘−2푖+1 푘−2푖+1 풞PL,푛(Li푘) = 푤1푤0 푓휏 /푘! + 푤푖푤0 푓휎2푖−1휏 푖=2 ⎛ 푘 ⎞ ⌈ 2 ⌉ 푘−1 푘 ∑︁ 푘−2푖 푘−2푖+1 = 푤0 ⎝푤1푤0 푓휏 /푘! + 푤푖푤0 푓휎2푖−1휏 ⎠ , 푖=2
u and since 푤0(푐−1) = 0, we have Li푘(훼−1) = 0.
62 For 푘 ≥ 3 odd, we have
푘 ⌈ 2 ⌉ # u 푘−1 푘 ∑︁ 푘−2푖+1 푘−2푖+1 풞PL,푛(Li푘) = 푤1푤0 푓휏 /푘! + 푤푖푤0 푓휎2푖−1휏 푖=2 ⎛ 푘 ⎞ ⌈ 2 ⌉−1 푘−1 푘 ∑︁ 푘−2푖 푘−2푖+1 푘+1 = 푤0 ⎝푤1푤0 푓휏 /푘! + 푤푖푤0 푓휎2푖−1휏 ⎠ + 푤 푓휎푘 , 2 푖=2
so by 푤0(푐−1) = 0, we have
u 푘+1 Li푘(훼−1) = 푤 (−1)푓휎푘 2 Liu(−1) = 푘 휁u(푘) 휁u(푘) u = Li푘(−1).
u u u u We have thus shown that log (훼−1) = 0, Li푘(훼−1) = 0 for 푘 even, and Li푘(훼−1) = Li푘(−1)
for 푘 odd. This shows that 훼−1 ∈ ΠPL,푛(퐴(푍)). 푝 The key fact is that Li푘(−1) = 0 for 푘 ≥ 2 even. This follows from the identity −푘 푝 2 푝 푝 2 Li푘(푧 ) = Li푘(푧) + Li푘(−푧), which is Proposition 6.1 of [Col82] for 푚 = 1. Indeed, setting 푝 −푘 푝 푝 푝 푧 = 1 in the identity shows Li푘(−1) = (2 − 1)Li푘(1), and since Li푘(1) = 휁 (푘), which is 0 푝 for 푘 even, we have Li푘(−1) = 0. 푝 푝 By Lemma 3.4.1, Li푘(−1) = 0 for 푘 ≥ 2 even. Since we also have log (−1) = 0, we get 푝 푝 that Li푘(−1) = log (−1) for 푘 ≥ 2. This implies that perp(훼−1) = 휅p(−1), as desired.
3.4.2 푆3-Symmetrization
MT We recall our strengthening of Conjecture 3.1.23. The 푆3-action on 푋 induces an 휋1 (푍)- un equivariant action on 휋1 (푋), and for a quotient Π and 휎 ∈ 푆3, we may refer to 휎(Π). We then write
푆3 ⋂︁ ⋂︁ 푋(푍p)PL,푛 := 휎(푋(푍p)PL,푛) = 푋(푍p)휎(Π). 휎∈푆3 휎∈푆3 Our symmetrized conjecture is that
63 푆3 Conjecture 3.4.3. 푋(푍) = 푋(푍p)PL,푛 for sufficiently large 푛.
Verification for 푍 = Spec Z[1/3]
We can use our computations in Section 3.3 to show verify a case of this conjecture:
Theorem 3.4.4. For 푍 = Spec Z[1/3] and 푝 = 5, 7, Conjecture 3.4.3 (and hence Conjecture 3.1.23) holds (with 푛 = 4).
Proof. By Theorem 3.3.7, 푋(푍p)PL,4 ⊆ {−1}. But −1 is not fixed by the action of 푆3, so
푆3 ⋂︁ 푋(푍p)PL,4 := 휎(푋(푍p)PL,4) = ∅ 휎∈푆3
for 푝 = 5, 7. In particular, Conjecture 3.4.3 and hence Conjecture 3.1.23 holds in these cases.
64 Part II
Obstructions to the Local-Global Principle
65 66 Chapter 4
Introduction and Setup
4.1 Introduction
Given a variety 푋 over a global field 푘, a major problem is to decide whether 푋(푘) = ∅. By [Poo09], it suffices to consider the case that 푋 is smooth, projective, and geometrically integral. As a first approximation one can consider the set 푋(A푘) ⊃ 푋(푘), where A푘 is the adele ring of 푘. It is a classical theorem of Minkowski and Hasse that if 푋 is a quadric, then
푋(A푘) ̸= ∅ ⇒ 푋(푘) ̸= ∅. When a variety 푋 satisfies this implication, we say that it satisfies the Hasse principle, or local-global principle. In the 1940s, Lind and Reichardt ([Lin40] [Rei42] ) gave examples of genus 1 curves that do not satisfy the Hasse principle. More counterexamples to the Hasse principle were given throughout the years, until in 1971 Manin [Man71] described a general obstruction to the Hasse principle that explained many of the counterexamples to the Hasse principle that were known at the time. The obstruction (known
Br as the Brauer-Manin obstruction) is defined by considering a certain set 푋(A푘) , such that Br 푋(푘) ⊂ 푋(A푘) ⊂ 푋(A푘). If 푋 is a counterexample to the Hasse principle, we say that it is Br accounted for or explained by the Brauer-Manin obstruction if ∅ = 푋(A푘) ⊂ 푋(A푘) ̸= ∅. In 1999, Skorobogatov ([Sko99]) defined a refinement of the Brauer-Manin obstruction known as the étale Brauer-Manin obstruction and used it to produce an example of a variety
Br ´et,Br 푋 such that 푋(A푘) ≠ ∅ but 푋(푘) = ∅. Namely, he described a set 푋(A푘) for which
67 ´et,Br Br ´et,Br 푋(푘) ⊂ 푋(A푘) ⊂ 푋(A푘) ⊂ 푋(A푘) and found a variety 푋 such that ∅ = 푋(A푘) ⊂ Br 푋(A푘) ̸= ∅. In his paper [Poo10], Poonen constructed the first example of a variety 푋 such that
´et,Br ∅ = 푋(푘) ⊂ 푋(A푘) ̸= ∅. However, Poonen’s method of showing that 푋(푘) = ∅ relies on the details of his specific construction and is not explained by a new finer obstruction. While it was hoped ([Pál10]) that an étale homotopy obstruction might solve the problem, it was shown ([HS13]) that this provides nothing new. Therefore, one wonders if Poonen’s counterexample can be accounted for by an additional
´et,Br refinement of 푋(A푘) . Namely, can one give a general definition of a set
new ´et,Br 푋(푘) ⊂ 푋(A푘) ⊂ 푋(A푘)
new such that Poonen’s variety 푋 satisfies 푋(A푘) = ∅. In this paper, we provide such a refinement and prove that it is necessary and sufficient over many number fields. The obstruction is essentially given by applying the finite abelian descent obstruction to open covers of 푋 or decompositions of 푋 as a disjoint union of locally closed subvarieties. We shall make this statement precise below. As an example of our results, we can prove the following:
Corollary 4.1.1 (of Theorem 4.3.3). Let 푋/Q be a variety for which 푋(Q) = ∅. Then there ⋃︁ Br is a Zariski open cover 푋 = 푈푖 such that 푈푖(A푘) = ∅ for all 푖. 푖
For a summary of all of the variants of this result, the reader may proceed to Section 4.3. Chapter 5 contains the main proofs of these results, one method unconditional and one method assuming the section conjecture in anabelian geometry. In Chapter 6, we examine the section conjecture more closely, in particular what happens to cuspidal sections in fibrations. Our results are of independent interest for the section conjecture, but they also allow us to reprove some of the results in Section 5 via a slightly different method (Corollary 6.3.16). In this chapter, we also derive an important result about the behavior of the étale-Brauer obstruction in fibrations (Theorem 6.2.10) that will be useful
68 in Chapter 7. Finally, as our discussion touches on the concept of cuspidal sections for higher dimensional varieties, we explore this topic in more detail in Section 5.3.1. Finally, in Chapter 7, we explicitly find a Zariski open cover of the examplePoo10 of[ ] and prove that its pieces have empty étale-Brauer set with the aid of one of our results on homotopy fixed points in fibrations (Theorem 6.2.10). In fact, we introduce a few variants of this result, depending on different hypotheses. Motivated by one of these examples, we also introduce the notion of quasi-torsors (Definition 7.2.1), which leads us to propose anew approach to computing obstructions for open subvarieties, known as the ramified étale-Brauer obstruction (Definition 7.2.2).
4.1.1 Notation and Conventions
Whenever we speak of a field 푘, we implicitly fix a separable closure 푘푠 throughout, which
produces a geometric point Spec(푘푠) → Spec(푘) of 푘. We write 푋/푘 to mean that 푋 is a 푠 scheme over Spec 푘, and we denote by 푋 the base change of 푋 to 푘푠, following [Poo17]. We
let 퐺푘 denote Gal(푘푠/푘). If 푋 is a scheme, 풪(푋) denotes 풪푋 (푋). All cohomology is taken in the étale topology unless otherwise stated, and fundamental group always denotes an étale fundamental group. The same is true for higher homotopy groups, once they are defined in Definition 6.1.12. If 푘 is a field, we write 퐻*(푘, −) for Galois cohomology of the field 푘.
For a global field 푘, we let A푘 denote the ring of adeles of 푘. For a place 푣 of 푘, we let 푘푣
denote the completion of 푘 at 푣, 풪푣 the ring of integers in 푘푣, m푣 the maximal ideal in 풪푣, 휋푣
a generator of m푣, F푣 the residue field of 풪푣, and 푞푣 the size of F푣. If 푆 is a subset of the set ′ ′ ∏︁ 푆 ∏︁ of places of 푘, we let A푘,푆 = (푘푣, 풪푣). This must not be confused with A푘 = (푘푣, 풪푣). 푣∈푆 푣∈ /푆 We let 풪푘,푆 denote the 푆-integers of 푘, i.e.„ in which primes in 푆 are inverted. In general, we 푓 replace 푆 by the letter “f” when 푆 is the set of finite places of 푘. We also use A푘 to denote
A푘,푆 when 푆 is the set of finite places, and we believe this does not lead to any confusion. We use loc to denote the map from the version of some set over a global field to the adelic version (or the product over all places of the local version), e.g. for Galois cohomology or
69 homotopy fixed points. We write loc푆 when we use the A푘,푆 version, and we write loc푣 to
denote the map from the version over the global field 푘 to that over 푘푣.
If 푋 is a variety over a local field 푘푣, we let 푋(푘푣)∙ denote the set of connected components ∏︁′ of 푋(푘푣) under the 푣-adic topology, and we similarly set 푋(A푘,푆)∙ = (푋(푘푣)∙, 푋(풪푣)∙). 푣∈푆
4.2 Obstructions to Rational Points
4.2.1 Generalized Obstructions
Let 푘 be a global field.
Definition 4.2.1. Let 휔 be a subfunctor of the functor 푋 ↦→ 푋(A푘) from 푘-varieties to 휔 sets. We write 푋(A푘) instead of 휔(푋). We say that 휔 is a generalized obstruction (to the 휔 local-global principle) if 푋(푘) ⊆ 푋(A푘) for every 푘-variety 푋.
휔 휔 If 푆 is a subset of the places of 푘, we define 푋(A푘,푆) to be the projection of 푋(A푘)
from 푋(A푘) to 푋(A푘,푆). We note the trivial but important lemmas:
′ 휔 휔′ Lemma 4.2.2. Let 휔 and 휔 be two generalized obstructions. If 푋(A푘) ⊆ 푋(A푘) , then 휔 휔′ 푋(A푘,푆) ⊆ 푋(A푘,푆) .
휔 휔′ Proof. The containment 푋(A푘) ⊆ 푋(A푘) implies that the containment remains true when we project from 푋(A푘) to 푋(A푘,푆).
′ 휔 Lemma 4.2.3. Let 푆, 푆 be two nonempty sets of places of 푘. Then 푋(A푘,푆) = ∅ if and 휔 only if 푋(A푘,푆′ ) = ∅.
′ 휔 Proof. It suffices to suppose 푆 is the set of all places of 푘. The set 푋(A푘,푆) is the projection 휔 of 푋(A푘,푆′ ) from 푋(A푘,푆′ ) to 푋(A푘,푆), so one is empty if and only if the other is.
휔 Lemma 4.2.4. The association 푋 ↦→ 푋(A푘,푆) is a subfunctor of the covariant functor from
푘-varieties to sets represented by Spec(A푘,푆).
70 휔 Proof. Let 푓 : 푋 → 푌 be a map of 푘-varieties. It suffices to show that 푓 maps 푋(A푘,푆) into 휔 휔 ′ 휔 ′ 푌 (A푘,푆) . Let 훼 ∈ 푋(A푘,푆) . Then 훼 is the projection of some 훼 ∈ 푋(A푘) . But 푓(훼 ) is in 휔 휔 푌 (A푘) and projects to 푓(훼), so 푓(훼) ∈ 푌 (A푘,푆) .
Very Strong Approximation
Definition 4.2.5. If 푘 is a global field, S is a nonempty set of places of k,and 휔 is a generalized obstruction, then we call (휔, 푆, 푘) an obstruction datum.
When 푆 is the set of finite places of 푘, we also write (휔, 푓, 푘) in this case. We will sometimes leave out 푆 and write (휔, 푘), in which case we understand 푆 to be the set of all places of 푘. We often write (휔, 푆) when 푘 is understood.
Definition 4.2.6. Let (휔, 푆, 푘) be an obstruction datum. We say that a variety 푋/푘 is or
휔 satisfies very strong approximation (VSA) for (휔, 푆) if 푋(푘) = 푋(A푘,푆) .
Lemma 4.2.7. Let 푊 ⊆ 푋 be be a (locally closed) subvariety of 푋 over 푘. If 푋 is VSA for (휔, 푆, 푘), then so is 푊 .
휔 휔 Proof. Let 훼 ∈ 푊 (A푘,푆) . Then 훼 ∈ 푋(A푘,푆) by functoriality. Since 푋 is VSA, 훼 ∈ 푋(푘); 훼 i.e., Spec A푘,푆 −→ 푊 factors through a map Spec 푘 → 푋.
Let 푣 ∈ 푆. Then the 푣-component 훼푣 of 훼 is a 푘푣-point of 푋 coming from a 푘-point of 푋.
But because 훼 ∈ 푊 (A푘,푆), this implies that 훼푣 ∈ 푊 (푘푣). But if it is a 푘-point as a point of
푋, then we also have 훼푣 ∈ 푊 (푘).
Remark 4.2.8. It is the need to prove Lemma 4.2.7 that prevents us from replacing the set of
points at the real places by the set of their connected components, as the map from 푊 (A푘)∙
to 푋(A푘)∙ need not be injective.
Lemma 4.2.9. Suppose 푋 and 푌 are 푘-varieties that are VSA for (휔, 푆, 푘). Then so is 푋 × 푌 .
휔 Proof. Let (푥, 푦) ∈ (푋 ×푌 )(A푘,푆) . By functoriality of 휔 applied to the projections 푋 ×푌 → 휔 휔 푋, 푌 , we have 푥 ∈ 푋(A푘,푆) and 푦 ∈ 푌 (A푘,푆) . By VSA for 푋 and 푌 , this implies 푥 ∈ 푋(푘) and 푦 ∈ 푌 (푘), so (푥, 푦) ∈ (푋 × 푌 )(푘).
71 4.2.2 Functor Obstructions
Let 푘 be a global field. Following the formalism in Section 8.1Poo17 of[ ], let 퐹 be a contravariant functor from schemes over 푘 to sets (a.k.a. a presheaf of sets on the category of 푘-schemes). For 푋 a 푘-scheme and 퐴 ∈ 퐹 (푋), we have the following commutative diagram
푋(푘) −−−→ 푋(A푘) ⎮ ⎮ ⎮ ⎮ ⌄ ⌄ loc ∏︁ 퐹 (푘) −−−→ 퐹 (푘푣), 푣
퐴 where the vertical arrows denote pullback of 퐴 from 푋 to 푘 or 푘푣. We define 푋(A푘) as the
subset of 푋(A푘) whose image in the lower right object is in the image of loc. We then define the obstruction set
퐹 ⋂︁ 퐴 푋(A푘) = 푋(A푘) . 퐴∈퐹 (푋) If ℱ is a collection of such functors 퐹 , we define
ℱ ⋂︁ 퐹 푋(A푘) = 푋(A푘) . 퐹 ∈ℱ
ℱ Lemma 4.2.10. The functor 푋 ↦→ 푋(A푘) is a generalized obstruction.
ℱ Proof. It is clear that 푋(푘) ⊆ 푋(A푘) , so it suffices to verify functoriality, which is a simply diagram chase.
′ ℱ ℱ ′ Lemma 4.2.11. Let ℱ ⊆ ℱ. Then 푋(A푘,푆) ⊆ 푋(A푘,푆) for all 푋.
ℱ 퐹 ′ Proof. If 훼 ∈ 푋(A푘) , then 훼 ∈ 푋(A푘) for all 퐹 ∈ ℱ, hence for all 퐹 ∈ ℱ . But this ℱ ′ ℱ ℱ ′ implies 훼 ∈ 푋(A푘) , i.e. 푋(A푘) ⊆ 푋(A푘) . The result for general 푆 follows by Lemma 4.2.2.
72 4.2.3 Descent Obstructions
Let 퐺 be a group scheme over 푘. We let 퐹퐺 denote the contravariant functor from 푘-schemes
to sets for which 퐹퐺(푋) is the set of isomorphism classes of 퐺-torsors for the fppf topology 1 over 푋. By Theorem 6.5.10(i) of [Poo17], we have 퐹퐺(푋) = 퐻fppf (푋, 퐺) when 퐺 is affine. If 퐺 is also smooth, it is isomorphic to the étale cohomology, denoted 퐻1(푋, 퐺) (c.f. Notation and Conventions). Let 풢 be a subset of the set of isomorphism classes of finite type group schemes over 푘.
For ℱ = {퐹퐺}[퐺]∈풢, we set 풢 ℱ 푋(A푘) = 푋(A푘) .
Usually, 풢 will consist only of algebraic groups over 푘, i.e., finite type separated group schemes over 푘.
Definition 4.2.12. For 풢 the set of smooth affine algebraic groups over 푘, we call this the
descent descent obstruction and denote it by 푋(A푘) (following [Poo17]).
Definition 4.2.13. For 풢 the set of finite smooth (étale) algebraic groups over 푘, we call
f−cov this the finite descent obstruction and denote it by 푋(A푘) (following [Sto07],[Sko09]).
Definition 4.2.14. For 풢 the set of finite solvable smooth algebraic groups over 푘, we call
f−sol this the finite solvable descent obstruction and denote it by 푋(A푘) (following [Sto07]).
Definition 4.2.15. For 풢 the set of finite commutative smooth algebraic groups over 푘, we
f−ab call this the finite abelian descent obstruction and denote it by 푋(A푘) (following [Sto07]).
Remark 4.2.16. By the equivalence between (i) and (i’) in Theorem 2.1 of [HS12], it suffices to take only finite constant algebraic groups over 푘 in Definition 4.2.13.
When we wish to refer to the obstruction and specify a specific 푆, we write (f − ab, 푆, 푘), etc.
descent f−cov f−sol f−ab Proposition 4.2.17. We have 푋(A푘) ⊆ 푋(A푘) ⊆ 푋(A푘) ⊆ 푋(A푘) .
73 Proof. This follows by Lemma 4.2.11 because the corresponding collections of isomorphism classes of group schemes become smaller as we progress from descent to f − cov to f − sol to f − ab.
Corollary 4.2.18. For fixed 푘 and 푆, VSA for f − ab implies VSA for f − sol, which implies VSA for f − cov, which implies VSA for descent.
Proof. This follows immediately from Proposition 4.2.17 and Lemma 4.2.2.
Alternative Description in Terms of Images of Adelic Points
Let 퐺 be a smooth affine algebraic group over 푘, and let 푍 ∈ 퐹퐺(푋) be given by, with abuse 휏 휏 of notation, 푓 : 푍 → 푋. If 휏 ∈ 퐹퐺(푘), one can define the twist 푓 : 푍 → 푋 as in Example 6.5.12 of [Poo17]. By Theorem 8.4.1 of [Poo17], if 푥 ∈ 푋(푘), then 푥 ∈ 푓 휏 (푌 휏 (푘)) if and only if 휏 is the pullback of 푍 under 푥: Spec 푘 → 푋. It follows that
⋃︁ 휏 휏 ⋃︁ 휏 휏 푋(푘) = 푓 (푍 (푘)) ⊆ 푓 (푍 (A푘,푆)).
휏∈퐹퐺(푘) 휏∈퐹퐺(푘)
Proposition 4.2.19. We have
⋃︁ 휏 휏 푍 푓 (푍 (A푘)) = 푋(A푘) .
휏∈퐹퐺(푘)
푍 Proof. Suppose 훼 ∈ 푋(A푘) . We think of this point as a map 훼: Spec A푘 → 푋, and we know by definition that the pullback of 푍 along this map is in the image of
1 loc ∏︁ 1 퐻 (푘, 퐺) −→ 퐻 (푘푣, 퐺), 푣
say of 휏 ∈ 퐻1(푘, 퐺). One can easily check that twisting commutes with pullback (from 푘
휏 to A푘). Thus the proof of 8.4.1 of [Poo17] tells us that the pullback of 푍 under this adelic
74 ∏︁ 1 휏 point is now the trivial element of 퐻 (푘푣, 퐺). But this means that the fiber of 푍 over 훼 푣 휏 휏 is a trivial torsor and thus contains an adelic point; i.e., 훼 ∈ 푓 (푍 (A푘)). 휏 휏 1 Conversely, suppose 훼 ∈ 푓 (푍 (A푘)) for some 휏 ∈ 퐻 (푘, 퐺). Then for each 푣, we have 휏 휏 훼푣 ∈ 푓 (푍 (푘푣)), so Theorem 8.4.1 tells us that the pullback of 푍 under 훼푣 : Spec 푘푣 → 푋
is loc푣(휏). But this implies that the pullback of 푍 under 훼 is in the image of loc; i.e., 푍 훼 ∈ 푋(A푘) .
4.2.4 Comparison with Brauer and Homotopy Obstructions
Br There is the Brauer-Manin obstruction set, 푋(A푘,푆) , which is the obstruction for 퐹 (−) = 2 퐻 (−, G푚). There is also the étale Brauer-Manin obstruction
´et,Br ⋂︁ ⋃︁ 휏 휏 Br 푋(A푘) = 푓 (푌 (A푘) ). finite ´etale 퐺 휏∈퐻1(푘,퐺) all 퐺−torsors푓:푌 →푋
f−cov By Section 4.2.3, this is contained in 푋(A푘,푆) . A series of obstructions is defined in [HS13] when 푘 is a number field:
ℎ ℎ,2 ℎ,1 푋(A푘) ⊆ · · · ⊆ 푋(A푘) ⊆ 푋(A푘) ⊆ 푋(A푘)
and
Zℎ Zℎ,2 Zℎ,1 푋(A푘) ⊆ · · · ⊆ 푋(A푘) ⊆ 푋(A푘) ⊆ 푋(A푘)
ℎ,푛 Zℎ,푛 Zℎ ⋂︁ Zℎ,푛 ℎ such that 푋(A푘) ⊆ 푋(A푘) for all 푛, 푋(A푘) = 푋(A푘) , and 푋(A푘) = 푛 ⋂︁ ℎ,푛 ℎ 푋(A푘) . The definition of 푋(A푘) will be discussed in more detail in Section 6.1.2. 푛 When discussing obstruction data, we write (ℎ, 푆, 푘) and (Zℎ, 푆, 푘). By Theorem 9.1 of [HS13] and Lemma 4.2.2, we have for a smooth geometrically connected
75 variety 푋 that:
ℎ ´et,Br 푋(A푘,푆) = 푋(A푘,푆)
Zℎ Br 푋(A푘,푆) = 푋(A푘,푆)
ℎ,1 f−cov 푋(A푘,푆) = 푋(A푘,푆)
Zℎ,1 f−ab 푋(A푘,푆) = 푋(A푘,푆)
In particular, we have the two inclusions:
´et,Br Br f−ab 푋(A푘,푆) ⊆ 푋(A푘,푆) ⊆ 푋(A푘,푆)
´et,Br f−cov f−ab 푋(A푘,푆) ⊆ 푋(A푘,푆) ⊆ 푋(A푘,푆) .
4.3 Statement of Our Main Results
For the rest of this section, we assume that 푘 is a number field. The main reason to care about these obstructions is that they help prove that a variety
´et,Br 푋/푘 has no rational points. The most powerful obstruction currently known is 푋(A푘) = descent 푋(A푘) . But, as stated in the introduction, even in this case, there is a variety 푋 with ´et,Br ∅ = 푋(푘) ⊆ 푋(A푘) ̸= ∅, as found in [Poo10]. In the method of proof that 푋(푘) = ∅, it is clear that the étale Brauer-Manin obstruction and its avatars still appear, but they are applied separately to different pieces of 푋. From this point of view, it’s natural to ask the following question:
Question 4.3.1. Let 푋 be a 푘-variety with 푋(푘) = ∅. Does there exist a finite open cover or stratification of 푋 for which each constituent part has empty étale-Brauer set? More strongly, is the same true for any of the other obstruction sets from Section 4.2.3?
If true, this proves that 푋(푘) = ∅, because if each constituent part has no rational points, then 푋 does not. One could ask for the stronger statement that each constituent part satisfies VSA:
76 Question 4.3.2. Let 푋 be a 푘-variety and (휔, 푆, 푘) an obstruction datum. Does there exist a finite open cover or stratification of 푋 for which each constituent part satisfies VSA for (휔, 푆, 푘)?
In this paper, we obtain the following result.
Theorem 4.3.3 (Corollary 5.2.4 and Corollary 5.2.9). The answer to Question 4.3.2 is yes for (f − ab, 푓, 푘) when 푘 is a imaginary quadratic or totally real number field.
This is actually a very strong result when it applies, because f − ab is weaker than the Brauer-Manin, finite descent, or étale Brauer-Manin obstruction. To see whether we expect this to hold for all number fields, we prove the following conditional result:
Theorem 4.3.4 (Corollary 5.3.27). Assuming Grothendieck’s section conjecture (5.3.25), the answer to Question 4.3.2 is yes for (f − cov, 푆, 푘), where 푆 is a nonempty set of finite places.
This leaves only the question for (f − ab, 푓, 푘) and arbitrary number fields 푘. Here, we can only make a conjecture:
Conjecture 4.3.5 (Conjecture 5.4.2). For any number field 푘, the answer to Question 4.3.2 is true for (f − ab, 푓, 푘)
The primary method of proof consists in reducing the problem to proving VSA for an
1 open subset of P , a condition we call 퐴(휔, 푓, 푘). Theorem 5.1.3 then tells us in general that this answers Question 4.3.2 (and therefore Question 4.3.1) affirmatively. This is described in Chapter 5. In Chapter 6, we examine the section conjecture more closely, in particular what happens to cuspidal sections in fibrations. Our results are of independent interest for the section conjecture, but they also allow us to reprove Theorem 4.3.4 in Corollary 6.3.16. While this might seem logically unnecessary if the theorem is already proven, the second proof provides a possibly different open cover than that of the first proof.
77 In Chapter 7, we review the original example of [Poo10] in terms of our results. We also introduce the notion of quasi-torsors (Definition 7.2.1), which leads us to propose the following conjecture:
Conjecture 4.3.6 (Conjecture 7.2.4). For any number field 푘 and variety 푋/푘 with 푋(푘) = ∅,
1 ∐︁ does there exist a stratification 푋 = 푋푖 and for each 푖, a quasi-torsor 푌푖 over the closure 푖 휎 Br 휎 푋푖 of 푋푖 such that 푌푖 restricts to a torsor over 푋푖 and 푌푖 (A푘) = ∅ for all twists 푌푖 of 푌푖.
4.3.1 A New Obstruction
Let X = {푋푖} be a finite collection of locally closed subvarieties of 푋 whose set-theoretic union is 푋 (e.g., a stratification or open cover) and (휔, 푆, 푘) an obstruction datum. We define
the following two sets, where unions take place in 푋(A푘,푆):
X ,휔 ⋃︁ 휔 푋(A푘,푆) = 푋푖(A푘,푆) 푖
Functoriality then gives us the inclusion:
X ,휔 휔 푋(푘) ⊆ 푋(A푘,푆) ⊆ 푋(A푘,푆) .
Finally, we must consider the case where C is a collection of finite collections of locally closed subvarieties. For example, C might be the collection of all open covers of 푋, denoted by 풪풫ℰ풩 , or the collection of all finite stratifications, denoted by 풮풯 ℛ풜풯 . We then define
C ,휔 ⋂︁ X ,휔 푋(A푘,푆) = 푋(A푘,푆) . X ∈C In fact, 풪풫ℰ풩 and 풮풯 ℛ풜풯 are really rules that assign such a C to each 푋/푘. We can then rephrase Theorem 5.1.3:
1We are defining a finite stratification of 푋 as a finite partially-ordered index set 퐼 along with a locally closed subset 푆푖 ⊆ 푋 for every 푖 ∈ 퐼 such that 푋 is the disjoint union of all 푆푖, and the closure of any given ⋃︁ 푆푖 in 푋 is the union 푆푗. 푗≤푖
78 Theorem 4.3.7. If 퐴(휔, 푆, 푘) holds, then for any 푋/푘, there is a finite Zariski open cover X such that X ,휔 푋(푘) = 푋(A푘,푆) .
As a result,
풪풫ℰ풩 ,휔 푋(푘) = 푋(A푘,푆) .
Just for completeness, we note a couple of basic properties of these types of obstructions.
Proposition 4.3.8. Let (휔, 푆, 푘) be an obstruction datum and 푓 : 푋 → 푌 a map of 푘-schemes.
풪풫ℰ풩 ,휔 풪풫ℰ풩 ,휔 풮풯 ℛ풜풯 ,휔 풮풯 ℛ풜풯 ,휔 Then 푓 maps 푋(A푘,푆) into 푌 (A푘,푆) and 푋(A푘,푆) into 푌 (A푘,푆) , respectively.
풪풫ℰ풩 ,휔 −1 Proof. Let 훼 ∈ 푋(A푘,푆) , and let {푌푖}푖 be a finite open cover of 푌 . Then {푓 (푌푖)}푖 is a −1 휔 휔 finite open cover of 푋, so 훼 ∈ 푓 (푌푖)(A푘,푆) . Then functoriality tells us that 푓(훼) ∈ 푌푖(A푘,푆) . 풪풫ℰ풩 ,휔 As {푌푖}푖 was arbitrary, 푓(훼) ∈ 푌 (A푘,푆) . 풮풯 ℛ풜풯 ,휔 The other part goes nearly word-for-word. Let 훼 ∈ 푋(A푘,푆) , and let {푌푖}푖 −1 be a finite stratification of 푌 . Then {푓 (푌푖)}푖 is a finite stratification of 푋, so 훼 ∈ −1 휔 휔 푓 (푌푖)(A푘,푆) . Then functoriality tells us that 푓(훼) ∈ 푌푖(A푘,푆) . As {푌푖}푖 was arbitrary, 풮풯 ℛ풜풯 ,휔 푓(훼) ∈ 푌 (A푘,푆) .
The following proposition explains why answering Question 4.3.2 positively for open covers rather than stratifications is more powerful:
Proposition 4.3.9. For any (휔, 푆, 푘), we have
풮풯 ℛ풜풯 ,휔 풪풫ℰ풩 ,휔 푋(A푘,푆) ⊆ 푋(A푘,푆)
풮풯 ℛ풜풯 ,휔 Proof. Suppose 훼 ∈ 푋(A푘,푆) . We must show that for every finite open cover 푛 ⋃︁ 휔 푋 = 푋푖, we have 훼 ∈ 푋푖(A푘,푆) for some 푖. 푖=1 We build a stratification out of this open cover as follows. For 푖 = 1, ··· , 푛, we let
⋃︁ 푆푖 = 푋푖 ∖ 푋푗. 푗<푖
79 ⋃︁ 휔 Then {푆푖} forms a stratification, with 푆푖 = 푆푗 for all 푖. Thus 훼 ∈ 푆푖(A푘,푆) for some 푖. 푗≥푖 휔 But 푆푖 ⊆ 푋푖, so functoriality tells us that 훼 ∈ 푋푖(A푘,푆) .
new If one wishes to define 푋(A푘) as in the introduction, the natural candidates are 풪풫ℰ풩 ,f−ab 풮풯 ℛ풜풯 ,f−ab 푋(A푘) and 푋(A푘) . The real power in our results, however, is that one need only take a single open cover.
Remark 4.3.10. It is interesting to consider the constructions in this subsection applied when
휔 휔 is such that 푋(A푘) = 푋(A푘) for all 푋. This is discussed in Appendix A.
Remark 4.3.11. This refinement of obstructions via open covers is related to the notion of cosheafification. This connection is discussed in Appendix B.
4.3.2 Relationship to the Birational Section Conjecture
The reader might think that the ideas in this paper can be used to prove the birational
section conjecture. The idea is this: let 푋 be a proper smooth hyperbolic curve over Q. If there is a birational Galois section (over Q), then the p-adic birational section conjecture
([Koe05]) shows that there is a Q푝-point for every place 푝. This provides an adelic point of 푋 whose associated birational section comes from a global section. In other words, it is in the finite descent set of every open subset of our curve, and by the results here, it isrational. The flaw in this argument is that an adelic point on 푋 might not be an adelic point of any proper open subvariety. In the language of Appendix B, this is the fundamental reason why
ℱ 푓 is not a cosheaf. Therefore, one would have to use the fact that its birational section is A푘 comes from a birational section over a global field in a deeper way to make such an argument work. All of this is in fact implicit in [Sti15] and Theorems 3.2, 3.3, 4.2, and 4.3 of [HS12].
80 Chapter 5
Main Result via Embeddings
5.1 General Setup
We set up the basic formalism for the various unconditional and conditional results proven via embeddings.
Definition/Theorem 5.1.1. Let (휔, 푆, 푘) be an obstruction datum. The following statements are equivalent:
1 (i) There is a nonempty open 푘-subscheme of P푘 that is VSA for (휔, 푆, 푘).
1 (ii) There is a nonempty open 푘-subscheme of A푘 that is VSA for (휔, 푆, 푘).
푛 (iii) For each positive integer 푛, there is a nonempty open 푘-subscheme of A푘 that is VSA for (휔, 푆, 푘).
푛 (iv) For each positive integer 푛, there is a nonempty open 푘-subscheme of P푘 that is VSA for (휔, 푆, 푘).
If this is the case, we say that the property 퐴(휔, 푆, 푘) is true.
1 Proof. If (i) holds, let this open subscheme be 푈. Then 푉 := 푈 ∩ A푘 is nonempty because any nonempty open is dense, and 푉 is then VSA by Lemma 4.2.7, so (ii) holds.
81 푛 푛 If (ii) holds, then 푉 is a nonempty open subscheme of A푘 for all 푛, which is VSA by Lemma 4.2.9, so (iii) holds.
푛 If (iii) holds, then the VSA nonempty open subscheme of A푘 is also a VSA nonempty 푛 open subscheme of P푘 , so (iv) holds. Finally, (iv) implies (i) by setting 푛 = 1.
1 푛 We note that PGL2(푘) acts on P푘 by 푘-automorphisms, and hence (PGL2(푘)) acts on 1 푛 (P푘) in the same way.
1 푛 Lemma 5.1.2. Let 푘 be an infinite field and 푥 a closed point of (P푘) . Then the orbit of 푥 푛 1 푛 under (PGL2(푘)) is Zariski dense in (P푘) .
1 푛 Proof. We choose an algebraic closure 푘 of 푘 and a point 푥 of (P (푘)) representing 푥. We 1 푛 푛 equivalently wish to show that for any open 푈 ⊆ (P푘) , the orbit of 푥 under (PGL2(푘)) intersects 푈(푘).
1 We first prove this for 푛 = 1. The points (푥 + 푎)푎∈푘 of P are distinct. As there are 1 infinitely many of them, and P is one-dimensional, they are Zariski dense. But these points are all in the orbit of 푥, so this orbit is Zariski dense.
1 Let 푥 = (푥1, ··· , 푥푛) with 푥푖 ∈ P푘(푘), and let 푆푖 be the PGL2(푘)-orbit of 푥푖. The previous 푛 1 ∏︁ 1 푛 paragraph implies that each 푆푖 is Zariski-dense in P (푘), so 푆푖 is dense in (P (푘)) . 푖=1 A less detailed and less general version of this argument is given in Lemma 6.3 of [SS16].
Theorem 5.1.3. Suppose that 퐴(휔, 푆, 푘) holds. Let 푋 be a 푘-variety. Then there exists a ⋃︁ finite affine open cover 푋 = 푉푖 such that 푉푖 is VSA for (휔, 푆, 푘) for every 푖. 푖 Proof. Let 푥 be any closed point. By the definition of a scheme, there is an affine open neighborhood of 푋 containing 푥, so we can assume that 푋 is affine.
푛 1 We now embed 푋 into A for some sufficiently large 푛. As A is an open subscheme of 1 푛 1 푛 1 푛 1 푛 P , we have an open inclusion A = (A ) ˓→ (P ) , so we get an embedding 휑 : 푋 ˓→ (P ) . 1 푛 Assuming 퐴(휔, 푆, 푘), there is an open subset 푈 ⊆ (P ) that is VSA for (휔, 푆, 푘). By 1 푛 Lemma 5.1.2, there is a 푘-automorphism 푔 of (P ) sending 휑(푥) into 푈. Then 푈 contains
82 푔(휑(푥)). But then 푈 ∩ 푔(휑(푋)) is an open subscheme of 푋 containing 푥 and is a locally closed subscheme of 푈. It is therefore VSA for (휔, 푆, 푘) by Lemma 4.2.7. Choose an affine
open neighborhood 푉푥 of 푥 in 푈 ∩ 푔(휑(푋)), and 푉푥 is again VSA. ⋃︁ We do this for every 푥, and we obtain an affine open cover 푋 = 푉푥 such that 푉푥 is 푥∈푋 VSA for (휔, 푆, 푘) for every 푥. As varieties are quasi-compact, this has a finite subcover, which proves the theorem.
Corollary 5.1.4. Let 푋 be a 푘-variety with 푋(푘) = ∅, and suppose 퐴(휔, 푆, 푘). Then there exists a finite open cover of 푋 such that each constituent open set has empty obstruction set for (휔, 푆, 푘). The same is true for any other nonempty set 푆′ of places of 푘. ⋃︁ Proof. Let 푉푖 be an open cover as in Theorem 5.1.3. As 푋(푘) = ∅, we have 푉푖(푘) = ∅. By 푖 휔 VSA, this implies 푉푖(A푘,푆) = ∅. 휔 By Lemma 4.2.3, we also have 푉푖(A푘,푆′ ) = ∅ for every 푖.
5.2 Embeddings of Varieties into Tori
For the rest of Section 5.2, we assume that 푘 is a number field.
Lemma 5.2.1. Let 푇 be an algebraic torus over 푘, let 푆 be a nonempty set of places of 푘, and let 훼 ∈ 푇 (A푘,푆). The first statement implies the second, and if 푆 consists only of finite places of 푘, the second implies the third:
f−ab (i) 훼 ∈ 푇 (A푘,푆) .
푛 (ii) For every 푛 ∈ Z≥0, there exists 푎푛 ∈ 푇 (푘) and 훽푛 ∈ 푇 (A푘,푆) for which 훼 = 푎푛(훽푛) .
(iii) 훼 ∈ 푇 (푘), with the closure taken in 푇 (A푘,푆).
[푛] Proof. (i) =⇒ (ii): For every 푛, there is a standard torsor 푇 −→ 푇 under the 푛-torsion scheme 푇 [푛] of 푇 given by the 푛th power map 푇 → 푇 . The Kummer map is the map sending [푛] 푥 ∈ 푇 (푘) to the pullback of 푇 −→ 푇 under 푥, the result of which is a torsor under 푇 [푛] over 푘. In this case, it is given by the boundary map 푇 (푘) → 퐻1(푘, 푇 [푛]) in Galois cohomology
83 coming from the short exact sequence 0 → 푇 [푛] → 푇 → 푇 → 0 of étale sheaves. In particular,
the image of the Kummer map is canonically 푇 (푘)/푛푇 (푘). The same holds with 푘푣 in place f−ab of 푘, and all maps respect the inclusions 푘 → 푘푣. If 훼 ∈ 푇 (A푘,푆) , it must be in the image of ∏︁ 푇 (푘)/푛푇 (푘) → 푇 (푘푣)/푛푇 (푘푣), 푣∈푆
푛 ∏︁ which amounts to saying that 훼 = 푎푛(훽푛) for 훽푛 ∈ 푇 (푘푣). As 푎푛, 훼 ∈ 푇 (A푘,푆), so is 훽푛. 푣∈푆 (ii) =⇒ (iii): Let 퐾 be any open subgroup of 푇 (A푘,푆). By Theorem 5.1 of [Bor63] (also 1 ℎ c.f. [Con06]) , 푇 (A푘,푆)/푇 (푘)퐾 is finite, say of order ℎ. We know that 훼 = 푎ℎ(훽ℎ) . But ℎ (훽ℎ) ∈ 푇 (푘)퐾, and therefore so is 훼. In other words, the coset 훼퐾 contains an element
of 푇 (푘). The set of all such 퐾 and their cosets form a basis for the topology of 푇 (A푘,푆) because 푆 only contains finite places. It follows that every open neighborhood of 훼 contains an element of 푇 (푘), or 훼 ∈ 푇 (푘).
Lemma 5.2.2. Let 푇 be an algebraic torus over 푘 and 퐾 an open subgroup of 푇 (A푘,푆) such 푓 that 푇 (푘) ∩ 퐾 is finite. Then 푇 (푘) is closed in 푇 (A푘). For example, this happens for 푆 = 푓 if 푇 has a model 풯 over 풪푘 with 풯 (풪푘) finite.
Proof. As 푇 (푘) is a subgroup, each coset of 퐾 also has finite intersection with 푇 (푘). The
topological space 푇 (A푘,푆) is the disjoint union of the cosets of 퐾. This space is 푇0, so finite sets are closed. Thus 푇 (푘) is closed in each coset of 퐾. This implies that it is closed in all of
푇 (A푘,푆). By the definition of the adelic topology, the subgroup
∏︁ 푓 풯 (풪̂︁푘) := 풯 (풪푣) ⊆ 푇 (A푘) 푣
is open. We can then conclude because 풯 (풪푘) = 풯 (푘) ∩ 풯 (풪̂︁푘).
1 In fact, [Bor63] only covers the case that A푘 = A푘,푆, but the result holds in general by considering the continuous projection A푘 → A푘,푆.
84 5.2.1 The Result When 푘 is Imaginary Quadratic
In this subsection, we prove 퐴(f − ab, 푓, 푘) when 푘 has finitely many units.
Remark 5.2.3. By Dirichlet’s unit theorem, 푘 has finitely many units if and only if 푘 is Q or imaginary quadratic.
Corollary 5.2.4. For 푘 as above, G푚 is VSA for (f − ab, 푓, 푘). In particular, 퐴(f − ab, 푓, 푘) holds.
Proof. That 푘 has finitely many units means that G푚(풪푘) is finite. By Lemma 5.2.2, this 푓 implies that G푚(푘) is closed in G푚(A푘). By Lemma 5.2.1, this implies that G푚(푘) = 푓 f−ab G푚(A푘) ; i.e., G푚 is VSA for (f − ab, 푓, 푘).
5.2.2 The Result When 푘 is Totally Real
We now prove 퐴(f − ab, 푓, 퐹 ) when 푘 is totally real. The material in this subsection is inspired by [Sti15].
1 In fact, we can choose the subscheme of P푘 as in Definition/Theorem 5.1.1(i) to be the 1 complement in P푘 of the vanishing scheme of any quadratic polynomial over 푘 with totally negative discriminant.
Definition 5.2.5. For 푘 a totally real number field and 퐸/푘 a totally imaginary quadratic extension, we define the norm one torus relative to 퐸/푘
푇 = 푇퐸/푘 = ker(N퐸/푘 : Res퐸/푘G푚 → G푚), which is a group scheme over 푘.
Proposition 5.2.6. Let 훼 ∈ 퐸 with minimal polynomial 푡2 + 푏푡 + 푐 over 푘. Let 푈 be the
1 2 complement in P푘 (with coordinate 푡) of the vanishing locus of 푡 +푏푡+푐. Then 푈 is isomorphic to 푇 .
Proof. As 퐸 = 푘(훼), and the norm of 푥 − 푦훼 is 푥2 + 푏푥푦 + 푐푦2, we can express 푇 as Spec 푘[푥, 푦]/(푥2 + 푏푥푦 + 푐푦2 − 1). This has projective closure Proj(푘[푥, 푦, 푧]/(푥2 + 푏푥푦 +
85 푐푦2 − 푧2)), which is a smooth projective conic. This conic has a point (푥, 푦, 푧) = (1, 0, 1), so
1 2 2 2 it is isomorphic to P푘. The complement of 푇 is given by 0 = 푧 = 푥 + 푏푥푦 + 푐푦 , which corresponds to the vanishing locus of 푡2 + 푏푡 + 푐 by setting 푡 = 푥/푦.
Proposition 5.2.7. For some integral model 풯 of 푇 , the set 풯 (풪푘) is finite. Thus 푇 is VSA for (f − ab, 푓, 푘) by Lemma 5.2.2.
Proof. The torus 푇 has an integral model 풯 = ker(N풪퐸 /풪푘 : Res푂푐퐸 /풪푘 G푚 → G푚). Thus
풯 (풪푘) is the kernel of N풪퐸 /풪푘 : G푚(풪퐸) → G푚(풪푘). N 풪퐸 /풪푘 2 The composition G푚(풪푘) ˓→ G푚(풪퐸) −−−−−→ G푚(풪푘) is 푥 ↦→ 푥 . Its image in the finitely
generated abelian group G푚(풪푘) therefore has full rank, and therefore so does the image
of 푁 : G푚(풪퐸) → G푚(풪푘). But 퐸 and 푘 have the same number of Archimedean places, so
G푚(풪퐸) and G푚(풪푘) have the same rank. But a map between finitely generated abelian
groups of the same rank whose image has full rank must have finite kernel. Therefore 풯 (풪푘) is finite.
Remark 5.2.8. One may alternatively prove Proposition 5.2.7 by noting that 풪퐾 is discrete in 푘 ⊗ R and that 푇 (푘 ⊗ R) is compact.
1 Corollary 5.2.9. If 푈 is the complement in P푘 of the vanishing locus of a quadratic polynomial 푡2 + 푏푡 + 푐 with totally negative discriminant, then 푈 is VSA for (f − ab, 푓, 푘). In particular, 퐴(f − ab, 푓, 푘) holds.
Proof. Let 퐸 be the splitting field of 푡2 + 푏푡 + 푐. By Proposition 5.2.6, 푈 is isomorphic to
푇퐸/푘, so by Proposition 5.2.7, 푈 is VSA for (f − ab, 푓, 푘).
5.3 Finite Descent and the Section Conjecture
We now prove 퐴(f − cov, 푓, 푘) for any number field 푘 assuming the section conjecture for
1 P푘 ∖ {0, 1, ∞}.
86 5.3.1 Grothendieck’s Section Conjecture
The Kummer Map
For the remainder of this subsection, we let 푋 denote a quasi-compact quasi-separated
geometrically connected scheme over a field 푘. We fix a separable closure 푘푠 of 푘, which
produces a geometric point Spec(푘푠) → Spec(푘) of 푘. We identify 퐺푘 := Gal(푘푠/푘) with the étale fundamental group of Spec 푘 based at this point. As in [Poo17], 푋푠 denotes the base
change 푋푘푠 .
푠 Definition 5.3.1. Let 푎 ∈ 푋(푘푠), which we may also view as an element of 푋 (푘푠) and
Spec 푘(푘푠). By [Sta17, Tag 0BTX], there is an exact sequence
푠 1 → 휋1(푋 , 푎) → 휋1(푋, 푎) → 퐺푘 → 1,
known as the fundamental exact sequence
Definition 5.3.2. The set of sections S휋1(푋/푘),푎 is defined to be the set of (multiplicative) 푠 sections of the surjection 휋1(푋, 푎) → 휋1(Spec 푘, 푎) modulo the action of 휋1(푋 , 푎) by conjuga- 푠 tion. If we fix a section, we get an action of 퐺푘 on 휋1(푋 , 푎), and S휋1(푋/푘),푎 is isomorphic to 1 푠 the nonabelian continuous Galois cohomology pointed-set 퐻 (푘, 휋1(푋 , 푎)) (c.f. “Generalized Sections” in 1.2 of [Sti13]).
Fact 5.3.3. As in Definition 23 ofSti13 [ ], the sets S휋1(푋/푘),푎 for different choices of 푎 are in
canonical bijection, so we may write S휋1(푋/푘) without ambiguity.
′ If 푘 /푘 is a field extension and 푋 a 푘-scheme, ′ denotes ′ . S휋1(푋/푘 ) S휋1(푋푘′ /푘 )
Definition 5.3.4 (Profinite Kummer Map). Given a point 푏 ∈ 푋(푘), there is a unique geometric basepoint 푏 lying over 푏 and compatible with the basepoint of Spec 푘. Then 푏 induces a pointed map of schemes Spec 푘 → 푋, which induces a map of fundamental groups
퐺푘 → 휋1(푋, 푏) compatible with the projection 휋1(푋, 푏) → 퐺푘, and hence an element of
87 . By Fact 5.3.3, this gives us a well-defined map S휋1(푋/푘),푏
휅 = 휅푋/푘 : 푋(푘) → S휋1(푋/푘), which we call the (profinite) Kummer .map
As with Definition 5.3.2, 휅 ′ denotes 휅 ′ . As well, if 푆 is a set of places of 푘, then 푋/푘 푋푘′ /푘 ∏︁ ∏︁ we set 휅푆 = 휅푋/A푘,푆 := 휅푋/푘푣 : 푋(A푘,푆) → 푋(ℎ푘푣), or 휅푋/A푘 when 푆 is all places. 푣∈푆 푣∈푆 A map 푓 : 푋 → 푌 of quasi-compact quasi-separated geometrically connected 푘-schemes
induces a map 푓 : S휋1(푋/푘) → S휋1(푌/푘) by choosing a compatible pair of geometric basepoints for 푋 and 푌 (but is easily seen to be independent of that choice). This map is compatible under the Kummer map with 푓 : 푋(푘) → 푌 (푘).
Remark 5.3.5. If we assume the existence of a (fixed) Galois-invariant basepoint 푎, we can
1 푠 푠 define 휅(푏) as the class in 퐻 (퐺푘, 휋1(푋 , 푎)) of the 퐺푘-equivariant torsor 휋1(푋 , 푎, 푏).
Remark 5.3.6. In Section 6.1, we will reformulate the section conjecture in more homotopical language, which will have the advantage of obviating the need for basepoints.
Conjecture 5.3.7 (Grothendieck [Gro97]). Let k be a number field and 푋/푘 a geometrically connected, smooth, proper curve of genus at least 2. Then the Kummer map 푋(푘) → S휋1(푋/푘) is a bijection.
Remark 5.3.8. This is false for all nontrivial abelian varieties, as shown in [CS15].
We will, however, need to extend this conjecture to the non-proper case, as in Chapter 18 of [Sti13]. To do so, we must introduce the notion of cuspidal sections.
Cuspidal Sections
Definition 5.3.9. A cuspidal datum (푋, 퐶, 푘) is a smooth geometrically connected variety
푋 over a field 푘 and a subset 퐶 ⊆ S휋1(푋/푘), called the set of cuspidal sections.
Definition 5.3.10. A cuspidal datum (푋, 퐶, 푘) is said to satisfy the surjectivity in the section conjecture if S휋1(푋/푘) ∖ 퐶 ⊆ 휅(푋(푘)).
88 Definition 5.3.11. Let 푎 be a geometric basepoint, and assume that the fundamental exact sequence for (푋, 푎) has a section 푠. We define the centralizer 푍(푠) as the set of elements of
푠 휋1(푋 , 푎) that commute in 휋1(푋, 푎) with 푠(푔) for all 푔 ∈ 퐺푘. If two sections are conjugate, their centralizers are conjugate. In particular, the property of having trivial centralizer is a
property of elements of S휋1(푋/푘).
푠 Remark 5.3.12. As in Definition 5.3.2, a section 푠 gives an action of 퐺푘 on 휋1(푋 , 푎). The 0 푠 centralizer of 푠 is then isomorphic to the cohomology group 퐻 (푘, 휋1(푋 , 푎)).
Definition 5.3.13. A cuspidal datum (푋, 퐶, 푘) is said to satisfy the injectivity in the section conjecture if
1. The map 휅: 푋(푘) → S휋1(푋/푘) is injective.
2. The sets 휅(푋(푘)) and 퐶 are disjoint.
3. The centralizer of every element of 휅(푋(푘)) is trivial
Remark 5.3.14. The reader might wonder what the last two conditions have to do with injectivity. Theorem 6.2.6 will show that this stronger definition of injectivity holds inductively in geometric fibrations, while the more naive version does not. We also note that this stronger version is known to hold for hyperbolic curves (Theorem 5.3.24).
Proposition 5.3.15. Let 푘′/푘 be an extension of characteristic 0 fields and 푋 as above.
′ ′ We choose a separable closure 푘푠 and an embedding of 푘푠 in 푘푠, which gives us a map ′ ′ ′ 퐺푘′ = 휋1(Spec 푘 , 푘푠) → 휋1(Spec 푘, 푘푠) → 휋1(Spec 푘, 푘푠) = 퐺푘. Then there is a base change map
′ S휋1(푋/푘) → S휋1(푋/푘 ) induced by precomposition with the map 퐺푘′ → 퐺푘 (defined by choosing a geometric basepoint
for 푋푘′ but independent of basepoint). Furthermore, the diagram
푋(푘) −−−→ 푋(푘′) ⎮ ⎮ 휅푋/푘⎮ ⎮휅푋/푘′ ⌄ ⌄
′ S휋1(푋/푘) −−−→ S휋1(푋/푘 )
89 is commutative.
Proof. The base change map is Definition 27 ofSti13 [ ]. Characteristic 0 is required to ensure
′ that 푋푘푠 and 푋푘푠 have the same étale fundamental group. (Remark: This also works in positive characteristic if 푋 is proper.)
Let 푥 ∈ 푋(푘) and 푥 an associated Galois-invariant basepoint of 푋푘′ . We may apply
휋1(−, 푥) to the commutative diagram of schemes
Spec 푘′ −−−→ Spec 푘 ⎮ ⎮ 푥⎮ ⎮푥 ⌄ ⌄
푋푘′ −−−→ 푋
to obtain the commutative diagram of profinite groups
퐺푘′ −−−→ 퐺푘 ⎮ ⎮ 휅 ′ (푥)⎮ ⎮휅 (푥) 푋/푘 ⌄ ⌄ 푋/푘
휋1(푋푘′ , 푥) −−−→ 휋1(푋, 푥)
As the base change map is obtained by precomposition with the top horizontal arrow, the commutativity of the original diagram is clear.
Definition 5.3.16. If (푋, 퐶, 푘) is a cuspidal datum and 푘′/푘 an extension of fields of
′ ′ characteristic 0, we say that a cuspidal datum (푋푘′ , 퐶 , 푘 ) is compatible with (푋, 퐶, 푘) if the base change map sends 퐶 into 퐶′.
′ ′ Lemma 5.3.17. With the notation of the previous definition, if (푋푘′ , 퐶 , 푘 ) satisfies the injectivity in the section conjecture, then so does (푋, 퐶, 푘).
′ Proof. As 푋(푘) ˓→ 푋(푘 ) and 휅푋/푘′ are injective, Proposition 5.3.15 tells us that 휅푋/푘 is
injective. As well, if there were 푥 ∈ 푋(푘) for which 휅푋/푘(푥) ∈ 퐶, then Proposition 5.3.15 ′ tells us that 휅푋/푘′ (푥) ∈ 퐶 , which is not the case. Finally, suppose there were an element 푠 of 휅(푋(푘)) with nontrivial centralizer. Then the
′ ′ ′ ′ ′ ′ image 푠 of 푠 in S휋1(푋/푘 ) lies in 휅(푋(푘 )), so by injectivity for (푋푘 , 퐶 , 푘 ), we know that the
90 centralizer of 푠′ is trivial. But a nontrivial element of the centralizer of 푠 is also a nontrivial element of the centralizer of 푠′, so 푠 must have trivial centralizer.
Curves
Definition 5.3.18. Let 푈 be a smooth geometrically connected curve over a field 푘 of characteristic 0, and let 푈 ˓→ 푋 be its unique compactification. Let 푌 = 푋 ∖ 푈 and 푦 ∈ 푌 (푘).
ℎ ℎ We fix a Henselization 풪푋,푦 and set 푋푦 = Spec 풪푋,푦. We let 푈푦 = 푋푦 ×푋 푈. By 18.3 of [Sti13], there is a short exact sequence
× × 0 → Hom(풪(푈푦) /풪(푋푦) , Ẑ︀(1)) → 휋1(푈푦) → 퐺푘 → 0,
and S휋1(푈푦/푘) is the set of sections of the surjection 휋1(푈푦) → 퐺푘 modulo conjugation by the
kernel. The map 푈푦 → 푈 induces a map S휋1(푈푦/푘) → S휋1(푈/푘), whose image 퐶푦 is known as
the packet of cuspidal sections at 푦, originally defined by Grothendieck. The set 퐶푈 of all cuspidal sections of the curve 푈 is defined to be
⋃︁ (︀ )︀ Im S휋1(푈푦/푘) → S휋1(푈/푘) ⊆ S휋1(푈/푘). 푦∈푌 (푘)
When we speak of cuspidal sections of a curve 푈/푘, we always mean (푈, 퐶푈 , 푘) unless otherwise stated.
Lemma 5.3.19. Let 푈/푘 be as in Definition 5.3.18, and let 푘′/푘 be a field extension. Then
′ ′ (푈, 퐶푈 , 푘) is compatible with (푈푘 , 퐶푈푘′ , 푘 ) in the sense of Definition 5.3.16.
Proof. Let 푋 be the compactification of 푈 over 푘, and let 푦 ∈ (푋 ∖ 푈)(푘). We can view 푦
′ × × × × as an element of (푋 ∖ 푈)(푘 ). The map 풪(푈푦) /풪(푋푦) → 풪((푈푘′ )푦) /풪((푋푘′ )푦) is an
isomorphism, so the map (푈푘′ )푦 → 푈푦 induces an isomorphism on geometric fundamental
groups. This produces a base change map → ′ . A diagram chase as in S휋1(푈푦/푘) S휋1((푈푘′ )푦/푘 )
91 Proposition 5.3.15 shows that the diagram
−−−→ ′ S휋1(푈푦/푘) S휋1((푈푘′ )푦/푘 ) ⎮ ⎮ ⎮ ⎮ ⌄ ⌄
−−−→ ′ S휋1(푈/푘) S휋1(푈푘′ /푘 )
commutes, which proves the desired compatibility.
Definition 5.3.20. Let 푈 be a connected smooth curve over a separably closed field 푘. Let 푈 ˓→ 푋 be the unique embedding of 푋 into a projective geometrically connected regular curve. Let 푌 := 푋∖푈, which is a zero dimensional scheme of some degree 푛. Let 푔 denote the genus of 푋. We define the Euler characteristic 휒(푈) of 푈 to be 2 − 2푔 − 푛.
Definition 5.3.21. Let 푈 be a geometrically connected smooth curve over an arbitrary field 푘. We set 휒(푈) = 휒(푈 푠).
Definition 5.3.22. Let 푈/푘 be a geometrically connected smooth curve (not necessarily projective). Then we say 푈 is hyperbolic if and only if 휒(푈) < 0.
푠 Remark 5.3.23. In characteristic 0, 푈 is hyperbolic if and only if 휋1(푈 ) is non-abelian.
Theorem 5.3.24. Let 푘 be a subfield of a finite extension of Q푝, and 푈/푘 a geometrically connected smooth hyperbolic curve. Then 푈 satisfies the injectivity in the section conjecture.
Proof. By Lemma 5.3.17 and Lemma 5.3.19, it suffices to prove the theorem for finite
extensions of Q푝. The injectivity of the Kummer map follows from Corollary 74 and Proposition 75 of [Sti13].
0 푠 The second part follows from Theorem 250 of [Sti13]. The fact that 퐻 (푘, 휋1(푈 , 푎)) = 0 is Proposition 104 of [Sti13].
The following is an extension of Conjecture 5.3.7 to non-proper curves:
92 Conjecture 5.3.25 (Grothendieck [Gro97]). Let k be a number field and 푋/푘 a geometrically connected, smooth hyperbolic curve (not necessarily proper). Then the surjectivity in the section conjecture holds for (푋, 퐶푋 , 푘).
5.3.2 VSA via the Section Conjecture
We now explain the relationship between the section conjecture and very strong approximation:
Proposition 5.3.26. Let 푆 be a nonempty set of finite places of a number field 푘 and 푋 a variety over 푘. Suppose 푋 satisfies the surjectivity in the section conjecture over 푘 and the injectivity in the section conjecture over 푘푣 for all 푣 ∈ 푆 (and some compatible choice of cuspidal data). Then 푋 satisfies VSA for (f − cov, 푆, 푘)
f−cov ′ f−cov Proof. Let 푃 ∈ 푋(A푘,푆) , and let 푃 ∈ 푋(A푘) project to 푃 . For each place 푣 of 푘, ′ ′ let 푠푣 : = 휅푋/푘푣 (푃푣) ∈ S휋1(푋/푘푣). Then 푃 satisfies (i) of Theorem 2.1 of[HS12] for 푈 the trivial group and 푆 (in the notation of [HS12]) the empty set. It therefore satisfies (iii) of the
same theorem; i.e., (푠푣) is image of some 푠 ∈ S휋1(푋/푘) under the bottom horizontal arrow of the following diagram: 푋(푘) −−−→ 푋(A푘) ⎮ ⎮ 휅푋/푘⎮ ⎮휅푋/ ⌄ ⌄ A푘 ∏︁ S휋1(푋/푘) −−−→ S휋1(푋/푘푣) 푣
If 푠 were cuspidal, 푠푣 it would be for each 푣. But 푠푣 is not cuspidal for every finite 푣 by the second part of injectivity in the section conjecture, so 푠 is not cuspidal. By surjectivity in the section conjecture, there is an element 푥 of 푋(푘) mapping to 푠. It has the same image
∏︁ ′ as 푃 under 휅푋/푘푣 . But Theorem 5.3.24 tells us that 휅푆 is injective, so 푃 equals 푥 at all 푣 푣 ∈ 푆. Thus, 푃 equals 푥 and therefore lies in 푋(푘).
This now allows us to answer Question 4.3.2.
Corollary 5.3.27. Conjecture 5.3.25 implies that the property 퐴(f − cov, 푆, 푘) holds for any number field 푘 and nonempty set 푆 of finite places.
93 1 Proof. By Conjecture 5.3.25 and Theorem 5.3.24, P ∖ {0, 1, ∞} satisfies the surjectivity in the section conjecture over 푘 and the injectivity in the section conjecture over 푘푣 for all finite places 푣 of 푘 (which holds for a compatible choice of cuspidal data by Lemma 5.3.19).
1 Proposition 5.3.26 then implies that P ∖ {0, 1, ∞} is VSA for (f − cov, 푆, 푘). By Theorem 5.1.1, 퐴(f − cov, 푆, 푘) holds.
5.4 Finite Abelian Descent for all 푘
Let 푘 be a number field. Lemma 5.2.1 states that the only reason tori do not satisfy VSA for (f − ab, 푓, 푘) is that the rational points are not closed in the adelic points. The following
1 result therefore suggests that P ∖ {0, 1, ∞} might play the role of tori over any number field 푘:
1 푓 Proposition 5.4.1. Let 푋 = P ∖ {0, 1, ∞}푘. The set 푋(푘) is closed in 푋(A푘).
푓 Proof. Let 훼 ∈ 푋(A푘) ∖ 푋(푘). We find a neighborhood of 훼 that does not intersect 푋(푘). ∏︁ ∏︁ There must be some finite set 푆 of places of 푘 for which 훼 ∈ 푈푆 := 푋(푘푣) × 푋(풪푣). 푣∈푆 푣∈ /푆 But
푋(푘) ∩ 푈푆 = 푋(풪푘,푆) is finite by Siegel’s Theorem, so 푈푆 ∖ 푋(푘) is an open set containing 훼.
This leads us to conjecture:
1 Conjecture 5.4.2. P ∖{0, 1, ∞}푘 is VSA for (f − ab, 푓, 푘), so 퐴(f − ab, 푓, 푘) for any number field 푘.
In fact, a weaker version of this follows from a conjecture of Harari and Voloch:
Proposition 5.4.3. Conjecture 2 of [HV10] implies that the answer to Question 4.3.1 is yes for the finite abelian descent obstruction over any number field 푘.
Proof. By the proof of Theorem 5.1.3, we can decompose any variety into affine open subsets
1 that embed in some power of (P ∖ {0, 1, ∞})푘. Therefore, it suffices to prove the result for
94 푛 1 푈/푘 for which there exists 푓 : 푈 → 푋 for some positive integer 푛 and 푋 = (P ∖ {0, 1, ∞})푘. f−ab Suppose that 푈(A푘) is nonempty; we wish to show that 푈 has a 푘-point. f−ab Br Let 훼 ∈ 푈(A푘) . We want to show that each coordinate of 푓(훼) is in 푋(A푘) . If 푋 were projective, this would follow from Corollary 7.3 of [Sto07]. But 푋 is not, so we must instead prove this using the étale homotopy obstruction of [HS13].
푛 f−ab 푛 Zℎ,1 We 푓(훼) ∈ 푋 (A푘) = 푋 (A푘) by Theorem 9.103 of [HS13]. It follows by Lemma Zℎ,1 푠 4.2.10 that every coordinate 푓푖(훼) (1 ≤ 푖 ≤ 푛) of 푓(훼) is in 푋(A푘) . As 푋 is connected, 푠 Zℎ,1 Zℎ and 퐻2(푋 ) = 0, we have 푋(A푘) = 푋(A푘) by Theorem 9.152 of [HS13]. Theorem 9.116 Zℎ Br Br of [HS13] tells us that 푋(A푘) = 푋(A푘) . Therefore, 푓푖(훼) ∈ 푋(A푘) . In particular, it pairs to 0 with every element of Br(푋).
Let 푆 be a finite set of places of 푘 large enough so that 푋 and 푓 are defined over 풪푘,푆, 1 and 푓(훼) is integral over 풪푘,푆. Let 풳 = (P ∖ {0, 1, ∞})풪푘,푆 . By forgetting the 푣-components 푆 푆 f−ab ∏︁ for 푣 ∈ 푆, we get a point 푓푖(훼) of 푋(A푘 ) ∩ 풳 (풪푣) for each 푖. The pairing of any 푣∈ /푆 푆 element of 퐵푆(푋) (in the notation of [HV10]) with 푓푖(훼) is the same as its pairing with 푆 푓푖(훼), for elements of 퐵푆(푋) pair trivially with every element of 푋(푘푣); in particular, 푓푖(훼) 푆 is orthogonal to all of 퐵푆(푋). But the conjecture implies that 푓푖(훼) comes from an element 푛 of 풳 (풪푘,푆). It follows that 푓(훼)푣 is in 푋 (푘) for every 푣∈ / 푆. Thus for any 푣∈ / 푆, we find
that 푓(훼)푣 ∈ 푓(푈(푘)).
Remark 5.4.4. It does not seem that Conjecture 2 of [HV10] implies Conjecture 5.4.2, for the former requires using only integral points, which in turn means that we must throw out a different set 푆 of primes depending on the point in question.
95 96 Chapter 6
The Fibration Method
In this chapter, we describe an alternative proof of a positive answer to Question 4.3.2 for (f − cov, 푓, 푘) that depends on the section conjecture but works over any number field 푘. The proof essentially uses the fibration method, due originally to M. Artin and first usedto prove many important properties of étale cohomology. We begin by reformulating the section conjecture in a form that works better in fibrations. We also prove some important auxiliary results on very strong approximation in fibration sequences (Proposition 6.2.10), which will be useful in Chapter 7, and on cuspidal sections for higher dimensional varieties (Theorem 6.4.6).
6.1 The Homotopy Section Conjecture
We will now reformulate the section conjecture in terms of homotopy fixed points before showing what happens to the section conjecture in fibration sequences in Section 6.2. There are three main reasons why homotopy fixed points work better than fundamental group
sections for use in fibration sequences: (1) If the base has a nontrivial 휋2, the ordinary section conjecture does not hold inductively in elementary fibrations (2) by working with homotopy types instead of fundamental groups, one need not deal with basepoints (3) the homotopy obstruction deals with connected components when the variety is not geometrically connected.
97 However, in most of our applications, the space is an étale 퐾(휋, 1), making (1) irrelevant. This begs the question why we did not use profinite groupoids instead of homotopy types, which would adequately deal with (2) while allowing us to dispose of Propositions 6.1.16 and 6.3.10. Aside from the natural inclination to prove results in the most general setup possible, we justify this choice by the fact that we also found it easier to cite references on homotopy fixed points of profinite spaces than of profinite groupoids. Unless stated otherwise, we let 푘 denote an arbitrary field. Let 풮 denote the category of simplicial sets.
Definition 6.1.1. The category 풮^ of profinite spaces is the category of simplicial objects in the category of profinite sets.
Definition 6.1.2. In Definition 4.4 of [Fri82], Friedlander defines the étale topological type 퐸푡(푋) of a scheme 푋, which is an object of Pro(풮) and represents the étale homotopy type of Artin-Mazur ([AM69]). This is functorial in the scheme 푋.
Definition 6.1.3. In Section 2.7 of [Qui08], Quick defines the profinite completion functor Pro(풮) → 풮^ and defines 퐸푡^ (푋) to be the profinite completion of 퐸푡(푋). The profinite completion of pro-spaces is defined by first taking profinite completion level-wise togeta pro-object in the category of profinite spaces and then taking the inverse limit in the category of profinite spaces.
The cohomology of this profinite space with finite coefficients recovers the étale cohomology of 푋, and the fundamental group of this space is the profinite étale fundamental group when 푋 is geometrically unibranch.
Definition 6.1.4. If 푋 is a 푘-scheme, the homotopy fixed points 퐸푡^ (푋푠)ℎ퐺푘 (Definition 2.22 of [Qui11]) are defined as follows. If 퐺 is a profinite group, one may use the bar construction
^ 푠 ℎ퐺푘 to define 퐸퐺 and 퐵퐺 as profinite spaces. Then 퐸푡(푋 ) is the space of 퐺푘-equivariant ^ 푠 maps from 퐸퐺푘 to 퐸푡(푋 ). (Compare with 9.3.2 of [HS13], 8.2 of [BS15], and introduction to [Pál15]). By the discussion immediately following Proposition 2.14 of [Qui08], this has the structure of a simplicial set. As is 퐸푡^ (푋푠), this is functorial in the 푘-scheme 푋.
98 ^ 푠 ℎ퐺푘 Definition 6.1.5. As in [HS13], [BS15], and [Pál15], we denote 휋0(퐸푡(푋 ) ) by 푋(ℎ푘) and refer to it as the homotopy sections of 푋/푘. This is again functorial in the 푘-scheme 푋.
′ ′ ^ 푠 ℎ퐺푘′ If 푘 /푘 is a field extension and 푋 a 푘-scheme, 푋(ℎ푘 ) still denotes 휋0(퐸푡(푋 ) ).
Definition 6.1.6. The homotopy profinite Kummer map 휅푋/푘 : 푋(푘) → 푋(ℎ푘) of Section 3.2 of [Qui11] is defined as follows. The set Spec 푘(ℎ푘) has one element, so by functoriality, an element of 푋(푘) gives rise to an element of 푋(ℎ푘). Compare with 9.3.2 of [HS13], 8.2 of
[BS15], and the introduction to [Pál15]. We write 휅푋 or even 휅 when 푘, 푋 are understood.
We also write 휅푆, 휅푋/A푘,푆 , and 휅푋/A푘 as in the case of Definition 5.3.4.
The section conjecture has been reformulated in homotopy-theoretic terms; see Section 3.2 of [Qui11] and Theorem 9.7(b) of [Pál15], or Section 2.6 of [Sti13] for a summary. However, these sources lack an analogue of cuspidal sections, without which one can only restrict to projective curves or conjecture something like the “homotopy section property” (HSP) of [Pál15], a local-to-global statement. We will fix this gap. In fact, HSP follows fromour versions of the section conjecture (although the former is expected to hold more generally), as proven below in Lemma 6.2.9. We now define the analogue of cuspidal sections:
Definition 6.1.7 (Analogue of Definition 5.3.9). A homotopy cuspidal datum (푋, 퐶, 푘) is a (smooth) variety 푋 over 푘 and a subset 퐶 ⊆ 푋(ℎ푘), called the set of cuspidal fixed points.
Remark 6.1.8. In practice (e.g. to reprove Corollary 5.3.27), we will only use cuspidal fixed points for varieties of dimension > 1 when they arise from good neighborhoods as in Definition 6.3.11. However, in [Cor17], we begin the study of a general notion of cuspidal sections for higher dimensional varieties similar to Definition 5.3.18.
Definition 6.1.9 (Analogue of Defintion 5.3.10). A homotopy cuspidal datum (푋, 퐶, 푘) is said to satisfy the surjectivity in the homotopy section conjecture if 푋(ℎ푘) ∖ 퐶 ⊆ 휅(푋(푘)).
Definition 6.1.10 (Analogue of Definition 5.3.13). A homotopy cuspidal datum (푋, 퐶, 푘) is said to satisfy the injectivity in the homotopy section conjecture if
1. The map 휅: 푋(푘) → 푋(ℎ푘) is injective.
99 2. The sets 휅(푋(푘)) and 퐶 are disjoint.
3. Every connected component of 퐸푡^ (푋푠)ℎ퐺푘 in the image of the Kummer map is simply connected.
6.1.1 Basepoints and Homotopy Groups
While the statements of the homotopical versions of the section conjecture avoid the use of basepoints, we sometimes need to use them. In particular, in order to discuss homotopy groups, we must discuss basepoints for 퐸푡^ (푋). On p.37 of [Fri82], it is stated that a geometric point 푎 of 푋 gives rise to a basepoint of ^ 퐸푡(푋), and this then gives rise to a basepoint of 퐸푡(푋), which we call 푏푎.
If this geometric point is a rational point, it is invariant under 퐺푘, so our basepoint of ^ 퐸푡(푋) is invariant under 퐺푘, i.e. we have a pointed 퐺푘-space. Sometimes 푋 does not have
any rational points, and yet we want a 퐺푘-invariant basepoint. This is provided by the following construction:
Construction 6.1.11. Let 푎 ∈ 푋(ℎ푘). This is a homotopy class of 퐺푘-equivariant maps ^ 푠 퐸퐺푘 → 퐸푡(푋 ). By taking a homotopy pushout of this map along the map from 퐸퐺푘 to ^ 푠 a point, we obtained a new model of 퐸푡(푋 ) as a profinite space with 퐺푘-action with a
퐺푘-invariant basepoint. Any such basepoint is denoted by 푏푎.
Definition 6.1.12. In Definition 2.15 of [Qui08] and Definition 2.12 ofQui11 [ ], Quick defines the homotopy groups of a pointed profinite space. If 푋 is a scheme, and 푎 is a geometric ^ 푒푡´ point or a homotopy section of 푋, we denote 휋푖(퐸푡(푋), 푏푎) by 휋푖 (푋, 푎) or simply 휋푖(푋, 푎) (c.f. Notation and Conventions). If 푎 is a Galois-invariant geometric point or a homotopy
section of a scheme 푋/푘, this basepoint is fixed by the action of 퐺푘, so there is a natural 푠 action of 퐺푘 on 휋푖(푋 , 푎).
Remark 6.1.13. Suppose that 푋 is geometrically unibranch (e.g., normal), Noetherian, and connected. By Proposition 2.33 of [Qui08], the homotopy groups of 퐸푡^ (푋) are isomorphic (as pro-groups) to the pro-homotopy groups of the profinite completion of the Artin-Mazur
100 homotopy type of 푋. By Theorem 11.2 of [AM69], this profinite completion is already isomorphic to the (non-completed) Artin-Mazur homotopy type as a pro-homotopy type. But 퐸푡(푋) represents the Artin-Mazur homotopy type (c.f. Corollary 6.3 of [Fri82]), so its homotopy groups are isomorphic to all of the above. More generally, if 푋 is not assumed to be connected (but still Noetherian), then it is a finite disjoint union of connected schemes. Thus the same result holds, as all constructions behave well with respect to finite coproducts.
^ Remark 6.1.14. If 푋 is connected, then 휋0(퐸푡(푋)) contains one element, so the homotopy groups are independent of the choice of basepoint up to isomorphism. This isomorphism is, however, only unique modulo the action of the fundamental group. This is also a subtlety in using different models for 퐸푡^ (푋푠), as in Construction 6.1.11. Furthermore, any two basepoints ^ 푠 of 퐸푡(푋 ) arising from the same element of 푋(ℎ푘) are homotopic in the 퐺푘-equivariant 푠 category, so the resulting actions on 휋푖(푋 ) are equivalent. This latter fact is important in the statement of Theorem 6.1.18. The reader may check that this is does not pose a problem in any of the theorems we prove.
Definition 6.1.15. A 푘-variety 푋 has the étale 퐾(휋, 1)-property or is an étale 퐾(휋, 1) if
푠 푋 is geometrically connected, and 휋푖(푋 , 푎) = 0 for 푖 ≥ 2 and some (equivalently, any) geometric point 푎.
Proposition 6.1.16. Let 푋 be a geometrically connected variety over 푘. Then there is a map Trunc1 : 푋(ℎ푘) → S휋1(푋/푘) (functorial in 푋) whose composition with the homotopy profinite Kummer map is the ordinary profinite Kummer map. Furthermore, if 푋 is an étale 퐾(휋, 1), the following are true:
∙ The map Trunc1 is a bijection
∙ A cuspidal datum (푋, 퐶, 푘) is the same as a homotopy cuspidal datum (푋, 퐶, 푘)
∙ 푋 satisfies the surjectivity in the section conjecture (5.3.10) if and only if itsatisfies the surjectivity in the homotopy section conjecture (6.1.9)
101 ∙ 푋 satisfies the injectivity in the section conjecture (5.3.13) if and only if it satisfies the injectivity in the homotopy section conjecture (6.1.10)
Remark 6.1.17. Related facts are discussed in Section 3.2 of [Qui11], Proposition 12.6(b) of [Pál15], and Section 2.6 of [Sti13], especially Equation (2.13).
Before proving Proposition 6.1.16, we must introduce the descent spectral sequence for homotopy fixed points. It is essentially Theorem 2.16Qui11 of[ ] for the pointed profinite ^ 푠 퐺푘-space 퐸푡(푋 ):
Theorem 6.1.18 (Theorem 2.16 of [Qui11]). For each 푎 ∈ 푋(ℎ푘), there is a spectral sequence
푝,푞 푝 푠 ^ 푠 ℎ퐺푘 퐸2 = 퐻 (퐺푘; 휋푞(푋 , 푎)) ⇒ 휋푞−푝(퐸푡(푋 ) , 푎), known as the descent spectral sequence. The basepoint 푎 of 퐸푡^ (푋푠)ℎ퐺푘 naturally results
푠 from the 퐺푘-invariant basepoint 푏푎 of Construction 6.1.11, and the 퐺푘-action on 휋푞(푋 , 푎) is well-defined by Remark 6.1.14.
Proof of Proposition 6.1.16. If 푋(ℎ푘) is empty, then the map Trunc1 automatically exists,
and it is functorial because there is a unique map from an empty set. It follows that Trunc1 is automatically a bijection. In this case, 푋(푘) and 퐶 are empty, so the surjectivity and injectivity in either version of the section conjecture always hold.
We now suppose that there is a homotopy fixed point 푎 ∈ 푋(ℎ푘). As 휋0(푋푠, 푎) and hence 0 푠 퐻 (퐺푘; 휋0(푋 , 푎)) is trivial, the spectral sequence produces a map 푋(ℎ푘) → S휋1(푋/푘). When 푋 = Spec 푘, this map is clearly an isomorphism. Functoriality of the spectral sequence and the definition of the Kummer map imply that for arbitrary 푋, the map
푋(ℎ푘) → S휋1(푋/푘) respects the Kummer map. 푠 We now suppose that 푋 is an étale 퐾(휋, 1). As 휋푖(푋 ) contains one element for 푖 ̸= 1, the
^ 푠 ℎ퐺푘 ∼ 푝 푠 spectral sequence tells us that 휋1−푝(퐸푡(푋 ) , 푎) = 퐻 (퐺푘; 휋1(푋 , 푎)). For 푝 = 1, this tells
us that Trunc1 is a bijection. (Alternatively, we could avoid the spectral sequence by noting that an étale 퐾(휋, 1) is the classifying space of its fundamental group and then applying Proposition 2.9 of [Qui15].)
102 The second bullet-point of the proposition follows from the first. The third bullet-point follows from the compatibility of the Kummer maps, as do the first two parts of injectivity (the fourth bullet-point). For third part of injectivity, we suppose that 푎 is in the image of the Kummer map.
^ 푠 ℎ퐺푘 ∼ 0 푠 Then the descent spectral sequence tells us that 휋1(퐸푡(푋 ) , 푎) = 퐻 (퐺푘; 휋1(푋 , 푎)). The 푠 0 푠 group 휋1(푋 , 푎) has 퐺푘-action arising from the section 푎, so 퐻 (퐺푘; 휋1(푋 , 푎)) is simply the
centralizer of Trunc1(푎).
Proposition 6.1.19. Let 푋 be a finite scheme over 푘. Suppose that 푋red is geometrically reduced (e.g., 푘 is perfect). Then 푋 satisfies the injectivity and surjectivity in the homotopy section conjecture.
Proof. We first suppose that 푋 is reduced. Such a scheme must be a disjoint union ⨿푖 Spec(푘푖),
푠 [푘푖:푘] where 푘푖/푘 is a finite separable extension. Then 푋 = ⨿푖 ⨿푗=1 Spec(푘푠), and 퐺푘 acts
[푘푖:푘] 푠 퐺푘 transitively on ⨿푗=1 Spec(푘푠) for each 푖. Thus 휋0(푋 ) is the set of all 푖 for which 푘푖 = 푘, 푠 which is the same as 푋(푘). But 휋푛(푋 ) is trivial for 푛 ≥ 1, so by Theorem 6.1.18, this is also 푋(ℎ푘). We have 푋(푘) = 푋(ℎ푘), and it is easy to check that this equality comes from the Kummer map, so we are done in this case.
We now reduce to the reduced case by considering the map 푋red → 푋. This map
is a universal homeomorphism, and it remains so after base change to 푘푠, so the map 푠 푠 ^ 푠 ^ 푠 푋red → 푋 induces an equivalence of étale sites. It follows that 퐸푡(푋red) → 퐸푡(푋 ) is an
equivalence, so the induced map 푋푟푒푑(ℎ푘) → 푋(ℎ푘) is an isomorphism. Furthermore, the
map 푋red(푘) → 푋(푘) is an isomorphism, so we are done.
6.1.2 Obstructions to the Hasse Principle
Proposition 6.1.20 (Analogue of Proposition 5.3.15). If 푘′/푘 is an extension of fields of characteristic 0, there is a base change map
푋(ℎ푘) → 푋(ℎ푘′)
103 Furthermore, the diagram 푋(푘) −−−→ 푋(푘′) ⎮ ⎮ 휅푋/푘⎮ ⎮휅푋/푘′ ⌄ ⌄ 푋(ℎ푘) −−−→ 푋(ℎ푘′) is commutative.
Proof. Proposition 5.4 in [Pál15] says that algebraically closed extension of base field in characteristic 0 does not change the étale homotopy type. The result is then clear because a map of groups gives a map on homotopy fixed points.
^ 푠 Let 푋 be a scheme over a field 푘. Then 퐸푡(푋 ) can be represented as a pro-space {푋훼}훼 ^ 푠 with 퐺푘-action, with each 푋훼 휋-finite (c.f. the proof of Lemma 6.2.4). As 퐸푡(푋 ) = holim푋훼, 훼 ℎ퐺푘 and homotopy limits commute with homotopy fixed points, we have 푋(ℎ푘) = 휋0(holim푋훼 ). 훼 We now recall that for a number field 푘 there is an étale homotopy obstruction to rational
ℎ points defined on p.314HS13 of[ ], denoted by 푋(A푘) . These are defined using another version of homotopy fixed points ([HS13], Definition 3.3), which we denote by 푋(ℎ푘)퐻푆. For
ℎ퐺푘 a pro-space {푋훼} with 퐺푘-action, it is given by taking lim 휋0(푋훼 ) (while [HS13] uses the 훼 ℎ퐺푘 푋훼 of [Goe95], it is easy to see that this is the same as that of [Qui11] by considering the
descent spectral sequence). There is a natural map 푋(ℎ푘) → 푋(ℎ푘)퐻푆, giving us a diagram
푋(푘) / 푋(A푘) 휅 휅 푋/푘 푋/A푘 loc ∏︀ 푋(ℎ푘) / 푣 푋(ℎ푘푣)
loc ∏︀ 푋(ℎ푘)퐻푆 / 푣 푋(ℎ푘푣)퐻푆, ℎ Then 푋(A푘) is the subset of the upper-right object of this diagram whose image in the lower-right object is contained in the image of the bottom horizontal arrow.
Lemma 6.1.21. The map 푋(ℎ푘) → 푋(ℎ푘)퐻푆 is surjective, and the map 푋(ℎ푘푣) →
푋(ℎ푘푣)퐻푆 is an isomorphism for each 푣.
^ 푠 Proof. As 푋 is defined over a number field, we can express 퐸푡(푋 ) as a pro-space {푋훼},
104 where 훼 ranges over a countable cofiltering category. It follows that we may replace itbya tower, so we may apply Proposition VI.2.15 of [GJ09] to get an exact sequence
1 ℎ퐺푘 * → lim 휋1푋훼 → 푋(ℎ푘) → 푋(ℎ푘)퐻푆 → * 훼
This implies the desired surjectivity.
Replacing 푘 by 푘푣, we note that we can assume (by Theorem 11.2 of [AM69]) that each
푋훼 has finite homotopy groups. As 퐺푘푣 is topologically finitely generated, this implies by
ℎ퐺푘 1 Theorem 6.1.18 that 휋1푋훼 is finite for each 훼. But this implies that our lim vanishes, so 훼 we get the desired isomorphism.
ℎ −1 Proposition 6.1.22. In the notation of the previous diagram, 푋( 푘) is 휅 (Im(loc)). A 푋/A푘
ℎ Proof. This follows immediately from Lemma 6.1.21 and the definition of 푋(A푘) .
Proposition 6.1.23 (Analogue of Proposition 5.3.26). Let 푘 be a number field, 푆 a nonempty set of places of 푘, and 푋 a smooth 푘-variety. If 푋 satisfies the surjectivity in the homotopy section conjecture over 푘 and the injectivity in the homotopy section conjecture over 푘푣 for all 푣 ∈ 푆, then it is VSA for (ℎ, 푆, 푘). If 푋 is geometrically connected, then the same is true for (´et − Br, 푆, 푘).
ℎ ′ ℎ Proof. Suppose 훼 ∈ 푋(A푘,푆) is the projection of 훼 ∈ 푋(A푘) . By the same argument as in Proposition 5.3.26, we find that 훼′ comes from a rational point of 푋 at every 푣 ∈ 푆, i.e. 훼 is a rational point. The last statement follows from Theorem 9.1 of [HS13].
6.2 Homotopy Section Conjecture in Fibrations
In this section, “Kummer map,” “cuspidal datum,” and “section conjecture ” will mean the homotopy versions as explained in Section 6.1. Of course, for an étale 퐾(휋, 1), Proposition 6.1.16 says that these are equivalent to the corresponding definitions in Section 5.3.1.
105 Definition 6.2.1 (Definition 1.1 of [Fri82]). A special geometric fibration is a morphism 푓 : 푋 → 푆 of schemes fitting into a diagram:
푗 푋 / 푋 o ? _푌 푖 푓 푓 푔 푆
satisfying the following conditions:
1. 푖 is a closed embedding
2. 푗 is a open immersion which is dense in every fiber of 푓, and 푋 = 푋 − 푌
3. 푓¯ is smooth and proper
4. 푌 is a union of schemes 푌1, ··· , 푌푚, with 푌푖 of pure relative codimension 푐푖 in 푋 over
푆, with the property that every intersection 푌푖1 ∩ · · · ∩ 푌푖푘 is smooth over 푆 of pure
codimension 푐푖1 + ··· + 푐푖푘 .
More generally, a geometric fibration is a map 푓 : 푋 → 푆 admitting a Zariski covering
{푉푖 → 푆} such that 푓푉푖 : 푋 ×푆 푉푖 → 푉푖 is a special geometric fibration for all 푖.
Lemma 6.2.2. Let 푓 : 푋 → 푆 be a geometric fibration and 푈 → 푆 a morphism of schemes. Then the base change
푓푈 : 푋 ×푆 푈 → 푈 is a geometric fibration.
Proof. This is clear because all of the properties in Definition 6.2.1 are stable under base extension.
Definition 6.2.3. Let 푘 be a field and 푆 a 푘-scheme. Let 푓 : 푋 → 푆 be a geometric fibration.
Suppose we are given cuspidal data (푆, 퐶푆, 푘) for 푆 and (푋푠, 퐶푋푠 , 푘) for the fiber 푋푠 above
every 푠 ∈ 푆(푘). Then we define the cuspidal datum (푋, 퐶푋,푓 , 푘) induced by 푓 as follows.
106 For 푠 ∈ 푆(푘), let 푋푠 denote the fiber of 푓 above 푠 and 휄푠 : 푋푠 → 푋 the inclusion. The map
푓 induces a map 푓 : 푋(ℎ푘) → 푆(ℎ푘), and the map 휄푠 induces a map 휄푠 : 푋푠(ℎ푘) → 푋(ℎ푘).
We define 퐶푋,푓 by
⎛ ⎞ −1 ⋃︁ ⋃︁ 퐶푋,푓 = 푓 (퐶푆) ⎝ 휄푠(퐶푋푠 )⎠ ⊆ 푋(ℎ푘). 푠∈푆(푘)
In particular, if 퐶푆 and 퐶푋푠 are all empty, which is typical for smooth proper curves, then
퐶푋,푓 is empty.
Lemma 6.2.4. Let 푓 : 푋 → 푆 be a map of 푘-schemes. Suppose that 푓 : 푋 → 푆 is a geometric fibration of Noetherian normal schemes with connected geometric fibers, and 푘 has characteristic 0. Then the sequence
^ 푠 ^ 푠 ^ 푠 퐸푡(푋푠 ) → 퐸푡(푋 ) → 퐸푡(푆 ) is a fibration sequence of profinite etale homotopy types (in the model structure of [Qui08]).
Proof. It follows by Theorem 11.5 of [Fri82] (also c.f. Theorem 3.7 of [Fri73]) that the ^ ^ ^ sequence 퐸푡(푋푠) → 퐸푡(푋) → 퐸푡(푆) is a fibration sequence, or equivalently that ^ ^ 퐸푡(푋푠) / 퐸푡(푋)
* / 퐸푡^ (푆) is a homotopy pullback diagram. We need to show that the profinite completion of this homotopy pullback diagram remains a homotopy pullback diagram. We may assume this is a diagram of fibrant cofibrant objects in Pro(풮), with the Isaksen model structure. By Theorem 11.2 of [AM69], each is isomorphic to a levelwise 휋-finite space (after possibly replacing each by its Postnikov tower, which is invariant under Quick’s
profinite completion). A 휋-finite space is fibrant in 퐿퐾휋 Pro(풮), in the notation of Chapter 7 of [BHH17], by Proposition 7.2.10 of loc.cit. As a limit of fibrant objects, it is itself fibrant,
i.e., this is a diagram of objects fibrant for 퐿퐾휋 Pro(풮). In particular, this means that it
107 is in the image of the inclusion (퐿퐾휋 Pro(풮))∞ ˓→ Pro(풮)∞ of infinity categories, which is induced by the identity functor on model categories. This inclusion of infinity categories is conservative and preserves limits, so it detects limits. In particular, the above diagram is a
pullback diagram in (퐿퐾휋 Pro(푆))∞, or equivalently, a homotopy pullback diagram in the
model category 퐿퐾휋 Pro(푆). Therefore, by Theorem 7.4.8 of [BHH17], applying Ψ퐾휋 to this
diagram yields a homotopy pullback diagram in Pro(푆휏 ), which by Proposition 7.4.1 of the same paper is the category 푆^ of profinite spaces, with Quick’s model structure.
We are done if we can identify the map Ψ = Ψ퐾휋 (these are the same functor on ordinary categories, just refer to different model structures) with the profinite completion functor of Definition 6.1.3. There is an adjunction Ψ: Pro 풮 ↽−⇀− 푆^ :Φ as well as an adjunction const: 풮 ↽−⇀− Pro(풮) :lim. By construction, lim ∘Φ takes a profinite space 푋 to an object of
Pro(풮휏 ) whose limit is 푋, then takes the limit of this pro-object as a pro-space. In particular, this is the forgetful functor from profinite spaces to simplicial sets of p.587 of[Qui08], and therefore Ψ ∘ const is Quick’s profinite completion from 풮 to 풮^. It remains to show that Ψ can be obtained by applying Ψ levelwise and taking the
limit. We therefore wish to show that Ψ preserves filtered limits. For this, let {푋푗}
be a filtered system in Pro(풮). For 푇 an object of 풮휏 , we have Hom(lim Ψ(푋푗), 푇 ) = 푗
colim푗 Hom(Ψ(푋푗), 푇 ) because 푇 is cocompact. But this latter is colim푗 Hom(푋푗, Φ(푇 )), which in turn is Hom(lim 푋푗, Φ(푇 )) = Hom(Ψ(lim 푋푗), 푇 ) because 휑(푇 ) is in 풮 and therefore 푗 푗
cocompact. As 풮휏 cogenerates Pro(풮휏 ), this shows that lim Ψ(푋푗) = Ψ(lim 푋푗) in Pro(풮휏 ), 푗 푗 and we are done.
Lemma 6.2.5. Let 푓 : 푋 → 푆 be a map of 푘-schemes. Let 푎 ∈ 푋(ℎ푘) and 푠 ∈ 푆(푘) such that
푓(푎) = 휅(푠). Let 푋푠 be the fiber of 푓 above 푠 (a geometrically connected variety). Suppose that ^ 푠 ^ 푠 ^ 푠 the sequence 퐸푡(푋푠 ) → 퐸푡(푋 ) → 퐸푡(푆 ) is a fibration sequence of profinite etale homotopy types (e.g., by Lemma 6.2.2v and Lemma 6.2.4, if 푓 is a geometric fibration of Noetherian normal schemes).
Then 푎 comes from an element of 푋푠(ℎ푘), also denoted by 푎, and there is a long exact
108 sequence
^ 푠 ℎ퐺푘 ^ 푠 ℎ퐺푘 ^ 푠 ℎ퐺푘 휋1(퐸푡(푋푠 ) , 푎) → 휋1(퐸푡(푋 ) , 푎) → 휋1(퐸푡(푆 ) , 휅(푠)) → 푋푠(ℎ푘) → 푋(ℎ푘) → 푆(ℎ푘), where the latter three are pointed sets based at 푎 and 휅(푠).
^ 푠 ℎ퐺푘 Proof. As homotopy fixed points commutes with homotopy limits, the sequence 퐸푡(푋푠 ) → 퐸푡^ (푋푠)ℎ퐺푘 → 퐸푡^ (푆푠)ℎ퐺푘 is a fibration sequence of simplicial sets. This implies that we get
an element of 푋푠(ℎ푘) mapping to 푎.
^ 푠 ℎ퐺푘 We then choose a basepoint 푏푎 of 퐸푡(푋푠 ) in the connected component represented
^ 푠 ℎ퐺푘 ^ 푠 ℎ퐺푘 ^ 푠 ℎ퐺푘 by 푎 ∈ 푋푠(ℎ푘), which makes 퐸푡(푋푠 ) → 퐸푡(푋 ) → 퐸푡(푆 ) a fibration sequence of pointed simplicial sets. The exact sequence then follows by the long exact sequence of homotopy groups for a pointed fibration.
Theorem 6.2.6. Let 푓 : 푋 → 푆 satisfy the hypotheses of Lemma 6.2.5. If 푆 and the fibers of 푓 above 푘-points of 푆 satisfy the injectivity in the section conjecture, then so does 푋.
Proof. We refer to 1, 2, and 3 of Definition 6.1.10.
1 for 푋: Let 푎, 푏 ∈ 푋(푘), and suppose 휅푋 (푎) = 휅푋 (푏). This implies 푓(휅(푎)) = 푓(휅(푏)), so
by 1 for 푆, 푓(푎) = 푓(푏). Let 푠 = 푓(푎). Then 푎, 푏 are both in 푋푠(푘). By Lemma 6.2.5 and 3
for 푆, the map 휄푠 : 푋푠(ℎ푘) → 푋(ℎ푘) is injective. Thus 휅푋푠 (푎) = 휅푋푠 (푏), so 1 for 푋푠 tells us that 푎 = 푏.
2 for 푋: Let 푎 ∈ 푋(푘) and 푠 = 푓(푎). Then 푓(휅푋 (푎)) = 휅푆(푠), so it is not in 퐶푆 by 2 for 푆, −1 hence not in 푓 (퐶푆). As in the previous paragraph, the map 휄푠 : 푋푠(ℎ푘) → 푋(ℎ푘) is injective. ′ As 푎 ∈ 푋푠(푘), 휅푋푠 (푎) ∈/ 퐶푋푠 by 2 for 푋푠, so 휅푋 (푎) ∈/ 휄푠(퐶푋푠 ). Suppose 푠 ∈ 푆(푘) ∖ {푠}. Then ′ ′ ′ ′ any element of 휄푠 (퐹푠 (ℎ푘)), in particular any element of 휄푠 (퐶퐹푠′ ), maps to 휅푆(푠 ) under 푓. ′ ′ By 1 for 푆, 휅푆(푠 ) ̸= 휅푆(푠), so 휅푋 (푎) ∈/ 휄푠 (퐶퐹푠′ ). Together, this implies that 휅(푎) ∈/ 퐶푋,푓 .
^ 푠 ℎ퐺푘 3 for 푋: Let 푎 ∈ 푋(푘) and 푠 = 푓(푎), so 푠 ∈ 푋푠(푘). We choose a basepoint of 퐸푡(푋푠 )
in the connected component 휅푋푠 (푎). Then 3 for 푋푠 and 푆 and Lemma 6.2.5 tell us that
^ 푠 ℎ퐺푘 휋1(퐸푡(푋 ) ) is trivial on the connected component 휅푋 (푎).
109 Theorem 6.2.7. Let 푓 : 푋 → 푆 satisfy the hypotheses of Lemma 6.2.5. If 푆 and the fibers of 푓 above 푘-points of 푆 satisfy the surjectivity in the section conjecture, then so does 푋.
Proof. Let 푎 ∈ 푋(ℎ푘) ∖ 퐶푋,푓 . Then 푓(푎) ∈/ 퐶푆, so 푓(푎) = 휅(푠) for 푠 ∈ 푆(푘). Then Lemma
6.2.5 gives us a long exact sequence. Thus 푎 comes from 푋푠(ℎ푘) and is not in 퐶푋푠 . By the
surjectivity in the section conjecture for 푋푠, it comes from a 푘-point of 푋푠.
We finally prove a general result about VSA for the étale homotopy obstruction thatwill be useful in Chapter 7. Before proving it, we need one definition.
Definition 6.2.8 (Notation 12 of [Pál15]). If 푋 is a scheme over a number field 푘, the Selmer
set Sel(푋/푘) is the subset of 푋(ℎ푘) whose image under loc is in the image of 푋(A푘) in the following diagram:
푋(푘) −−−→ 푋(A푘) ⎮ ⎮ ⎮ ⎮ ⌄ ⌄ loc ∏︁ 푋(ℎ푘) −−−→ 푋(ℎ푘푣), 푣 Clearly 휅(푋(푘)) ⊆ Sel(푋/푘).
Lemma 6.2.9. Suppose 푋 is a variety over a number field 푘 that satisfies the surjectivity in the section conjecture over 푘 and the injectivity in the section conjecture over 푘푣 for some place 푣 of 푘. Then 휅(푋(푘)) = Sel(푋/푘).
Proof. As mentioned in Definition 6.2.8, 휅(푋(푘)) ⊆ Sel(푋/푘). Now suppose 푠 ∈ Sel(푋/푘).
It suffices to show that 푠 is not cuspidal. But if it were cuspidal, its base change to 푘푣 would
be cuspidal, contradicting injectivity over 푘푣, so it is not.
Theorem 6.2.10. Let 푓 : 푋 → 푆 be a geometric fibration of smooth varieties over a number field 푘 with connected geometric fibers, let 푇 be a nonempty set of places of 푘. Suppose that the follows conditions hold:
1. 푆 is VSA for (ℎ, 푇, 푘).
110 2. One of the following is true:
(a) 푆 satisfies the surjectivity in the section conjecture over 푘 and the injectivity in
the section conjecture over 푘푣 for some 푣.
(b) The image of 푆(푘) in 푆(ℎ푘) under 휅 is Sel(푆/푘), and 푆 satisfies the injectivity in
the section conjecture over 푘푣 for some 푣 ∈ 푇 . ∏︁ (c) The localization map loc푇 : 푆(ℎ푘) → 푆(ℎ푘푣) is injective on Sel(푆/푘). 푣∈푇
3. For all finite 푣∈ / 푇 , 휅푆/푘푣 is injective (vacuous if 푇 = 푓).
푠 ℎ퐺 4. For all finite 푣, every connected component of 퐸푡^ (푆 ) 푘푣 in the image of the Kummer map is simply connected (e.g., by Proposition 104 of [Sti13] and Proposition 6.1.16, 푆
1 is a smooth geometrically connected curve not isomorphic to P ).
5. For every real place 푣 of 푘, every 푎 ∈ 푆(푘), and every 푏 ∈ 푋푎(푘푣), the map 휋1(푋(푘푣), 푏) →
휋1(푆(푘푣), 푎) is surjective and 휋1(푋푎((푘푣)푠)) is trivial (where 휋1 denotes the topological fundamental group under the 푣-adic topology).
6. For every 푎 ∈ 푆(푘), the fiber 푋푎 of 푓 above 푎 is VSA for (ℎ, 푇, 푘).
Then 푋 is VSA for (ℎ, 푇, 푘). One may replace (ℎ, 푇, 푘) by (´et − Br, 푇, 푘) whenever the variety in question is smooth and geometrically connected.
Proof. We first note that 2a implies 2b, which implies 2c. The first implication isLemma
6.2.9. For the second implication, suppose 훼, 훽 ∈ Sel 푆/푘 with loc푣(훼) = loc푣(훽). Then there
are 푎, 푏 ∈ 푆(푘) such that 훼 = 휅(푎) and 훽 = 휅(푏). Therefore 푎, 푏 ∈ 푆(푘푣) have the same image
under 휅푋/푘푣 , so injectivity tells us that 푎 = 푏. This implies that 훼 = 훽, so 2c holds. ℎ ′ Now let 훼 ∈ 푋(A푘) project to 훼 ∈ 푋(A푘,푆), and let 훽 ∈ 푋(ℎ푘) such that loc(훽) = 휅(훼). ℎ Then 휅(푓(훼)) = loc(푓(훽)), so 푓(훼) ∈ 푆(A푘) . By 1, there is 푎 ∈ 푆(푘) such that 푎 = 푓(훼)푣 for all 푣 ∈ 푇 .
111 For 푣 ∈ 푇 , we have loc푣(푓(훽)) = 휅푆/푘푣 (푓(훼)푣) = 휅푆/푘푣 (푎) = loc푣(휅푆/푘(푎)). Thus
loc푇 (푓(훽)) = loc푇 (휅푆/푘(푎)). It’s clear that 푓(훽), 휅푆/푘(푎) ∈ Sel(푆/푘), so 2c implies that
푓(훽) = 휅푆/푘(푎).
This implies by Lemma 6.2.5 that there exists 훾 ∈ 푋푎(ℎ푘) such that 푖푎(훾) = 훽, where 푖푎
denotes the inclusion 푋푎 ˓→ 푋.
We now know that 휅푆/푘푣 (푓(훼)푣) = loc푣(훽) = 휅푆/푘푣 (푎) for all 푣. By 3, we know that
푎 = 푓(훼)푣 for all finite 푣. For 푣 infinite, Theorem 1.2Pál15 of[ ] implies that 푎 and 푓(훼)푣 are
in the same connected component of 푆(푘푣). As 푓 is an elementary fibration and therefore a
fiber bundle for the 푣-adic topology, we can modify 훼 so that 푓(훼)푣 = 푎 and 휅(훼) remains the same. We do this at all infinite places, so that 푓(훼) = 푎, yet still 휅(훼) = loc(훽).
We now have 훼 ∈ 푋푎(A푘) such that 푖푎(loc(훾)) = 푖푎(휅푋푎/A푘 (훼)). We need to show that
푖푎 : 푋푎(ℎ푘푣) → 푋(ℎ푘푣) is injective for all 푣. By Theorem 6.2.5, 4 implies injectivity for all finite 푣. At all complex 푣, the geometric
connectedness assumption on 푋푎 implies that 푋푎(ℎ푘푣) is a point, so injectivity follows.
Let 푣 be a real place. By Theorem 1.2 of [Pál10], the map 푋푎(푘푣)∙ → 푋푎(ℎ푘푣) is a
bijection, and by Theorem 1.2 of [Pál15], the map 푋(푘푣)∙ → 푋(ℎ푘푣) is injective. It therefore
suffices to prove that 푋푎(푘푣)∙ → 푋(푘푣)∙ is injective. But 푋푎(푘푣) → 푋(푘푣) → 푆(푘푣) is a fibration sequence of real analytic varieties, so the condition on topological fundamental groups guarantees the desired injectivity.
We conclude that loc(훾) = 휅푋푎/A푘 (훼). We conclude by 6 that there is 푐 ∈ 푋푎(푘) such that 푐 = 훼′.
6.3 Elementary Fibrations
This section gives us a tool for finding open subsets of arbitrary smooth geometrically connected varieties that satisfy the section conjecture. The following definition is similar to one from [sga73].
112 Definition 6.3.1 (Definition 11.4 of [Fri82]). An elementary fibration is a geometric fibration whose fibers are geometrically connected affine curves.
Remark 6.3.2. Assuming 푓 is a special geometric fibration, 푌 defines a relatively ample divisor of 푋, which means that 푓 is automatically projective.
Remark 6.3.3. In Definition 6.2.3, if 푓 : 푋 → 푆 is an elementary fibration, it is understood
unless mentioned otherwise that 퐶푋푠 is taken as in Definition 5.3.18.
Definition 6.3.4. In the notation of the previous definition, if the geometric fibers of 푓 are hyperbolic (Definition 5.3.22), we say that 푓 is a hyperbolic elementary fibration.
Lemma 6.3.5. Let 푘 be a perfect field. Then a map 푋 → Spec 푘 is a (hyperbolic) elementary fibration if and only if 푋 is a (hyperbolic) smooth geometrically irreducible curve.
Proof. It is clear that if 푋 → Spec 푘 is a (hyperbolic) elementary fibration then 푋 is a (hyperbolic) smooth geometrically irreducible curve. To show the converse, note that every smooth curve 푋/푘 has a unique smooth compactification 푋¯. We take 푓¯ : 푋¯ → Spec 푘 and 푌 := 푋¯∖푋 and note that 푌 is smooth and finite and thus étale over Spec 푘.
Lemma 6.3.6. Let 푓 : 푋 → 푆 be a (hyperbolic) elementary fibration and 푈 → 푆 a map of schemes. Then the base change
푓푈 : 푋 ×푆 푈 → 푈 is a (hyperbolic) elementary fibration.
Proof. This is an immediate corollary of Lemma 6.2.2 and the fact that every geometric fiber
of 푓푈 is also a geometric fiber of 푓.
Lemma 6.3.7. Let 푋, 푆 be two smooth geometrically connected varieties over an infinite field 푘, and let 푓 : 푋 → 푆 be an elementary fibration over 푘. Let 푥 be a closed point of 푋. Then there exist nonempty open subsets 푈푋 ⊂ 푋, 푈푆 ⊂ 푆, with 푥 ∈ 푈푋 , such that 푓(푈푋 ) ⊂ 푈푆, and the restriction 푓|푈푋 : 푈푋 → 푈푆 is an hyperbolic elementary fibration.
113 Proof. The problem is local on 푆, so we may assume that 푆 is quasi-projective and that 푓 is a special geometric fibration with compactification 푓 : 푋 → 푆. By Remark 6.3.2, 푓 is projective, so 푋 is quasi-projective by EGA II.5.3.4(ii) ([DG67]), meaning we can embed 푋
푁 into P푘 for some 푁 ∈ N. Let 퐹 denote the fiber of 푓 over 푓(푥). We need to find a hyperplane section 푌 ′ of 푋 that is smooth, intersects 퐹 transversely (i.e., with smooth scheme-theoretic intersection), and does not intersect 푌 ∩ 퐹 or 푥. The last condition defines a subscheme of the space of hyperplanes containing a dense open subset, and Bertini’s Theorem (Théorème 6.3 of [Jou83]) implies that the first two conditions do as well. Since 푘 is infinite, such a hyperplane exists over 푘, providing our 푌 ′. It follows that there is a neighborhood of 푓(푥) in 푆 over which the map 푌 ∪ 푌 ′ → 푆 is étale. Since 푌 ′ is closed in 푋, its projection to 푆 is proper, so this map is finite étale. We now replace 푌 by 푌 ∪ 푌 ′. Either the fibers of 푓 are now hyperbolic, in which case we are done, or the fibers have two punctures, in which case we repeat our procedure to finally obtain a hyperbolic elementary fibration around 푥.
Lemma 6.3.8. Let 푥 be a closed point of a smooth geometrically connected variety 푋 over an infinite field 푘. Then there is an open subset 푈 of 푋 containing 푥 and an elementary fibration 푓 : 푈 → 푆 for some smooth geometrically connected variety 푆 over 푘.
Proof. The proof of Lemma 6.3 in [SS16] explains how to extend the argument of [sga73] to the case of a closed point over an infinite field.
6.3.1 Good Neighborhoods
Definition 6.3.9. An 푆-scheme 푋 is called a (hyperbolic) good neighborhood if there exists a sequence of 푆-schemes
푋 = 푋푛, ..., 푋0 = 푆
and (hyperbolic) elementary fibrations
푓푖 : 푋푖 → 푋푖−1, 푖 = 1, ..., 푛.
114 Following [Hos14], we refer to such a sequence of 푆-schemes as a sequence of parametrizing morphisms.
Proposition 6.3.10. Let 푓 : 푋 → 푆 be an elementary fibration over a field of characteristic 0. If 푆 is an étale 퐾(휋, 1), then so is 푋. In particular, a good neighborhood is an étale 퐾(휋, 1). Under this condition, the results of Section 6.2 hold for the classical section conjecture.
Proof. Lemma 2.7(a) of [SS16] implies that the geometric fibers are étale 퐾(휋, 1)’s. The result then follows from the long exact sequence of Theorem 11.5 of [Fri82], noting that Remark 6.1.13 implies that different notions of étale homotopy group coincide for normal schemes.
Definition 6.3.11. Let 푓 : 푋 → 푆 be a good neighborhood with a sequence of parametrizing morphisms. Suppose we are given a cuspidal datum (푆, 퐶, 푘) for 푆. Then we define the cuspidal datum (푋, 퐶푋,푓 , 푘) induced by 푓 and the sequence of parametrizing morphisms to be the cuspidal datum induced successively by the elementary fibrations.
Remark 6.3.12. By an abuse of notation, we do not include the sequence of parametrizing
morphisms in the notation for 퐶푋,푓 , but we believe this should not lead to too much confusion.
In Conjecture 6.4.1, we conjecture that 퐶푋,푓 does not depend on the sequence of parametrizing morphisms.
Corollary 6.3.13. Let 푋 be a hyperbolic good neighborhood over a sub-푝-adic field 푘. Then
(푋, 퐶푋,푓 , 푘) satisfies the injectivity in the (homotopy) section conjecture (for any sequenceof parametrizing morphisms).
Proof. By Proposition 6.1.16 and repeated application of Proposition 6.3.10, the ordinary and homotopy section conjectures are equivalent. The result then follows by repeated application of Theorem 6.2.6 along with Theorem 5.3.24.
Corollary 6.3.14. Let 푋 be a hyperbolic good neighborhood. If Conjecture 5.3.25 is true over 푘, then (푋, 퐶푋,푓 , 푘) satisfies the surjectivity in the (homotopy) section conjecture (for any sequence of parametrizing morphisms).
115 Proof. As in Corollary 6.3.13, the ordinary and homotopy section conjectures are equivalent. The result follows by repeated application of Theorem 6.2.7.
Lemma 6.3.15. Every smooth geometrically connected variety 푋 over an infinite field 푘 has an open cover by hyperbolic good neighbourhoods over 푘.
Proof. Let 푥 be a closed point of 푋. We proceed by induction on the dimension of 푋. If 푋 has dimension 0, it is Spec 푘, so we are done. Otherwise, Lemma 6.3.8 implies that there is a neighborhood 푈 of 푥 and an elementary fibration 푓 : 푈 → 푆 over 푘. By Lemma 6.3.7, we can replace 푈 and 푆 so that 푓 is now a hyperbolic elementary fibration. Finally, by the induction hypothesis, there is an open neighborhood 푉 of 푓(푥) that is a hyperbolic good neighborhood over 푘. By Lemma 6.3.6, the pullback of 푓 from 푆 to 푉 is still a hyperbolic elementary fibration, and this pullback is our desired hyperbolic good neighborhood.
We can now answer Question 4.3.2 for (f − cov, 푆, 푘) while circumventing Theorem 5.1.3:
Corollary 6.3.16. Let 푋 be a smooth geometrically connected variety over a number field 푘, ⋃︁ and assume Conjecture 5.3.25. Then 푋 has a Zariski open cover 푋 = 푈푖 such that 푈푖 is 푖 VSA for (f − cov, 푆, 푘) for any nonempty set 푆 of finite places of 푘.
Proof. By Proposition 6.3.15, 푋 has an open cover by hyperbolic good neighborhoods 푈푖. By
Corollary 6.3.14, 푈푖 satisfies the surjectivity in the section conjecture over 푘, and by Corollary
6.3.13, 푈푖 satisfies the injectivity in the section conjecture over 푘푣 for all finite places 푣 of 푘.
By Proposition 5.3.26, 푈푖 satisfies VSA for (f − cov, 푆, 푘).
6.4 Cuspidal Sections in Arbitrary Dimension
Conjecture 6.4.1. Let 푓 : 푋 → 푆 be a good neighborhood and (푆, 퐶, 푘) a cuspidal datum
for 푆. Then the cuspidal datum (푋, 퐶푋,푓 , 푘) induced by 푓 and a sequence of parametrizing morphisms is independent of the sequence of parametrizing morphisms.
If we assume the section conjecture (Conjecture 5.3.25), we may then define 퐶푋,푓 as the complement in 푋(ℎ푘) of 휅(푋(푘)), which is clearly independent of the sequence of
116 parametrizing morphisms. We therefore expect Conjecture 6.4.1 to be true, at least for finitely generated fields of characteristic 0. In order to attempt to answer this question, it seems natural to construct a general definition of cuspidal sections for arbitrary smooth varieties (similar to Definition 5.3.18)and then prove that this construction is stable under Definition 6.2.3. We survey some current ideas of the author in this direction. To do this, we must work with ordinary rather than homotopy cuspidal data, as in Section 5.3.1. Doing so allows us to ignore boundary components of codimension greater than one, which would make it easier to prove a claim such as Claim 6.4.6. While this might not seem completely ideal, it suffices for the application to Conjecture 6.4.1, by Proposition 6.3.10.
6.4.1 Cuspidal Sections Associated to a Compactification
Definition 6.4.2. Let 푈 ⊆ 푋 be an open immersion of good 푘-schemes. Let 푌 = 푋 ∖ 푈
ℎ ℎ and 푦 ∈ 푌 (푘). We fix a Henselization 풪푋,푦 and set 푋푦 = Spec 풪푋,푦. We let 푈푦 = 푋푦 ×푋 푈, which is known as the scheme of nearby points to 푈 at 푦.
In this case, 푈푦 is a good 푘-scheme, so we may write S휋1(푈푦/푘).
Definition 6.4.3. The map 푈푦 → 푈 induces a map S휋1(푈푦/푘) → S휋1(푈/푘), whose image 퐶푦 is known as the packet of cuspidal sections at 푦.
Definition 6.4.4. Given a smooth variety 푈 over a field 푘, we refer to an open inclusion 푈 ˓→ 푋 into a smooth proper variety 푋 with complement 푌 a smooth normal crossings divisor in 푋 as a good compactification.
If 푈 ⊆ 푋 is a good compactification, we let 푈푦 be the scheme of nearby points to 푈 at 푦. By 18.3 of [Sti13], there is a short exact sequence
× × 0 → Hom(풪(푈푦) /풪(푋푦) , Ẑ︀(1)) → 휋1(푈푦) → 퐺푘 → 0.
Definition 6.4.5. Let 푈 be a smooth geometrically connected variety over a field 푘 of characteristic 0, and let 푈 ˓→ 푋 be a good compactification. Let 푌 = 푋 ∖ 푈 and 푦 ∈ 푌 (푘).
117 We define the set 퐶푈,푋 of all cuspidal sections of the variety 푈 as
⋃︁ 퐶푈,푋 : = 퐶푦 ⊆ S휋1(푈/푘). 푦∈푌 (푘)
In a forthcoming paper, we intend to prove the following.
Claim 6.4.6. The subset 퐶푈,푋 ⊆ S휋1(푈/푘) is independent of 푋, and we denote it by 퐶푈 .
6.4.2 Cuspidal Sections in Fibrations
Conjecture 6.4.7. Let 푓 : 푋 → 푆 be a special geometric fibration of smooth étale 퐾(휋, 1) varieties, let 퐶푆, 퐶푋푠 , and 퐶푋 be as in Definition 6.4.5, and let
⎛ ⎞ −1 ⋃︁ ⋃︁ 퐶푋,푓 = 푓 (퐶푆) ⎝ 휄푠(퐶푋푠 )⎠ ⊆ S휋1(푋/푘). 푠∈푆(푘)
Then 퐶푋,푓 = 퐶푋 .
The above conjecture clearly implies Conjecture 6.4.1. In the same forthcoming paper, we also intend to prove:
Claim 6.4.8. With the notation of Conjecture 6.4.7, 퐶푋 ⊆ 퐶푋,푓 .
118 Chapter 7
Examples
7.1 Poonen’s Counterexample
In [Poo10], Poonen found the first example of a variety with no rational points that does not satisfy VSA for the étale-Brauer obstruction. In this section, we review the construction of this variety and then explicitly find an open cover with empty étale-Brauer set.
7.1.1 Conic Bundles
We now present some general notions about conic bundles, as described in §4 of [Poo10]. We base our notation on [Poo10] and then add some notation of our own. Let 푘 be a field. From now on, let 휀 equal 1 if 푘 has characteristic 2 and 0 otherwise. Let 퐵 be a nice 푘-variety. Let ℒ be a line bundle on 퐵. Let ℰ be the rank 3 locally free sheaf 풪 ⊕ 풪 ⊕ ℒ
on 퐵. Let 푎 ∈ 푘×, and let 푠 ∈ Γ(퐵, ℒ⊗2) be a nonzero global section. Consider the section
휀 ⊕ 1 ⊕ (−푎) ⊕ (−푠) ∈ Γ(퐵, 풪 ⊕ 풪 ⊕ 풪 ⊕ ℒ⊗2) ⊂ Γ(퐵, Sym2 ℰ), where the first 풪 corresponds to the product of the first two summands of ℰ, and the last three
119 terms 풪 ⊕ 풪 ⊕ ℒ⊗2 ⊂ Sym2 ℰ are the symmetric squares of the three individual summands
푣 of ℰ. The zero locus of 휀 ⊕ 1 ⊕ (−푎) ⊕ (−푠) in Pℰ is a projective geometrically integral 푘-variety 푋 = 푋(퐵, ℒ, 푎, 푠) with a morphism 훼 : 푋 → 퐵.
Definition 7.1.1. We call an element
(ℒ, 푠, 푎) ∈ Pic 퐵 × Γ(퐵, ℒ⊗2) × 푘×, where 푠 ̸= 0, a conic bundle datum on 퐵 and 푋 the total space of (ℒ, 푠, 푎). We denote
푋 = 푇 표푡퐵(ℒ, 푠, 푎).
∼ If 푈 is a dense open subscheme of 퐵 with a trivialization ℒ|푈 = 풪푈 , and we identify 2 2 푠|푈 with an element of Γ(푈, 풪푈 ), then the affine scheme defined by 푦 + 휀푦푧 − 푎푧 = 푠|푈 in 2 A푈 is a dense open subscheme of 푋. We therefore refer to 푋 as the conic bundle given by 푦2 + 휀푦푧 − 푎푧2 = 푠.
1 1 In the special case where 퐵 = P , ℒ = 풪(2), and the homogeneous form 푠 ∈ Γ(P , 풪(4)) is separable, 푋 is called the Châtelet surface given by 푦2 + 휀푦푧 − 푎푧2 = 푠(푥), where 푠(푥) ∈ 푘[푥] denotes a dehomogenization of 푠. Returning to the general case, we let 푍 be the subscheme 푠 = 0 of 퐵. We call 푍 the degeneracy locus of the conic bundle (ℒ, 푠, 푎). Each fiber of 훼 above a point of 퐵 − 푍 is a smooth plane conic, and each fiber above a geometric point of 푍 is a union of two projective lines crossing transversally at a point. A local calculation shows that if 푍 is smooth over 푘, then 푋 is smooth over 푘.
7.1.2 Poonen’s Variety
× ˜ ˜ Let 푘 be a global field, let 푎 ∈ 푘 , and let 푃∞(푥), 푃0(푥) ∈ 푘[푥] be relatively prime separable 2 2 ˜ degree 4 polynomials such that the (nice) Châtelet surface 풱∞ given by 푦 +휀푦푧−푎푧 = 푃∞(푥)
over 푘 satisfies 풱∞(A푘) ̸= ∅ but 풱∞(푘) = ∅. Such Châtelet surfaces always exist : see [[Poo10], Proposition 5.1 and 11] if the characteristic of 푘 is not 2 and [Vir12] otherwise. If 푘 = Q, one ˜ 2 2 may use the original example from [Isk71] with 푎 = −1 and 푃∞(푥) := (푥 − 2)(3 − 푥 ).
120 ˜ ˜ Let 푃∞(푤, 푥) and 푃0(푤, 푥) be the homogenizations of 푃∞ and 푃0. Let ℒ = 풪(1, 2) on 1 1 P × P and define
2 2 1 1 ⊗2 푠1 := 푢 푃∞(푤, 푥) + 푣 푃0(푤, 푥) ∈ Γ(P × P , ℒ ),
1 where the two copies of P have homogeneous coordinates (푢 : 푣) and (푤 : 푥), respectively. 1 1 1 Let 푍1 ⊂ P × P be the zero locus of 푠1. Let 퐹 ⊂ P be the (finite) branch locus of the first 1 projection 푍1 → P . i.e.,
{︀ 1 2 2 }︀ 퐹 := (푢 : 푣) ∈ P |푢 푃∞(푤, 푥) + 푣 푃0(푤, 푥) has a multiple root .
1 1 2 2 Let 훼1 : 풱 → P × P be the conic bundle given by 푦 + 휀푦푧 − 푎푧 = 푠1, a.k.a. the conic 1 1 bundle on P × P defined by the datum (푂(1, 2), 푎, 푠1). 1 1 1 1 Composing 훼1 with the first projection P × P → P yields a morphism 훽1 : 풱 → P whose fiber above ∞ := (1 : 0) is the Châtelet surface 풱∞ defined earlier. Now let 퐶 be a nice curve over 푘 such that 퐶(푘) is finite and nonempty. Choose a dominant
1 morphism 훾 : 퐶 → P , étale above 퐹 , such that 훾(퐶(푘)) = {∞}. Define 푋 := 풱 ×P1 퐶 to 1 1 be the fiber product with respect to the maps 훽1 : 풱 → P , 훾 : 퐶 → P , and consider the morphisms 훼 and 훽 as in the diagram:
푋 / 풱
훼 훼1 1 (훾,1) 1 1 훽 퐶 × P / P × P 훽1
p푟1 p푟1 훾 Ð 퐶 / P1 The variety 푋 is the one constructed in §6 of [Poo10]. The same paper proves that
´et,Br 푋(A푘) ̸= ∅ (Theorem 8.2) and 푋(푘) = ∅ (Theorem 7.2). We present the proof that 푋(푘) = ∅ because it is short and simple.
Proposition 7.1.2. If 푋 is the variety constructed above, then 푋(푘) = ∅.
121 −1 Proof. Assume 푥0 ∈ 푋(푘); we have 푐0 := 훽(푥0) ∈ 퐶(푘), but then 푥 ∈ 훽 (푐0). By the −1 −1 −1 ∼ construction of 푋, 훽 (푐0) is isomorphic to 훽1 (훾(푐0)) = 훽1 (∞) = 풱∞, but 풱∞(푘) = ∅ by construction.
Note that 푋 can also be considered as the variety corresponding to the datum (풪(1, 2), 푎, 푠1) 1 pulled back via (훾, 1) to 퐶 × P .
7.1.3 VSA Stratifications and Open Covers
We now show, under some conditions, that there is a finite stratification X of 푋 with X ,´et,Br 푋(A푘) = ∅.
Proposition 7.1.3. Suppose 푘 is a number field. Let 퐽 be the Jacobian of 퐶, and suppose that 퐽(푘) is finite and that the Tate-Shafarevich group of 퐽 has trivial divisible subgroup.
Br Br Then there is a stratification 푋 = 푋1 ∪ 푋2 such that 푋1(A푘) = ∅ and 푋2(A푘) = ∅.
−1 Proof. Let 푈 denote the complement in 퐶 of 퐶(푘). Let 푋1 be 훽 (푈), and let 푋2 be 훽−1(퐶(푘)). By Theorem 8.1 of [Sto07], we have
Br 퐶(푘) = 퐶(A푘)∙ .
푓 Br 푓 Br Br In particular, we find that 퐶(푘) = 퐶(A푘) , and therefore that 푈(A푘) = 푈(A푘) = ∅. This Br implies that 푋1(A푘) = ∅ as well. Br We also know by construction that 푋2 is a union of copies of 풱∞. As 풱∞(A푘) ) = ∅, we Br find that 푋2(A푘) = ∅.
In this subsection, we present an explicit proof that 푋, as constructed in Section 7.1.2, has a Zariski open cover with empty étale-Brauer set using Theorem 6.2.10. Section 7.2 will then prove a stronger result, namely that there is a finite ramified cover with empty étale-Brauer set.
122 We let 푘 be a number field in this section. From now on, we shall assume that 퐶(푘) is finite. Let 푈 denote the complement in 퐶 of 퐶(푘). We now have the following:
Theorem 7.1.4. Suppose that every nonempty open subvariety of 퐶 satisfies Conjecture
5.3.25. Then there is an open subvariety 푆 ⊆ 퐶 such that 푆 ∪ 푈 = 퐶, and 푋푆 := 푆 ×퐶 푋 satisfies VSA for (´et − Br, 푓, 푘).
Proof. Let 퐹 ′ := 훾−1(퐹 ), and let 퐶′ := 퐶 ∖ 퐹 ′. Then 훽 is smooth over 퐶′.
′ Let 푣 be a real place of 푘. As 퐶 is smooth, 퐶 (푘푣) is a disjoint union of components each homeomorphic to a line or circle. For each circle component, we wish to remove a closed non-rational point of 퐶′ that meets that component.
To do this, we need to show that each component contains a 푘푠-point that is not a 푘-point. alg Let 푘푣 denote the algebraic closure of 푘 in 푘푣. By the completeness of the theory of real alg closed fields, any first-order formula true of 푘푣 is also true of 푘푣 . By Theorems 2.4.4 and 2.4.5 of [BCR98], the property that a point lies in a given connected component is a semi-algebraic condition. As well, the property that a point is not equal to any element of 퐶′(푘) is also a
semi-algebraic condition. As each component has uncountably many 푘푣-points and hence at ′ alg least one point not in 퐶 (푘), each component has a 푘푣 -point not in 퐶(푘). We choose one for each component and remove the corresponding closed points of 퐶′. We let 푆 denote the resulting open subvariety of 퐶. It is clear that 푆(푘) = 퐶(푘), so that 푆 ∪ 푈 = 퐶.
It suffices to verify the hypotheses of Theorem 6.2.10 for the projection 푓 = 훽 |푋푆 : 푋푆 → 푆 and 푇 = 푓. By Lemma 4.2.7, 푆 is VSA for (ℎ, 푓, 푘), so 1 holds. By Conjecture 5.3.25 and Theorem 5.3.24, 2a holds. 3 and 4 follow immediately because 푇 = 푓 and because 푆 is a smooth geometrically connected curve, respectively.
By construction, every connected component of 푆(푘푣) is simply connected. As well, every
geometric fiber is a Châtelet surface, so 휋1(푋푎((푘푣)푠)) is trivial, and hence 5 is verified.
123 Finally, every fiber over a rational point of 푆 is isomorphic to 풱∞. But we assumed that Br ´et,Br 풱∞(푘) = ∅, so by [CTSSD87a, CTSSD87b], we have 풱∞(A푘) = 풱∞(A푘) = ∅. This shows that 6 holds.
Corollary 7.1.5. Under the hypotheses and notations of Theorem 7.1.4, the variety 푋 has an open cover 푋 = 푋푆 ∪ 푋푈 such that 푋푆 and 푋푈 := 푈 ×퐶 푋 both have empty étale-Brauer set.
Proof. We have already shown that 푋푆(푘) ⊆ 푋(푘) = ∅, so by Theorem 7.1.4, we have ´et,Br 푋푆(A푘) = ∅. By Conjecture 5.3.25 and Proposition 5.3.26 applied to 푈, we find that f−cov f−cov ´et,Br 푈(A푘) = ∅, which implies that 푋푈 (A푘) = 푋푈 (A푘) = ∅.
Remark 7.1.6. For the proof of Corollary 7.1.5, one may avoid using Conjecture 5.3.25 for 푈 by assuming that the Jacobian of 퐶 has finitely many rational points and Tate-Shafarevich
Br group with trivial divisible subgroup. For then, as in Proposition 7.1.3, 푈(A푘) = ∅. In particular, 푈 satisfies VSA for (Br, 푓, 푘) and hence for (´et − Br, 푓, 푘), and the fact that
´et,Br ´et,Br 푈(A푘) = ∅ implies that 푋푈 (A푘) = ∅ as well.
7.2 The Brauer-Manin obstruction applied to ramified covers
In this section, we introduce the notion of the ramified étale-Brauer obstruction. In a sense, this lies between the obstructions from Section 4.3.1 and the ordinary étale-Brauer obstruction. We also formulate a conjecture analogous to Theorems 4.3.3 and 4.3.4. In Section 7.2.2, we then explain how to apply this to Poonen’s example.
7.2.1 Quasi-Torsors and the Ramified Etale-Brauer Obstruction
We shall now define slight generalizations of the concepts of torsors and the étale Brauer-Manin obstruction.
124 Definition 7.2.1. Let 푋 be a variety over a field 푘, 퐺 a finite smooth 푘-group, and 퐷 ⊆ 푋 an a closed subvariety. A 퐺-quasi-torsor over 푋 unramified outside 퐷 is a map 휋 : 푌 → 푋 and a 퐺-action on 푌 respecting 휋 such that
1. 휋 is a surjective and quasi-finite morphism.
2. The pullback of 휋 from 푋 to 푋 ∖ 퐷 is a 퐺-torsor over 푋 ∖ 퐷.
We call 푑 = |퐺| the degree of 푌 .
Given 휌: 푍 → 푋 an arbitrary morphism of 푘-varieties, the pullback 휌−1(푌 ) is a 퐺-quasi- torsor unramified outside 휌−1(퐷). Let 휋 : 푌 → 푋 be a 퐺-quasi-torsor over 푋 unramified outside 퐷. As in the case of a usual 퐺-torsor, one can twist 휋 : 푌 → 푋 by any 휎 ∈ 퐻1(푘, 퐺) and get a 퐺휎-quasi-torsor 휋휎 : 푌 휎 → 푋, also unramified outside 퐷. If we assume that 퐷(푘) = ∅, then as in Section 4.2.3, we get:
⋃︁ 푋(푘) = 휋휎(푌 휎(푘)) 휎∈퐻1(푘,퐺)
If 푘 is a global field, it follows that:
휋,Br ⋃︁ 휎 휎 Br 푋(푘) ⊂ 푋(A푘) := 휋 (푌 (A푘) ). 휎∈퐻1(푘,퐺) Definition 7.2.2. Letting 휋 range over all quasi-torsors over 푋 unramified outside 퐷, we define the (퐷-)ramified étale-Brauer obstruction by
´et,Br∼퐷 ⋂︁ 휋,Br 푋(A푘) := 푋(A푘) ⊂ 푋(A푘) 휋 ´et,Br ´et,Br∼퐷 It follows from Section 4.2.3 that (푋 ∖ 퐷)(A푘) ⊆ 푋(A푘) . Until now, we have left the fact that 퐷(푘) = ∅ as a black box. The idea behind this construction is that 퐷 has smaller dimension, and it therefore might be easier to show that 퐷 has empty étale-Brauer set. More generally, one might, e.g., find a subvariety 퐸 of 퐷, prove
´et,Br∼퐸 ´et,Br that 퐷(A푘) = ∅, and finally prove that 퐸(A푘) = ∅. We formalize this as follows.
125 Definition 7.2.3. We say that the absence of rational points is explained by the ramified ∐︁ étale-Brauer obstruction if there exists a stratification 푋 = 푋푖 and for each 푖, we have 푖 ´et,Br∼퐷푖 푋푖(A푘) = ∅, where 퐷푖 := 푋푖 ∖ 푋푖.
This is similar to the method of considering the étale-Brauer sets of dense open subsets, for
´et,Br∼퐷 ´et,Br 푋(A푘) = ∅ implies that (푋 ∖ 퐷)(A푘) = ∅. In particular, if the absence of rational points is explained by the ramified étale-Brauer obstruction, then the answer to Question 4.3.1 is yes. However, one crucial difference is that the ramified étale-Brauer obstruction only requires proving emptiness of Brauer sets of proper varieties. Inspired by Theorem 4.3.3, we conjecture that the absence of rational points is always explained by the ramified étale-Brauer obstruction and that one may furthermore take a single quasi-torsor for each stratum:
Conjecture 7.2.4. For any number field 푘 and variety 푋/푘 with 푋(푘) = ∅, there exists a ∐︁ stratification 푋 = 푋푖 and for each 푖, a quasi-torsor 푌푖 over the closure 푋푖 of 푋푖 such that 푖 휎 Br 휎 푌푖 restricts to a torsor over 푋푖 and 푌푖 (A푘) = ∅ for all twists 푌푖 of 푌푖.
This conjecture still does not imply that there is an algorithm to determine whether 푋(푘)
is empty, as one may need to consider infinitely many twists of a quasi-torsor 푌푖. Nonetheless, we now show that it is computable in the example of [Poo10] (in particular, see Remark 7.2.8 to see why we need only finitely many twists). In particular, we know of no counterexamples to Conjecture 7.2.4.
7.2.2 Quasi-torsors in Poonen’s Example
In this subsection, we show (under some conditions) that for the variety 푋 defined in Section
Br ´et,Br∼퐷 7.1.2, one can choose a divisor 퐷 ⊆ 푋 such that 퐷(A푘) = ∅, and 푋(A푘) = ∅. More specifically, we assume as in Remark 7.1.6 that the Jacobian of 퐶 has finitely many rational points and trivial divisible subgroup of its Tate-Shafarevich group, which implies by Theorem 8.1 of [Sto07] that
Br 퐶(푘) = 퐶(A푘)∙
126 and
Br 푈(A푘) = ∅.
As before, let 퐹 ′ := 훾−1(퐹 ) ⊆ 퐶 and 퐶′ := 퐶∖퐹 ′. Note that 퐶′ is a non-projective curve. Now let 퐷 := 훽−1(퐹 ′). Note that ∞ ̸∈ 퐹 , so that 퐶(푘) ∩ 퐹 ′ = ∅. As 퐷 maps to 푈 and
Br Br 푈(A푘) = ∅, we have 퐷(A푘) = ∅. We will now spend the rest of Section 7.2.2 proving:
Theorem 7.2.5. With notations as above, the absence of rational points on 푋 is explained by the ramified étale-Brauer obstruction if 푘 is a global field with no real places (i.e., a function field or a totally imaginary number field).
1 Now 푋 is a family over 퐶 of conic bundles over P . The fiber over any element of 퐶(푘) is ′ isomorphic to the Châtelet surface 풱∞. All the fibers over 퐶 are smooth conic bundles. ′ 1 1 4 Let 퐸 ⊂ (P ∖퐹 ) × (P ) be the curve defined by
2 2 푢 푃∞(푤푖, 푥푖) + 푣 푃0(푤푖, 푥푖) = 0, 1 ≤ 푖 ≤ 4
(푤푖 : 푥푖) ̸= (푤푗 : 푥푗), 푖 ̸= 푗, 1 ≤ 푖, 푗 ≤ 4,
1 where (푢 : 푣) are the projective coordinates of P ∖퐹 and (푤푖 : 푥푖), 1 ≤ 푖 ≤ 4 are the projective 1 ˜ ˜ coordinates of the four copies of P . Since 푃∞(푥) and 푃0(푥) are separable and coprime, we ′ ′ 1 ′ have that 퐸 is a smooth connected curve and that the first projection 퐸 → P ∖퐹 gives 퐸 1 the structure of an étale Galois covering of P ∖퐹 with automorphism group 퐺 = 푆4 that acts on the fibres by permuting
(푤푖 : 푥푖), 1 ≤ 푖 ≤ 4.
Since every birationality class of curves contains a unique projective smooth member, one
1 ′ can construct an 푆4-quasi-torsor over 퐸 → P unramified outside 퐹 which gives 퐸 when 1 restricted to P ∖퐹 . 1 1 The 푘-twists of 퐸 → P are classified by 퐻 (푘, 푆4) which (since the action of Γ푘 on 푆4 is
trivial) coincides with the set Hom(Γ푘, 푆4)/ ∼ of homomorphisms up to conjugation. More
127 concretely, for every homomorphism 휑 :Γ푘 → 푆4, define 퐸휑 to be the 푘-form of 퐸 with Galois action that restricts to the action
휎 : ((푢 : 푣), ((푤1 : 푥1), (푤2 : 푥2), (푤3 : 푥3), (푤4 : 푥4))) ↦→
휎 ((푢 : 푣), ((푤휑휎(1) : 푥휑휎(1)), (푤휑휎(2) : 푥휑휎(2)), (푤휑휎(3) : 푥휑휎(3)), (푤휑휎(4) : 푥휑휎(4)))) on 퐸′.
1 For every 휑 :Γ푘 → 푆4, we set 퐶휑 := 퐶 ×P1 퐸휑 (relative to 훾 : 퐶 → P and the first 1 projection 휋휑 : 퐸휑 → P ) and 푋휑 := 푋 ×퐶 퐶휑 (relative to 훽 : 푋 → 퐶 and the projection
퐶휑 → 퐶). 1 1 Since the maps 훾 : 퐶 → P and 퐸 → P have disjoint ramification loci, all 퐶휑 are
geometrically integral, and so are all the 푋휑.
Then 푋휑 is a complete family of twists of a quasi-torsor over 푋 of degree 24 unramified Br outside 퐷. As we already know that 퐷(A푘) = ∅, it suffices for the proof of Theorem 7.2.5 to show that
Br 푋휑(A푘) = ∅
1 for every 휑 ∈ 퐻 (Γ푘, 푆4). We devote the rest of Section 7.2.2 to proving this fact.
Reduction to 푋휑∞
1 Br Lemma 7.2.6. For every 휑 ∈ 퐻 (푘, 푆4), we have 퐶휑(푘) = 퐶휑(A푘) ∙.
Br f−ab Br f−ab Proof. By Corollary 7.3 of [Sto07], we have 퐶휑(A푘)∙ = 퐶휑(A푘)∙ and 퐶(A푘)∙ = 퐶(A푘)∙ . Br Then Proposition 8.5 of [Sto07] and the fact that 퐶(푘) = 퐶(A푘)∙ implies the result.
1 Denote by 휑∞ ∈ 퐻 (푘, 푆4) the map Γ푘 → 푆4 defined by the Galois action on the four
roots of 푃∞.
1 Lemma 7.2.7. Let 휑 ∈ 퐻 (Γ푘, 푆4) be such that 휑 ̸= 휑∞. Then 퐶휑(푘) = ∅.
−1 Proof. Since 휑 ≠ 휑∞ we get that 퐸휑(푘) ∩ 휋휑 (∞) = ∅. Since 훾(퐶(푘)) = ∞, we get that
퐶휑(푘) = ∅.
128 1 Let 휌휑 : 푋휑 → 퐶휑 denote the map defined earlier. For every 휑 ∈ 퐻 (푘, 푆4), we have
Br Br 휌휑(푋휑(A푘) ∙) ⊆ 퐶휑(A푘) ∙ = 퐶휑(푘) = ∅,
Br so 푋휑(A푘) = ∅ for 휑 ̸= 휑∞.
Remark 7.2.8. It is Lemma 7.2.7 along with VSA for 퐶 coming from our condition on the Tate-Shafarevich group that lets us avoid all but one twist. In fact, this has a conceptual explanation. While infinitely many twists might have adelic points (as 훽−1(퐶′) is not proper), all possible elements of the Brauer set lie in the fiber over 퐶(푘), which is in fact proper. There are therefore finitely many twists that might have adelic points in this fiber.
Br The proof that 푋휑∞ (A푘) = ∅
Br Br In this subsection, we shall prove that if 푘 has no real places, then 푋휑∞ (A푘) = 푋휑∞ (A푘)∙ = ∅. Let 푝 ∈ 퐶 (푘). The fiber 휌−1 (푝) is isomorphic to the Châtelet surface 풱 . We shall 휑∞ 휑∞ ∞ denote by 휌 : 풱 → 푋 the corresponding natural isomorphism onto the fiber 휌−1 (푝). 푝 ∞ 휑∞ 휑∞ Br Recall that 풱∞ satisfies 풱∞(A푘) = ∅.
Br Lemma 7.2.9. Let 푘 be a global field with no real places. Let 푥 ∈ 푋휑∞ (A푘)∙ . Then there exists 푝 ∈ 퐶휑∞ (푘) such that 푥 ∈ 휌푝(풱∞(A푘)∙).
Proof. From functoriality and Lemma 7.2.6 we get
Br Br 휌휑∞ (푥) ∈ 휌휑∞ (푋휑∞ (A푘)∙ ) ⊂ 퐶휑∞ (A푘) ∙ = 퐶휑∞ (푘)
′ We denote 푝 = 휌휑∞ (푥) ∈ 퐶휑∞ (푘). Now it is clear that in all but maybe the infinite places
푥 ∈ 휌푝(풱∞(A푘)). Hence it remains to deal with the infinite places, which by assumption
are all complex. But since both 푋휑∞ and 풱∞ are geometrically integral, taking connected
components reduces 푋(C) and 풱∞(C) to a single point.
129 Lemma 7.2.10. Let 푝 ∈ 퐶휑∞ (푘) be a point. Then the map
* 휌푝 : Br(푋휑∞ ) → Br(풱∞) is surjective.
We will prove Lemma 7.2.10 in Section 7.2.2.
Br Lemma 7.2.11. Let 푘 be global field with no real places. Then 푋휑∞ (A푘)∙ = ∅.
Br Br Proof. Assume that 푋휑∞ (A푘)∙ ̸= ∅. Let 푥 ∈ 푋휑∞ (A푘)∙ . By Lemma 7.2.9 there exists a
푝 ∈ 퐶휑∞ (푘) such that 푥 ∈ 휌푝(풱∞(A푘)∙). Let 푦 ∈ 풱∞(A푘)∙ be such that 휌푝(푦) = 푥. We shall Br show that 푦 ∈ 풱∞(A푘)∙ . ˜ ′ * ˜ Indeed, let 푏 ∈ Br(풱∞). By Lemma 7.2.10 there exists a 푏 ∈ Br(푋휑∞ ) such that 휌푝(푏) = 푏. Now * ˜ ˜ ˜ (푦, 푏) = (푦, 휌푝(푏)) = (휌푝(푦), 푏) = (푥, 푏) = 0
Br ˜ But by assumption 푥 ∈ 푋휑∞ (A푘)∙ , so we have (푦, 푏) = (푥, 푏) = 0. Thus we have 푦 ∈ Br 풱∞(A푘)∙ = ∅ which is a contradiction.
* The surjectivity of 휌푝
In this subsection, we shall prove the statement of Lemma 7.2.10. We switch gears for a moment and let 훼: 푋 → 퐵 be an arbitrary conic bundle given by datum (ℒ, 푠, 푎).
푠 푠 푠 1 푠 Lemma 7.2.12. The generic fiber 푋휂 of 푋 → 퐵 is isomorphic to P휅(퐵푠), where 휅(퐵 ) is the field of rational functions on 퐵푠 .
Proof. It is a smooth plane conic and it has a rational point since 푎 is a square in 푘푠 ⊂ 휅(퐵푠).
Lemma 7.2.13. Denote the generic point of 퐵 by 휂. Let 푍 be the degeneracy locus. Assume
푠 푠 ⋃︁ that 푍 is the union of the irreducible components 푍 = 푍푖. Then there is a natural 1≤푖≤푟 exact sequence of Galois modules.
130 휌4 휌1 푠 + − 휌2 푠 q 푠 푍 / Pic 푋 / Pic 푋 / 0 ⊕ 푍 ⊕ 푍 / Pic 퐵 / 0 Z 푖 Z 푖 Z 푖 휌3 휂
푑푒푔 % Z
where 휌4 is a natural section of 휌3.
Proof. Call a divisor of 푋푠 vertical if it is supported on prime divisors lying above prime
푠 ± divisors of 퐵 , and horizontal otherwise. Denote by 푍푖 the divisors that lie over 푍푖 and √ defined by the additional condition that 푦 = ± 푎푧. Now define 휌1 by
+ − 휌1(푍푖) = (−푍푖, 푍푖 , 푍푖 )
and 휌2 by * 휌2(푀, 0, 0) = 훼 푀
+ + 휌2(0, 푍푖 , 0) = 푍푖
− − 휌2(0, 0, 푍푖 ) = 푍푖
푠 푠 Let 휌3 be the map induced by 푋휂 → 푋 . Each 휌푖 is Γ푘-equivariant. Given a prime divisor 퐷 푠 푠 on 푋휂 , we take 휌4(퐷) to be its Zariski closure in 푋 . It is clear that 휌3 ∘ 휌4 = Id, so 휌3 is indeed surjective.
The kernel of 휌3 is generated by the classes of vertical prime divisors of 푋. In fact, there 푠 is exactly one above each prime divisor of 퐵, except that above each 푍푖 ∈ Div 퐵 we have + − 푠 푠 both 푍푖 , 푍푖 ∈ Div 푋 . This proves exactness at Pic 푋 . Now, since 훼 : 푋푠 → 퐵푠 is proper, a rational function on 푋푠 with a vertical divisor must
푠 be the pullback of a rational function on 퐵 . Using the fact that the image of 휌2 only contains vertical divisors, we prove exactness at
− + 푠 Pic 퐵 ⊕ Z푍푖 ⊕ Z푍푖
131 The injectivity of 휌1 is then trivial.
We switch gears once again and let 푋 be as in Poonen’s example.
Lemma 7.2.14. Let 푝 ∈ 퐶휑∞ (푘) and 휌푝 : 풱∞ → 푋휑∞ be the corresponding map as above. Then the map of Galois modules
* 푠 푠 휌푝 : Pic(푋휑∞ ) → Pic(풱∞)
has a section.
1 1 Proof. Consider the map 휑푝 : P → P × 퐶휑∞ defined by 푥 ↦→ (푥, 푝). It is clear that the map 1 휌푝 : 풱∞ → 푋휑∞ comes from pulling back the conic bundle datum defining 푋휑∞ over P × 퐶휑∞ 1 by this map. Let 퐵 = P × 퐶휑∞ , and consider the following commutative diagram with exact rows
푠 p − + 푠 0 / Z푍푖 / Pic 퐵 ⊕ Z푍푖 ⊕ Z푍푖 / Pic 푋휑 / Z / 0 J J ∞ 푑푒푔 * 푠1 푠2 휌푝 + − q 푊 / Pic 풱푠 / / 0 ⊕ 푊 ⊕ 푊 / Pic 1 / 0 Z 푖 P푘푠 Z 푖 Z 푖 ∞ 푑푒푔 Z
1 where 푍 is the degeneracy locus of 푋휑∞ over 퐵, 푊 is the degeneracy locus of 풱∞ over P , 푠 ⋃︁ 푠 ⋃︁ and 푍 = 푍푖 and 푊 = 푊푖 are decompositions into irreducible components. The 1≤푖≤푟 1≤푖≤푟 * existence of a section for 휌푝 follows by diagram chasing and the existence of the compatible
sections 푠1 and 푠2.
Every 푊푖 (1 ≤ 푖 ≤ 4 ) is a point that corresponds to a different root (푤푖 : 푥푖) of the 푠 polynomial 푃∞(푥, 푤). We can choose 푍푖 ⊂ 퐵 to be Zariski closure of the zero set of 푤푖푥−푥푖푤, ± 푠 √ and similarly 푍푖 ⊂ 푋휑∞ to be Zariski closure of the zero set of 푦 ± 푎푧, 푤푖푥 − 푥푖푤. ± ± 1 푠 Now we define: 푍푖 = 푠1(푊푖) and 푍푖 = 푠2(푊푖 ) and the map 푠2 : Pic P푘푠 → Pic 퐵 is 1 1 defined by the unique section of the map 휑푝 : P → P × 퐶휑∞ .
It is clear that 푠1 and 푠2 are indeed group-theoretic sections. To prove that 푠1 and 푠2 also
132 respect the Galois action, we can write
0 0 0 0 0 0 0 0 1 푝 = (푐, ((푥1 : 푤1), (푥2 : 푤3), (푥2 : 푤3), (푥2 : 푤3))) ∈ 퐶(푘) ×P (푘) 퐸휑∞ (푘),
0 0 0 0 0 0 0 0 and since 훾(퐶(푘)) = {∞}, the four points {(푥1 : 푤1), (푥2 : 푤3), (푥2 : 푤3), (푥2 : 푤3)} are
exactly the four different roots of 푃∞(푥, 푤) .
Lemma 7.2.15 (Lemma 7.2.10). Let 푝 ∈ 퐶휑∞ (푘). Then the map
* 휌푝 : Br(푋휑∞ ) → Br(풱∞) is surjective.
푠 Proof. Denote by 푠푝 : Pic (풱∞) → Pic(푋휑∞ ) the section of
* 푠 푠 휌푝 : Pic(푋휑∞ ) → Pic(풱∞)
It is clear that 푠푝 induces a section of the map
** 1 푠 1 푠 휌푝 : 퐻 (푘, Pic(푋휑∞ )) → 퐻 (푘, Pic(풱∞))
By the Hochschild–Serre spectral sequence for 푋, we have:
퐻1(푘, Pic(푋푠)) = ker[Br푋 → Br푋푠]/Im [Br푘 → Br푋]
Letting
푠 Br1(푋) := ker[Br푋 → Br푋 ],
* 푠 we get that the map 휌푝 : Br1(푋휑∞ ) → Br1(풱∞) is surjective. But since 풱∞ is a rational 푠 surface (it is a Châtelet surface), we have Br풱∞ = 0, and thus Br1(풱∞) = Br(풱∞). So we get * that 휌푝 : Br(푋휑∞ ) → Br(풱∞) is surjective.
133 134 Chapter 8
Appendices to Part II
8.1 Appendix A: Obstructions Without Functors
As the main novelty of our methods is to introduce decompositions into subvarieties, one might wonder what happens if one tries to build local-global obstructions using only such decompositions. We extend the notation from above by omitting 휔:
X ⋃︁ 푋(A푘,푆) = 푋푖(A푘,푆) 푖
C ⋂︁ X 푋(A푘,푆) = 푋(A푘,푆) . X ∈C For any 휔, we then have:
X ,휔 X 휔 푋(푘) ⊆ 푋(A푘,푆) ⊆ 푋(A푘,푆) , 푋(A푘,푆) .
The main point of this appendix is the following, open-ended question:
풮풯 ℛ풜풯 풪풫ℰ풩 Question 8.1.1. For a variety 푋, describe 푋(A푘,푆) and 푋(A푘,푆) .
1 In particular, for 푋 = A , this provides two interesting subsets of the adele ring A푘,푆. Is either a subring? As a first step, we note that neither of these subsets is always equal to the set of rational
135 points. We prove the following result modulo a lemma, then provide the lemma:
Proposition 8.1.2. Let 푋/푘 be integral. If 푋/푘 has a generic 푘푣-point for every 푣 ∈ 푆 (e.g., 풮풯 ℛ풜풯 풪풫ℰ풩 if 푋 is a 푘-rational variety), we have 푋(푘) * 푋(A푘,푆) ⊆ 푋(A푘,푆) .
Proof. It suffices to find a 푘-algebra homomorphism 푘(푋) → A푘,푆, for such a homomorphism
gives an A푘,푆-point of 푋 that extends to every nonempty open in 푋. As the existence of such a homomorphism depends only on 푘(푋), it suffices to assume that 푋 is affine and smooth.
× To do this, we need to find a point 훼 ∈ 푋(A푘,푆) such that 푓(훼) ∈ A푘,푆 for every nonzero 푓 ∈ 풪(푋).
By spreading out, there exists a finite type model 풳 of 푋 over 풪푘. As every element × × of 풪(푋) is a 푘 -multiple of an element of 풪(풳 ), it suffices to check 푓(훼) ∈ A푘,푆 for every
nonzero 푓 ∈ 풪(풳 ). This is equivalent to saying that 푓(훼푣) ̸= 0 for all 푣, and 푓(훼푣) ∈ 풪푣 ∖ m푣 for all but finitely many 푣.
We choose an enumeration of 풪(풳 ) ∖ {0} by the positive integers, where to 푖 ∈ Z>0 we × associate 푓푖 ∈ 풪(풳 ). We thus wish to choose 훼 ∈ 풳 (A푘,푆) such that 푓푖(훼) ∈ A푘,푆 for all
푖 ∈ Z>0.
For 푚 ≥ 1 let 푅푚 = {푓푖}1≤푖≤푚. By Lemma 8.1.4, there is some constant, which we denote
by 퐶푚, such that for all 푣 of norm at least 퐶푚, there is a smooth F푣-point of 풳 at which no
element of 푅푚 vanishes. We choose each 퐶푚 to be as small as possible for that 푚, so that
퐶푚 ≤ 퐶푚+1 for all 푚. We then set 퐷푚 = 푚 + 퐶푚, so that 퐷푚 < 퐷푚+1, and lim 퐷푚 = ∞. 푚→∞ Starting with 푚 = 1, for each place 푣 with norm in [퐷푚 + 1, 퐷푚+1], we choose 푎푣 ∈ 풳 (F푣)
such that 푓푖(푎푣) ̸= 0 for all 1 ≤ 푖 ≤ 푚. It follows that for every 푚, the value of 푓푚(푎푣) is
nonzero in F푣 for 푣 of norm greater than 퐷푚.
For every 푣 ∈ 푆 of norm > 퐷1, we choose an 훼푣 ∈ 풳 (풪푣) ⊆ 푋(푘푣) reducing to 푎푣 modulo
m푣 and which maps to the generic point of 푋. This exists by Lemma 8.1.3.
For every 푣 ∈ 푆 of norm ≤ 퐷1, we choose an arbitrary 훼푣 ∈ 푋(푘푣) that maps to the generic point of 푋. ∏︁ We set 훼 = 훼푣. Then for every nonzero 푓 ∈ 풪(푋), the value 푓(훼푣) is nonzero for all 푣∈푆 푣 and invertible in 풪푣 for all but finitely many 푣. That is, 푓(훼) is invertible in A푘,푆, so 훼
136 gives us our desired homomorphism from 푘(푋) to A푘,푆.
Lemma 8.1.3. Let 풳 be a finite type scheme over 풪푘 with irreducible generic fiber 푋 of dimension 푑. Let 푎 be a smooth point in 풳 (F푣) for a finite place 푣 of 푘. Then there is an
풪푣-point 훼 of 풳 lifting 푎 whose generic fiber is a generic point of 풳 .
Proof. Let 푥1, ··· , 푥푑 be local parameters at 푎. Then Hensel’s lemma says that there is a 푑 bijection between lifts of 푎 to 훼 ∈ 풳 (풪푣) and (풪푣) , given by
훼 ↦→ (푥1(훼)/휋푣, ··· , 푥푑(훼)/휋푣).
Let us choose an 훼 associated to a set of 푑 elements of 풪푣 that are algebraically independent
over 푘. Tensoring over 풪푘 with 푘, we get a homomorphism 풪(푋) → 풪푣 ⊗풪푘 푘 = 푘푣 whose
image in 푘푣 generates a field of transcendence degree 푑. This implies that the map 풪(푋) → 푘푣 is injective; i.e., the point is generic.
Lemma 8.1.4. Let 풳 be a finite type integral scheme over 풪푘 with generic fiber of dimension 푑 ≥ 1 that is geometrically integral and affine. Let 푅 be a finite subset of 풪(풳 ) ∖ {0}. Then
for almost all places 푣 of 푘, there is a smooth F푣-point of 풳 at which no element of 푅 vanishes.
Proof. By replacing 푅 with the one-element set containing the product of all elements of 푅, we may assume that 푅 has a single element, call it 푓. By the chart in Appendix C of [Poo17], the properties “flat,” “affine,” and “geometrically integral fibers” satisfy spreading out. We can therefore findaset 푇 of places of 푘 so that the
fiber of 풳 over 풪푘,푇 has geometrically integral fibers and is flat and affine. By flatness, allof
its fibers have the same dimension, which must be 푟. We replace 풳 by its fiber over 풪푘,푇 at the expense of modifying finitely many places. The zero set of 푓 is a closed subscheme 풴 of 풳 with generic fiber of dimension 푑 − 1. We can enlarge 푇 so that 풴 is flat, ensuring that the dimension is constant. Let 풵 denote the open subscheme of 풳 obtained by removing 풴.
Its fibers are geometrically irreducible of dimension 푑. We need to show that 풴(F푣) * 풳 (F푣),
or that 풵(F푣) ̸= ∅, for almost all 푣.
137 Let us apply the Lang-Weil bounds, i.e., Theorem 7.7.1(iv) of [Poo17], with 푌 = Spec 풪푘,푇
and 푋 = 풵. Noting that 풵F푣 is geometrically integral, 풵 has a smooth F푣-point for 푞푣 sufficiently large, so the result is true.
8.2 Appendix B: Reformulation in Terms of Cosheaves
Definition 8.2.1. Let 퐶 be a site. Then a precosheaf ℱ on 퐶 is a functor ℱ : 퐶 → Set.
Definition 8.2.2. Let 퐶 be a site. Then a cosheaf ℱ on 퐶 is a precosheaf such that for any
푈 ∈ ob(퐶) and covering {푈푖}푖 of 푈, the natural map
(︃ )︃ ∐︁ −→ ∐︁ 푔 colim ℱ(푈푖 ×푈 푈푗) −→ ℱ(푈푖) −→ℱ(푈) 푖,푗 푖 is an isomorphism.
Definition 8.2.3. In the notation of Definition 8.2.2, if 푔 is only assumed to be injective, we call 퐶 a separated precosheaf.
For any scheme 푋, we let 푋Zar denote the (big or small) Zariski site of 푋.
Definition 8.2.4. Let 푅 be a 푘-algebra and 푋 a scheme over 푘. There is a precosheaf ℱ푅 on 푋Zar associating to every 푈 ∈ ob(푋Zar) the set 푈(푅).
Lemma 8.2.5. The precosheaf ℱ푅 is always separated. If 푅 is a local ring (e.g. a field), then ℱ푅 is a cosheaf.
Proof. Since the lemma is about any scheme 푋, it suffices to consider Zariski open covers
{푈푖}푖 of the whole scheme 푋.
Consider any two elements of the domain of 푔 whose images in ℱ푅(푋) are equal. Let ∐︁ 푥1, 푥2 ∈ ℱ푅(푈푖) be representatives for these. Suppose 푥1 ∈ ℱ푅(푈푖) and 푥2 ∈ ℱ푅(푈푗). The 푖 universal property of fiber products of schemes then gives us a map 푥1×푥2 : Spec 푅 → 푈푖×푋 푈푗. ∐︁ Thus 푥1 and 푥2 each come from an element of ℱ(푈푖 ×푈 푈푗), so they correspond to the 푖,푗 same element of the domain of 푔. It follows that 푔 is injective, so ℱ푅 is separated.
138 Now suppose that 푅 is a local ring. Let 푥 : Spec 푅 → 푋. Then the closed point of Spec 푅
maps to some physical point of 푋, which must be contained in 푈푖 for some 푖. Thus the
preimage of 푈푖 under 푥 contains the closed point of Spec 푅, so it is all of Spec 푅. It follows
that 푥 factors as Spec 푅 → 푈푖 → 푋, i.e., comes from ℱ푅(푈푖). As 푥 was arbitrary, the map 푔
from Definition 8.2.2 is surjective. As 푔 is surjective and injective, it is bijective, so ℱ푅 is a cosheaf.
Example 8.2.6. Let 푋 = 1 . If 푆 contains at least two elements, then ℱ is not a cosheaf. PQ A푘,푆
Indeed, let 푣1, 푣2 ∈ 푆. Let 훼 ∈ 푋(A푘,푆) have coordinate 0 in the 푣1-component and ∞ in
the 푣2-component. Then 훼 is a A푘,푆-point of neither 푋 ∖ {0} nor 푋 ∖ {∞}, so the cosheaf condition is violated.
Definition 8.2.7. Let (휔, 푆, 푘) be an obstruction datum and 푋/푘 a scheme. By Proposition
휔 4.3.8, there is a precosheaf 푋(A푘,푆) associating to 푈 ∈ ob(푋Zar) the set
휔 푈(A푘,푆) .
C ,휔 Furthermore, in the terminology of Section 4.3.1, we denote by 푋(A푘,푆) the precosheaf associating to 푈 the set
C ,휔 푈(A푘,푆) .
Definition 8.2.8. By Theorem 2.1(a) of [Pra16], the inclusion functor from cosheaves on 퐶 to precosheaves on 퐶 has a right adjoint, known as cosheafification.
풪풫ℰ풩 ,휔 Proposition 8.2.9. Suppose that 푋(A푘,푆) is a separated precosheaf (e.g., suppose the 풪풫ℰ풩 ,휔 answer to Question 4.3.2 is yes for (휔, 푆, 푘)). Suppose 푋 is a variety over 푘 or 푋(A푘,푆) 휔 풪풫ℰ풩 ,휔 is a cosheaf. Then the cosheafification of 푋(A푘,푆) is 푋(A푘,푆) .
풪풫ℰ풩 ,휔 Proof. We first must show that 푋(A푘,푆) is a cosheaf. Let 푈 ∈ ob(푋Zar), and let
{푈푖}푖∈퐼 be a finite open cover of 푈. We need to show that the associated 푔 from Definition 8.2.2 is surjective and injective. We start by showing that it is surjective.
풪풫ℰ풩 ,휔 풪풫ℰ풩 ,휔 We therefore need to show that any 훼 ∈ 푈(A푘,푆) is in 푈푖(A푘,푆) for some 푖. If we are not already assuming this fact, we are assuming that 푋 is a variety. As 푘 is
139 countable, there are countably many Zariski open subsets of 푋 and therefore countably
many finite open covers of 푈. We enumerate all finite open covers of 푈 as 풲푘 = {푊푗푘}푗 for
푘 ∈ Z>0. For each positive integer 푘, we let 풯푘 = {푇푗푘}푗 be the common refinement of 풲푙 for
1 ≤ 푙 ≤ 푘, so that 풯푘+1 is always a refinement of 풯푘, and the system is still cofinal. Finally, we let 푉푖푗푘 = 푈푖 ∩ 푇푗푘, and we let 풱푘 denote the open cover {푉푖푗푘}푖푗 (where 푘 is fixed).
As a result, the system 풱푘 of open covers satisfies the following properties:
∙ The system is cofinal as 푘 ranges over the positive integers
∙ The covering 풰푘+1 is a refinement of 풰푘
∙ For each 푖 and 푘, the collection {푉푖푗푘}푗 is a covering of 푈푖
Furthermore, for fixed 푖, the open covers {푉푖푗푘}푗 form a cofinal system of open covers of
푈푖 as 푘 varies, and each such covering is a refinement of the previous one.
For every 푘 ∈ Z>0, let 퐼푘 ⊆ 퐼 be the set of 푖 ∈ 퐼 for which there exists 푗 such that 풪풫ℰ풩 ,휔 훼 ∈ 푉푖푗푘(A푘,푆). Because 훼 ∈ 푈(A푘,푆) , the set 퐼푘 is nonempty for all 푘. Furthermore, ⋂︁ 퐼푘+1 ⊆ 퐼푘, as 풰푘+1 is a refinement of 풰푘. As 퐼 is finite, the set 퐼푘 is nonempty. Choose 푖 푘 in this intersection.
풪풫ℰ풩 ,휔 We conclude that 훼 ∈ 푈푖(A푘,푆) , since the open covers {푉푖푗푘}푗 form a cofinal system
of open covers of 푈푖. It follows that 푔 is surjective. 풪풫ℰ풩 ,휔 Finally, 푔 is injective by assumption. It follows that 푋(A푘,푆) is in fact a cosheaf. 휔 Now let 풢 be a cosheaf. It suffices to show that every precosheaf map 푓 : 풢 → 푋(A푘,푆) 풪풫ℰ풩 ,휔 comes from a unique map 풢 → 푋(A푘,푆) . Uniqueness follows from the injectivity of 풪풫ℰ풩 ,휔 휔 푋(A푘,푆) → 푋(A푘,푆) . It is simply necessary to show that the image of any such 푓 풪풫ℰ풩 ,휔 lands in 푋(A푘,푆) . 휔 Let 푈 ∈ ob(푋Zar), and let 푠 ∈ 풢(푈). We want to show that 푓(푠) ∈ 푈(A푘,푆) actually 풪풫ℰ풩 ,휔 lies in the subset 푈(A푘,푆) . For this, let {푈푖} be a covering of 푈. We want to show 휔 that 푓(푠) ∈ 푈푖(A푘,푆) for some 푖. By the cosheaf property, there exists 푖 such that 푠 is the 휔 corestriction of some element 푠푖 ∈ 풢(푈푖). But then 푓(푠푖) is the element of 푈푖(A푘,푆) we seek, so we are done.
140 풪풫ℰ풩 ,휔 Under the assumption about Question 4.3.2, Lemma 8.2.5 implies that 푋(A푘,푆) is a cosheaf.
Our main results can therefore be thought of as a description of the cosheafification of
ℱ(f−ab,푓,푘) or ℱ(f−cov,푓,푘).
Furthermore, the interest of Question 8.1.1 is based on the fact that ℱA푘,푆 is not a cosheaf.
Finally, in this language, this cosheafification contains ℱ푘 precisely because ℱ푘 maps to
ℱA푘,푆 and is a cosheaf.
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