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Obstructions to Rational and Integral Points David Alexander Corwin

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Obstructions to Rational and Integral Points David Alexander Corwin 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 S-unit equation, which asks for solutions to x + y = 1 with x and y both S-units, or units in Z = Z[1=S]. This is equivalent to finding the set of Z-points 1 of P n f0; 1; 1g. 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 p-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=S]. One of the reasons for doing this is to verify cases of Kim’s conjecture, which states that these p-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 n f0; 1; 1g 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 A(Z) ............................. 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 Z = Spec Z[1=S] ....................... 52 3.3.1 Abstract Coordinates for Z = Spec Z[1=S] ............... 52 3.3.2 The Geometric Step for Z[1=`] in Half-Weight 4 ............ 55 3.3.3 Coordinates on the Galois Group for Z = Z[1=`] ............ 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 S3-Symmetrization . 61 3.4.1 Answer to Question 3.1.24 . 61 3.4.2 S3-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 k is Imaginary Quadratic . 85 5.2.2 The Result When k 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 k ......................... 94 6 The Fibration Method 97 6.1 The Homotopy Section Conjecture . 97 6.1.1 Basepoints and Homotopy Groups .
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