Restriction of Scalars, the Chabauty–Coleman Method, and P

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Restriction of Scalars, the Chabauty–Coleman Method, and P1 r {0, 1, ∞} by Nicholas George Triantafillou Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019

○c Nicholas George Triantafillou, MMXIX. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Author...... Department of Mathematics April 18, 2019

Certified by...... Bjorn Poonen Claude Shannon Professor of Mathematics Thesis Supervisor

Accepted by...... Davesh Maulik Chairman, Department Committee on Graduate Theses 2 Restriction of Scalars, the Chabauty–Coleman Method, and

P1 r {0, 1, ∞} by Nicholas George Triantafillou

Submitted to the Department of Mathematics on April 18, 2019 , in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

Abstract We extend Siksek’s development of Chabauty’s method for restriction of scalars of

curves to give a method to compute the set of 푆-integral points on certain 풪퐾,푆- models 풞 of punctured genus 푔 curves 퐶 over a number field 퐾. Our assumptions on 풞 guarantee that it carries a morphism 푗 : 풞 → 풥 to a commutative group scheme 풥 over 풪퐾,푆 which is analogous to the Abel-Jacobi map from a proper curve of positive genus to its Jacobian.

While Chabauty’s method (generally) requires that rank 풥 (풪퐾,푆) ≤ dim 풥퐾 − 1 in order to compute a finite subset 푝-adic points on 풞 containing 풞(풪퐾,푆), Chabauty’s method for restriction of scalars computes a subset Σ풞 of 푝-adic points of Res 풞 which contains 풞(풪퐾,푆). Naïvely, one might expect that Σ풞 is finite whenever the RoS inequality is satisfied. However, even if this rank 풥 (풪퐾,푆) ≤ [퐾 : Q](dim 풥퐾 − 1) inequality is satisfied, Σ풞 can be infinite for geometric reasons, which we call base change obstructions and full Prym obstructions. When attempting to compute the -points of 1 , we show that 풪퐾,푆 풞 = P r {0, 1, ∞} 풞 can be replaced with a suitable descent set D of covers 풟, such that for each 풟 ∈ D the RoS Chabauty inequality holds for 풟. Although we do not prove that the Σ풟 are finite, we do prove that the Σ풟 are not forced to be infinite for any of the known geometric reasons. In other words, there are no base change or full Prym obstructions to RoS Chabauty for 풟. We also give several examples of the method. For instance, when both 3 splits completely in and is prime to we show that 1 . We 퐾 [퐾 : Q] 3 (P r {0, 1, ∞})(풪퐾 ) = ∅ also give new proofs that 1 is finite for several classes of number (P r {0, 1, ∞})(풪퐾 ) fields 퐾 of low degree. These results represent the first infinite class of cases where Chabauty’s method for restrictions of scalars is proved to succeed where the classical Chabauty’s method does not.

Thesis Supervisor: Bjorn Poonen Title: Claude Shannon Professor of Mathematics

3 4 Acknowledgments

First and foremost, I would like to thank my advisor, Bjorn Poonen. Bjorn, it is no exaggeration to say that this thesis would not exist without you. Over the past five years, I have learned so much from you, both about mathematics and how to be a good mentor. Thank you for your many patient hours answering my mathematical questions, for sharing your ideas, and for helping me to refine my own ideas. Thank you for letters of recommendation (too often at the last minute), for your mathematical guidance, and for giving me the freedom and support to explore and grow both mathematically and personally. And thank you for believing in me even when I had trouble believing in myself. I would not be the mathematician that I am without you. Among the many things I will miss about MIT, our weekly meetings to discuss my progress will be near the top of the list.

Next and equally much, I would like to thank my parents, Jean Farrington and George Triantafillou. Mom and Dad, thank you so much for being there forme throughout my life. Thank you for answering my first-grade request for extra math homework and for always nurturing my love of learning. Thank you for always being there with wise advice and a listening ear when I needed it. Thank you for understanding when the demands of graduate school kept me from spending as much time with you (or even on the phone with you) as we would all have liked. Thank you for listening and caring about what I do, even when you don’t understand most of what I’m talking about. I couldn’t have asked for better parents. I love both of you to the moon and back. Thank you also to all the rest of my family, especially my Yiayia and my aunt Maryrose. You have all been a tremendous source of support.

I have many mathematical mentors to thank. Thank you to my committee mem- bers, Wei Zhang and Andrew Sutherland, for taking the time to read my thesis and for your thoughtful questions. Thank you to Jennifer Balakrishnan and Michael Stoll for introducing me to Chabauty’s method as a topic of active research. Thank you to Henry Cohn, for teaching me about sphere packing and how ideas from pure and applied mathematics can work together. Thank you to Stephen DeBacker, Steven

5 Miller, and all of my other undergraduate mentors, for nurturing my love of mathematics and for doing the same for many others. Without your mentorship, I expect my career path would have been very different. And thank you to John Voight, David Zurieck-Brown, Joe Rabinoff, Barry Mazur, Tim Dokchitser, Noam Elkies, David Vo- gan, George Lustig, Edgar Costa, Adam Morgan, all of those listed above, and many more for helpful mathematical conversations, and for their belief in me in graduate school.

My fellow math students have been a huge source of support and knowledge thoughout my time in graduate school. Padma Srinivasan, thank you for always being there to remind me of the sense of wonder that mathematics inspires. Isabel Vogt, thank you for pushing from the very start of graduate school to learn more mathematics. You set a great example, helping me to overcome burn-out and to keep learning as I started graduate school. Ruthi Hortsch, thank you for convincing me that MIT would be a good fit for me for graduate school. Renee Bell, Vishal Arul, and Borys Kadets, thank you for listening to my frequent rants, both mathematical and non-mathematical. I realize I must have often been a distraction in the office - I hope it was more commonly welcome than unwelcome. To those above, Alex Best, Atticus Christiansen, Alex Smith, Sachi Hashimoto, and many others, thank you for many helpful mathematical conversations at tea, at conferences, at STAGE, and more. Thank you to Amelia Perry, who left us too soon. And to all of those above, Lynnelle Ye, Kevin Sackel, Paul Gallagher, Hood Chatham, Geoff Smith, Gwen McKinley, David Rolnick, and many more, thank you for your friendship and for helping me to stay sane throughout graduate school.

The Sidney-Pacific Graduate Community did so much to help me maintain balance in graduate school. Thank you to Rachael Harding and Sherry Hall for introducing me to the community. Thank you to Dan Kolodrubetz and Fabian Kozynski for inspiring me to be the best coffee hour chair I could be and for showing mehow I could give back to the community. Thank you to Boris Braverman and Ahmed Helal, for your sage advice and for setting an incredible example of how to support a community, finish a PhD and have a great time while doing it. Thank youto

6 Jenny Wang for holding me accountable and making sure I did what was best for the community. Thank you to Andrea Carney for always being willing to listen. Thank you to Alberto Rodriguez, Nuria Jane, Julie Shah, Neel Shah, and Brian Ward for their mentorship and support. Thank you to Eric Stansifer, Leo Barlach, Jit Hin Tan, Hsin-Yu Lai, Zelda Mariet, Fabian, and Ahmed, for many great hours spent on board games and puzzle hunts. Thank you to the rest of my SPEC, Greg Izatt, Ben Yuan, Akshay Mohan, and Richard Swartwout, for being a great team. It was a real pleasure to work with all of you, pulling each other along when we needed it, and I would do it again in a heartbeat. Thank you to all of our officers who turned our ideas into reality and to the SPECs, Trustees, and officers throughout SP’s history who created this great community. And thank you again to Kevin and Greg for being great roommates and friends. I hope we’ll all cross paths often in the future. Thank you to Peng Shi, Annie Chen, Sam Elder, Grace Goon, and many more from the Graduate Christian Fellowship, for showing me how to pair a life of science and faith. Thank you to Peter Su, Sarah Goodman, Daniel Curtis, Lisa Guay, Orpheus Chatzivasileiou, Malvika Verma, and many more for many good times and much good work in service of the graduate community on the Graduate Student Council. Thank you to Ian Waitz and Jag Patel and many other MIT administrators for your mentorship and support of student advocacy. And thank you to those I have forgotten or omitted for space. I have been blessed in graduate school and life to have many wonderful mentors and friends. I feel like I could have filled a novel with the names of those who have made a lasting positive impression on my life. I hope those deserving folks that I have omitted here will forgive me for the omission and know that I value them.

7 8 Contents

1 Introduction 11 1.1 Overview of methods and results...... 11 1.2 Chabauty in dimension greater than 1: History...... 21 1.3 Other improvements to Chabauty’s method...... 25

1.4 History of the 푆–unit equation...... 29 1.5 Outline of this thesis...... 31

2 Preliminaries 33 2.1 Notation and conventions...... 33 2.2 Generalized Jacobians...... 36 2.3 Descent...... 46

3 The Method of Chabauty-Coleman-Skolem 53 3.1 Chabauty’s result...... 53

3.2 The 푝-adic logarithm map...... 55 3.3 Coleman’s approach...... 58

4 Chabauty for Restrictions of Scalars 61 4.1 Introduction to RoS Chabauty...... 61 4.2 Base change obstructions...... 65 4.3 Full Prym obstructions...... 67 4.4 Simplifying regular differentials via base change...... 72

9 5 RoS Chabauty and Genus 0 Descent: General Theory 75 5.1 Classical Chabauty and genus 0 descent...... 76 5.2 RoS Chabauty and genus 0 descent...... 78

6 RoS Chabauty and P1 r {0, 1, ∞}: Computations 93 6.1 RoS Chabauty and P1 r {0, 1, ∞}: Generalities...... 93 6.2 Finiteness of the Chabauty set: Examples...... 98 6.3 Fields where 3 splits completely...... 108

10 Chapter 1

Introduction

1.1 Overview of methods and results

Let 푋 be a smooth, proper, geometrically integral curve over a number field 퐾. When the genus 푔 of 푋 is at least 2, Faltings’ theorem says that the set 푋(퐾) is finite. Unfortunately, Faltings’ theorem is ineffective. While it is possible to extract

an (extremely large) bound on #푋(퐾) from a careful analysis of Faltings’ method, it seems implausible that Faltings’ approach could be used to compute the set 푋(퐾). Fortunately, for many curves 푋, there is another way. Building on earlier work of Chabauty, Coleman proved:

Theorem 1.1.1 ([Col85, Theorem 4]). Suppose that 퐾 = Q, that 푔 > 1, and that 푝 is a prime of good reduction for 푋 with 푝 > 2푔. Let 퐽 be the Jacobian of 푋. Suppose

that rank 퐽(Q) ≤ 푔 − 1. Then

(1.1.2) #푋(Q) ≤ #푋(F푝) + (2푔 − 2) .

The great strength of the Chabauty–Coleman method over Faltings’ proof is that the bound (1.1.2) is sometimes sharp [GG93], in which case Theorem 1.1.1 can be

used to compute 푋(Q). Even when (1.1.2) is not sharp, the method computes (to any desired -adic precision) a finite subset of which contains the set . 푝 푋(Q푝) 푋(Q) In combination with tools like the Mordell–Weil sieve, it is often possible to compute

11 the set 푋(Q) exactly. For instance, Stoll has used this approach to compute 푋(Q) for all 46,436 curves with Jacobian of rank 1 in a database of ‘small’ genus 2 curves [Sto09]. For the sake of exposition, we now give a high-level overview of the Chabauty– Coleman method. We give a more detailed exposition (in a more general context) in Chapter3.

Suppose that 푋(Q) ̸= ∅ so that the Abel–Jacobi map 푗 : 푋 ˓→ 퐽 is defined over Q. From the commutative diagram

푋(Q) 푋(Q푝)

퐽(Q) 퐽(Q푝) we see that inside , 퐽(Q푝)

푋(Q) ⊆ 푋(Q푝) ∩ 퐽(Q) .

Considering as a -adic Lie group, the ( -adic) closure of is a -adic 퐽(Q푝) 푝 푝 퐽(Q) 퐽(Q) 푝 Lie group of some dimension 휌 ≤ rank 퐽(Q). Now, has dimension as a -adic Lie group. For dimension-counting rea- 퐽(Q푝) 푔 푝 sons, it is reasonable to hope that if 휌 ≤ 푔 − 1 then the intersection

푋(Q푝) ∩ 퐽(Q) is finite. Indeed, this is what Chabauty proved. Later, Coleman developed atheory of 푝-adic integration. Using this theory in Chabauty’s setup, Coleman showed how to compute the set explicitly as the vanishing locus of a -adic analytic 푋(Q푝) ∩ 퐽(Q) 푝 function on . Such a function is given locally by single-variable -adic power 푋(Q푝) 푝 series, so methods from the theory of Newton polygons can be used to bound the size of its set of zeroes.

The restriction 퐾 = Q in Theorem 1.1.1 is not serious. So long as rank 퐽(퐾) ≤ 푔− 1, Coleman’s result generalizes to bound #푋(퐾) for any number field 퐾. Replacing the Jacobian with a generalized Jacobian, Theorem 1.1.1 can also be generalized to bound the set of 푆-integral points on an 푆-integral model of a punctured curve. (See

12 Chapter3.)

Unfortunately, the restriction rank 퐽(Q) ≤ 푔 − 1 (or more precisely 휌 ≤ 푔 − 1) is much more serious. Several methods have been proposed to extend the method of Chabauty-Coleman to handle the case where rank 퐽(퐾) ≥ 푔. In all cases, the idea is to replace the Jacobian with something ‘larger.’ We give three examples:

1. Restriction of Scalars Chabauty: If 퐾 is a number field of degree 푑 > 1, replace and with the Weil restrictions of scalars and . 푋 퐽 Res퐾/Q 푋 Res퐾/Q 퐽

2. Descent + Chabauty: Using the theory of descent, given any isogeny from an

abelian variety to 퐽, it it possible to construct a finite set of covering curves

D := {푓푖 : 푋푖 → 푋}

such that ⋃︀ . Then apply Chabauty’s method to each 푋(퐾) = 푖(푓푖(푋푖(퐾)))

curve 푋푖.

3. Non-abelian Chabauty-Kim: Replace 퐽 with a “Selmer variety” constructed from a quotient of the pro-unipotent étale fundamental group of 푋.

In this thesis, we study the power of the first two approaches, especially in the context of computing 푆-integral points on P1 r{0, 1, ∞}. Although it is not the focus of our study, the Chabauty-Kim approach plays an important motivating role for this project, as we discuss at the end of Section 1.4.

Write 푑 = [퐾 : Q]. In the method of restriction of scalars Chabauty (or RoS Chabauty for short), attributed by Siksek to a talk of Wetherell [Sik13, Wet00], one replaces the inclusion

푋(퐾) ⊆ 푋(퐾p) ∩ 퐽(퐾)

13 inside 퐽(퐾p) with the inclusion

(1.1.3) 푋(퐾) = (Res퐾/Q 푋)(Q) ⊆ (Res퐾/Q 푋)(Q푝) ∩ (Res퐾/Q 퐽)(Q) inside the -dimensional -adic Lie group . In order to prove that 푔푑 푝 (Res퐾/Q 퐽)(Q푝) 푋(퐾) is finite, we aim to show that the right-hand side is finite. Unfortunately, the right-hand side of (1.1.3) can have positive-dimension as a naïve 푝-adic analytic variety. In particular, it is infinite if this happens. So we wish to understand when this intersection is finite. As a first pass, we note that if this intersection is finite, it will typically have expected dimension less than or equal to zero. More precisely, taking dimensions of

푝-adic analytic varieties, we must have

dim(Res퐾/Q 푋)(Q푝) + dim (Res퐾/Q 퐽)(Q) ≤ dim(Res퐾/Q 퐽)(Q푝) , or equivalently

(1.1.4) dim (Res퐾/Q 퐽)(Q) ≤ 푑(푔 − 1) .

Since

dim (Res퐾/Q 퐽)(Q) ≤ rank(Res퐾/Q 퐽)(Q) = rank 퐽(퐾) , we see that (1.1.4) holds whenever

rank 퐽(퐾) ≤ 푑(푔 − 1) . (1.1.5)

If (1.1.5) is true, we say that the RoS Chabauty inequality holds for 푋. Informally, the RoS Chabauty inequality holds for 푋 exactly when we ‘expect for dimension reasons’ that RoS Chabauty will prove that 푋(퐾) is finite. As with Chabauty’s method, when the RoS Chabauty inequality holds for 푋, the intersection can be computed as the vanishing set of (Res퐾/Q 푋)(Q푝)∩(Res퐾/Q 퐽)(Q) several 푑-variable 푝-adic analytic functions, locally expressible as 푝-adic power series

14 in variables. Within each residue disc, the number of isolated -valued points in 푑 C푝 this vanishing locus is finite. In contrast to what happens in the Chabauty-Coleman method, the set

(Res퐾/Q 푋)(Q푝) ∩ (Res퐾/Q 퐽)(Q)

may not be zero-dimensional even if the RoS Chabauty inequality holds for 푋. In particular, this set may be infinite even when the RoS Chabauty inequality holds. Indeed, if this set were always finite, we could attempt to prove all of Faltings’ Theorem as follows:

1. Given 푋/Q of genus 푔 with Jacobian 퐽, apply Chabauty’s theorem if possible.

2. Otherwise, produce a sequence of number fields 퐾푖 of degree tending to infinity with . (It is likely, although not proven that such can be 퐽(Q) = 퐽(퐾푖) 퐾푖 found.)

3. Then, for some 퐾 := 퐾푖, the base change 푋퐾 satisfies the RoS Chabauty inequality. Apply RoS Chabauty to show that 푋(퐾) is finite.

4. Conclude that 푋(Q) ⊂ 푋(퐾) is also finite.

Unfortunately, there is a problem with this argument. In this situation, the set

in (Res퐾/Q 푋퐾 )(Q푝) ∩ (Res퐾/Q 퐽)(Q) = 푋(퐾 ⊗ Q푝) ∩ 퐽(퐾) 퐽(퐾 ⊗ Q푝) will contain a subset isomorphic to . Here, he failure of RoS Chabauty for 푋(Q푝) 푋퐾 is a sort of certificate of the failure of the Chabauty-Coleman method to compute the

set 푋(Q). This principle applies more broadly: If there is some 휅 ⊂ 퐾 and a curve 푌/휅 such that both

∼ 1. 푌퐾 = 푋 and

2. is infinite (Res휅/Q 푌 )(Q) ∩ (Res휅/Q 퐽푌 )(Q)

15 then will be infinite as well. When this occurs we (Res퐾/Q 푋)(Q) ∩ (Res퐾/Q 퐽푋 )(Q) say that 푋 has a base change obstruction to RoS Chabauty. See Section 4.2 for a more detailed discussion. Recently, [Dog19] pointed out another geometric situation which can force (Res퐾/Q 푋)(Q)∩ to be infinite. If there is a curve and a morphism (Res퐾/Q 퐽푋 )(Q) 푌/퐾 푓 : 푋 → 푌 such that

1. * has finite index in * and (Res퐾/Q 퐽푋 /푓 (퐽푌 ))(Q) (Res퐾/Q 퐽푋 /푓 (퐽푌 ))(Q푝)

2. is infinite (Res휅/Q 푌 )(Q) ∩ (Res휅/Q 퐽푌 )(Q)

then will be infinite as well. When this occurs we (Res퐾/Q 푋)(Q) ∩ (Res퐾/Q 퐽푋 )(Q) say that 푋 has a full Prym obstruction to RoS Chabauty. See Section 4.3 for a more detailed discussion.

It seems natural to hope that RoS Chabuaty can prove that 푋(퐾) is finite whenever 푋 satisfies the RoS Chabauty inequality and has no base change obstructions or full Prym obstructions. While there are no known counterexamples, the hope seems extremely difficult to prove. While general results may be out of range of current technology, RoS Chabauty is useful in particular examples. Siksek [Sik13] gives a sufficient criterion to determine if the intersection

(Res퐾/Q 푋퐾 )(Q푝) ∩ (Res퐾/Q 퐽)(Q) √ 3 is finite and uses it to compute the setof Q( 2)-points on several genus 2 curves.

Let 푆 be a finite set of finite places of 퐾 and let 푂퐾,푆 be the ring of 푆-integers of 퐾. In this work, we develop the method of RoS Chabauty to compute integral points on curves, in particular to compute the set ( 1 {0, 1, ∞})(풪 ). This is the set P풪퐾,푆 r 퐾,푆 of solutions to the 푆-unit equation, i.e., the set

× × {(푥, 푦) ∈ 풪퐾,푆 × 풪퐾,푆 : 푥 + 푦 = 1} .

The 푆-unit equation is a classical and well-studied number theoretic object. Work of Mahler [Mah33] using diophantine approximation in the style of Siegel and Thue

16 gave the first proof that this set is finite. The theory of linear forms in logarithms can

be used to give explicit bounds on the size of the set. Let 푟1 and 푟2 be the number of real embeddings and conjugate pairs of complex embeddings of 퐾. Then, the current state-of-the-art upper bound, due to Evertse [Eve84], is 3 · 72#푆+3푟1+4푟2 . This theory has been developed to give efficient algorithms, implemented in Sage, to compute

solutions to 푆-unit equations [AKM+18, Sma99]. We give a more detailed review of results on 푆-unit equations and their application in Section 1.4. Although the 푆-unit equation has been well-studied, it remains an important testing ground for new approaches to reprove Faltings’ Theorem. Both Kim’s nonabelian

Chabauty program [Kim05] and Lawrence and Venkatesh’s recent 푝-adic proof of Faltings’ Theorem [LV18] first found success in proving finiteness of solutions toan

푆-unit equation. And in both cases, studying the example of P1 r {0, 1, ∞} helped to shine a light on the full theory.

We hope that our study of P1 r {0, 1, ∞} will do the same for RoS Chabauty. While RoS Chabauty alone is not enough to solve the 푆-unit equation, we are able to use the method (and related ideas) to recover several finiteness results for unit- equations with stronger bounds than are given by Evertse’s more general theorem. For example, we prove

Theorem 6.3.1. Suppose that [퐾 : Q] is not divisible by 3 and that 3 splits completely in . Then there is no pair × such that . Equivalently, 퐾 푥, 푦 ∈ 풪퐾 푥 + 푦 = 1

1 (P r {0, 1, ∞})(풪퐾 ) = ∅.

While Theorem 6.3.1 holds independent of the degree of the field, our other results require strong assumptions on the size of the field. In a series of examples (Corollar- ies/Propositions 6.2.1, 6.2.3, 6.2.4, and 6.2.5) we show that if 퐾 is a real quadratic field, mixed cubic field, or totally complex quartic field, or a mixed quartic fieldwith a totally real subfield, then

1 (1.1.6) #(P r {0, 1, ∞})(풪퐾 ) < ∞ .

17 This result provides the first infinite collection of examples of fields for whichRoS Chabauty has been proven to show finiteness of integral or rational points on a curve but a straightforward application of classical Chabauty cannot prove finiteness. Notably, for quadratic fields, the proof is sufficiently uniform in the field thatit allows us to recover the elementary fact that there are finitely many solutions to the unit equation in the union of all quadratic fields [Nag64]. (In fact, there are exactly

8 such solutions!) Dan-Cohen and Wewers [DCW15] also recover a version of this result when developing explicit algorithms for computing 푆-unit equations based on Kim’s nonabelian Chabauty. Having discussed RoS Chabauty, we now turn to our second ingredient, the method of descent. Before stating our results, we need a definition

Definition 2.3.1. A descent set for (풞, 풪퐾,푆) is a finite set D of clearance hole curves

풟 over 풪퐾,푆 (see Definition 2.1.1) equipped with morphisms 푓풟 : 풟 → 풞 such that

⋃︁ 풞(풪퐾,푆) ⊆ 푓풟(풟(풪퐾,푆)) . 풟∈D

If, moreover, the generic fibers 풟퐾 of all of the curves 풟 are punctured genus 0 curves, we say that D is a genus 0 descent set.

We make an analogous definition for sets of covering curves of a proper curve 푋 over 퐾. Methods for constructing descent sets are well-established. We discuss the method of descent in detail in Section 2.3.

If 푋 has genus at least 2, the Riemann-Hurwitz formula implies that the curves

푋푖 in a descent set D for 푋 will have higher genus than 푋. Alternatively, if 풳 is a punctured genus zero curve and D is a genus 0 descent set, the curves 풳푖 will have more punctures than 풳 . As in section 2.2, define the generalized Jacobian of a punctured curve 풳 to be a certain 풪퐾,푆-model for the generalized Jacobian of the projective closure of the special fiber of 풳 with modulus equal to the divisor consisting of the sum of the punctures, each with multiplicity 1.

18 Then, in either case, the dimension of the (generalized) Jacobian of the 풳푖 will be higher than the dimension of the (generalized) Jacobian of the 풳 . If the ranks of the generalized Jacobians do not grow as quickly as the dimensions, one could attempt to apply Chabauty’s method to the curves 풳푖 and use this to determine the set 풳 (풪퐾,푆). When 퐾 = Q and 푆 is arbitrary, Poonen observed in a July 2005 email to Kim that one can use this approach to prove that

1 (P r {0, 1, ∞})(풪Q,푆) is finite. The following result shows that with only descent bygenus 0 covers and the classical Chabauty-Chabauty method (as described in Chapter3), one cannot do much better.

Corollary 5.1.3. Suppose that we are not in the following situations: (i) 퐾 = Q, (ii) 퐾 a real quadratic field and #푆 ≤ 1, (iii) 퐾 an imaginary quadratic or totally real cubic field and #푆 = 0. Then the classical Chabauty inequality (3.2.6) is not satisfied by any descent set consisting of genus zero covers of 1 {0, 1, ∞}. Under Leopoldt’s Conjecture, this P풪퐾,푆 r implies that the combination of classical Chabauty and descent by genus zero covers is insufficient to prove that

( 1 {0, 1, ∞})(풪 ) P풪퐾,푆 r 퐾,푆 is finite.

While the classical Chabauty inequality is never satisfied by descent sets consisting of punctured genus zero covers of 1 {0, 1, ∞}, it is not difficult to construct P풪퐾,푆 r such descent sets where the RoS Chabauty inequality is satisfied. So, it is reasonable to hope that the method of RoS Chabauty with descent could be used to prove that

1 is finite. (P r {0, 1, ∞})(풪퐾,푆) It is difficult to prove in full generality when the (analogue of the) right-hand side of (1.1.3) is finite (so that RoS Chabauty will suffice to prove finiteness ofrational

19 points). However, in all known cases where the right-hand side is infinite, the curve

풞, this infinitude can be attributed to a combination of base change obstructions (see Section 4.2) and full Prym obstructions (see Section 4.3.) In several circumstances, we prove that it is possible to avoid base change obstructions and Prym obstructions using descent.

To state our results, we need a definition. We say that a number field 휅 is CM if 휅 is totally complex and is a degree 2 extension of a totally real number field.

Theorem 5.2.2. The punctured curve 1 {0, 1, ∞} has a descent set consisting P풪퐾,푆 r D of genus 0 sound clearance hole curves 풟 = 1 Γ (see Definition 2.1.1) such P풪퐾,푆 r 풟 that I. the RoS Chabauty inequality

rank 풥풟(풪퐾,푆) ≤ 푑(#Γ풟(퐾) − 2). (5.2.2) holds for all 풟 ∈ D. II. Under the further assumption that 퐾 does not contain a CM field, D can be chosen so that there is no base change obstruction to RoS Chabauty for (풟, 풪퐾,푆) for any 풟 ∈ D.

Theorem 5.2.4. Let 퐾 be a number field which does not contain a CM subfield and let 푞 be a prime. Fix an 훼 ∈ 퐾 which is not a 푞th power in 퐾 and let

풞 := 1 {푥 ∈ 퐾 : 푥푞 − 훼 = 0} . P풪퐾,푆 r

For 푞 sufficiently large (depending only on 퐾 and 푆), there is no full Prym obstruction to RoS Chabauty for 풞. Moreover, if 풞 is the base change of some curve 풟, there is no full Prym obstruction to RoS Chabauty for 풟. Also, there is no base change obstruction to RoS Chabauty for 풞.

While Theorems 5.2.2 and 5.2.4 may not give much new information about solutions to the 푆-unit equation, we emphasize that our primary goal in this thesis is not to prove new bounds on the number of solutions the 푆-unit equation, but rather

20 to understand the power of RoS Chabauty in combination with descent. We focus on genus 0 covers of P1 r {0, 1, ∞} because studying the ranks of their Jacobians is possible. On the other hand, understanding ranks of Jacobians of higher genus curves seems well beyond the range of current mathematical technology. Our work also helps to clarify what can be said about the 푆-unit equation using purely 푝-adic methods. It would be interesting to combine our ideas with stronger results from the theory of linear forms in 푝-adic logarithms to understand this more completely.

1.2 Chabauty in dimension greater than 1: History

Although Chabauty’s method is traditionally discussed in context of rational points on curves, the high-level idea of the method also applies to more general projective varieties 푉 equipped with a morphism 푗 : 푉 → 퐴 to an abelian variety 퐴 such that dim(푉 ) + rank 퐴(퐾) ≤ dim(퐴) and such that we understand the rational points of the fibers of 푗. Of course, the restriction that we understand the rational points of the fiber of 푗 rules out many classes of variety 푉 (for instance K3 surfaces, which have only constant maps to abelian varieties). Both restrictions of scalars of curves and symmetric powers of curves are examples of varieties 푉 which are well-suited to Chabauty’s method. In this section, we briefly summarize past results in these settings. Given a curve 푋/Q and a degree 푑 extension 푑 퐾 of Q, we also compare the application of Chabauty’s method to Sym 푋 and . Res퐾/Q 푋

1.2.1 RoS Chabauty and variants

In this thesis, we focus on the variant of Chabauty’s method applied to the restriction of scalars of a curve. In this situation, [Sik13] gives an explicit criterion to determine if several multivariable 푝-adic power series have a single zero in a residue disc. For several genus 2 curves √ 3 푋, he is able to use this criterion and RoS Chabauty to compute the set 푋(Q( 2)). This allows him to determine the set of (푥, 푦, 푧) ∈ Z3 such that 푥2 + 푦3 = 푧10. Siksek

21 attributes the idea to a talk of [Wet00], where the set of Q[푖]-points on a bielliptic curve was computed. Unfortunately, the results of [Wet00] have never been published, even as a preprint.

Given number fields 퐾 ⊂ 퐿 and curves 퐵/퐾 and 퐶/퐿 with a non-constant mor-

phism ℎ: 퐶 → 퐵퐿,[FT15] uses related ideas (involving both restriction of scalars and Chabauty-like techniques) to bound/compute the subset of 푃 ∈ 퐶(퐿) such that ℎ(푃 ) ∈ 퐵(퐾).

1.2.2 Comparison with symmetric powers Chabauty

Several others [Sik09], [Par16], [Kla93], [GM17] have considered a similar variant of Chabauty’s method that applies to symmetric powers of a curve.

Recall that RoS Chabauty starts with a variety 푋 over a degree 푑 number field 퐾 and attempts to bound the set

(Res퐾/Q 푋)(Q푝) ∩ (Res퐾/Q 퐽)(Q)

inside of , in order to compute . We ‘expect’ that RoS Chabauty (Res퐾/Q 퐽)(Q푝) 푋(퐾) will prove that 푋(퐾) is finite when rank 퐽(퐾) ≤ 푑(푔 − 1).

Similarly, symmetric powers Chabauty starts with a curve 푋 defined over Q (say with a rational point 푃 ∈ 푋(Q).) The Abel-Jacobi map 푋 → 퐽 extends to a map 푗 : Sym푑 푋 → 퐽. Geometrically, the fibers of 푗 are projective spaces whose dimensions are well-understood. In particular, we understand their rational points well. Symmetric powers Chabauty attempts to bound the set

푑 푗(Sym 푋)(Q푝) ∩ 퐽(Q)

inside in order to bound the number of fibers of which contain rational points. 퐽(Q푝) 푗 One might hope to use this bound to bound 푑 . #(Sym 푋)(Q푝) 푑 Unfortunately, (Sym 푋)(Q) is frequently infinite. For example, 푋 has a mor- 푑 phism of degree ≤ 푑 to P1 or an elliptic curve 퐸 of positive rank then (Sym 푋)(Q)

22 will be infinite.

if 푑 is greater than either the gonality of 푋, it will contain a copy of the projective line. If 푑 is greater than the degree of a map from 푋 to an elliptic curve of positive rank. Nevertheless, if one excludes a special set 풮(Sym푑 푋), we ‘expect’ that symmetric power Chabauty will prove that

푑 푑 (Sym 푋 r 풮(Sym 푋))(Q) is finite when rank 퐽(Q) ≤ 푔 − 푑 − 1. Following similar lines to our development of RoS Chabauty, it should be possible to adapt Chabauty’s method for symmetric powers to study integral points on symmetric powers of affine curves.

For 푋/Q, symmetric powers Chabauty may seem strictly superior to RoS Chabauty for computing 푋(퐾) when both methods apply. After all, it gives information about

푋(퐾) for all étale Q-algebras of degree 푑 simultaneously. Moreover, symmetric powers Chabauty may apply even if the RoS Chabauty inequality is not satisfied. Despite these flaws, RoS Chabauty still provides additional value. First, RoS

Chabauty gives access to the 퐾-points in the special set. For example, in the case where [퐾 : Q] = 2 and 푋 is a smooth proper hyperelliptic curve with affine patch 푦2 = 푓(푥), RoS Chabauty gives information about how many points with 푥-coordinate in Q belong to 푋(퐾). Symmetric power Chabauty has nothing to say about this, since all such points are defined over some quadratic extension. Second, RoS Chabauty may apply even when symmetric power Chabauty does not. Finally, RoS Chabauty includes symmetric power Chabauty as a special case in the following sense. Fix a nice curve 푋/Q and let 퐽 be the Jacobian of 푋. For any number field of degree , there is a norm map . 퐾 푑 Nm : Res퐾/Q 퐽퐾 → 퐽 Let be the map induced by the Abel-Jacobi map. 푗 : Res퐾/Q 푋퐾 → Res퐾/Q 퐽퐾 Now, we claim that the composition Nm ∘푗 factors through Sym푑 푋. Indeed, given a -algebra , any maps to a -tuple of ‘conjugate’ elements of Q 퐴 푃 ∈ (Res퐾/Q 푋)(퐴) 푑 푋(퐾 ⊗퐴), or equivalently to an element of (Sym푑 푋)(퐴). The Abel-Jacobi map from

23 (Sym푑 푋)(퐴) → 퐽(퐴) is just the sum of the images of the elements in the 푑-tuple. The composition clearly agrees with Nm ∘푗. As a result, there is a commutative cube

(Res퐾/Q 푋)(Q) (Res퐾/Q 푋)(Q푝)

푑 푑 (Sym 푋)(Q) (Sym 푋)(Q푝)

(Res퐾/Q 퐽)(Q) (Res퐾/Q 퐽)(Q푝)

퐽(Q) 퐽(Q푝)

Inside , we have 퐽(Q푝)

푑 퐽(Q) ∩ (Res퐾/Q 푋)(Q푝) ⊆ 퐽(Q) ∩ (Sym 푋)(Q푝) .

In particular, this implies that the bound from RoS Chabauty on #푋(퐾) will never be worse than the bound which could be extracted naïvely from the result of symmetric powers Chabauty. This observation can be interpreted as saying that symmetric powers Chabauty is

a version of RoS Chabauty that holds uniformly for every base change of 푋 to étale

algebra 퐾 with 푑 = [퐾 : Q]. We will see a further example of this when we consider 1 for a (real) quadratic field in section 6.2.1. (P r {0, 1, ∞})(풪퐾 ) 퐾 Having introduced and compared RoS Chabauty and symmetric powers Chabauty, we now survey results from both approaches. The first result on symmetric powers Chabauty appears in[Kla93]. In [Sik09], Siksek proved a sufficient criterion for a residue disk to contain asingle

2 rational point, computed (Sym 푋)(Q) in two explicit examples, and a hypothesis on the gonality of 푋 required in [Kla93]. Park developed this work further in [Par14] and [Par16], proving a bound on

푑 푑 #(Sym 푋 r풮(Sym 푋))(Q) depending only on 푔, 푑, and a prime 푝 of good reduction, under a difficult-to-verify technical assumption on 푋. Unfortunately, there is an error in the proof of [Par16, Theorem 4.15 and Theorem 4.18] (and there are counter-

24 examples to the statements.) Most recently, Gunther and Morrow [GM17] further refined [Par16] and combined this work with results from [BG13] on average ranks of hyperelliptic curves. For a

positive proportion of genus 푔 hyperelliptic curves over Q with a rational Weierstrass point, they give an explicit bound for the number of quadratic points not obtained

by pulling back points of P1(Q). Unfortunately, their bounds appear to depend on [Par16, Theorem 4.15 and Theorem 4.18].

1.3 Other improvements to Chabauty’s method

Many other authors have considered variants of Chabauty’s method in an attempt

to compute the 퐾 or 풪퐾,푆 points on curves. We give a brief survey of some such results here. Although a comprehensive survey would run very long, we hope this section will serve as a taste for what can be done with Chabauty’s method and will encourage the reader to explore the area further.

Lorenzini and Tucker showed how to adapt Coleman’s method to a prime 푝 of bad reduction (as in [LT02, Corollary 1.11]). Let 풳 to be a minimal proper regular model for 푋 over , and let #풳 sm. be the smooth locus of its special fiber. When Z푝 F푝 rank 퐽(Q) < 푔, they show that

#푋( ) ≤ #풳 sm.( ) + (2푔 − 2) . Q F푝 F푝

Notably, the bounds of [Col85] and [LT02] depend on 푝. A priori, the smallest bound obtained as 푝 varies may be quite large if all of the small primes are primes of bad reduction for 푋. Some recent developments in Chabauty’s method involve uniform bounds on rational points on curves satisfying a more restrictive rank hypothesis than the rank hypothesis appearing in classical Chabauty. The first such uniform bound appears in[Sto19] and applies to hyperelliptic curves

푋 whose Jacobian 퐽 satisfies with rank 퐽(Q) ≤ 푔 − 3. Perhaps the most significant

25 result in this direction is

Theorem 1.3.1 ([KRZB16]). Let 푋 be any smooth curve of genus 푔 and let 푟 =

rank 퐽(Q). Suppose 푟 ≤ 푔 − 3. Then

2 #푋(Q) ≤ 84푔 − 98푔 + 28 .

It is also known that in certain families, ‘most’ curves have very few rational

points. An argument involving equidistribution results on 2-Selmer groups from [BG13] leads to

Theorem 1.3.2 ([PS14, Theorem 10.6]). For 푔 > 1, the lower density of degree 2푔+1 hyperelliptic curves having just one rational point (namely, the point at infinity) is

at least 1 − (12푔 + 20)2−푔. In particular, as 푔 tends to infinity, the lower density of degree 2푔 + 1 hyperelliptic curves having just one rational point tends to 1.

An analogous result for even degree hyperelliptic curves is proved in [BG13]. Other results have focused on making it easier to apply the Chabauty-Coleman method in practice. For instance, one usually assumes that generators for (a finite

index subgroup of) 퐽(Q) are known when applying the method. In practice, it can be difficult to prove that a proposed generating set is complete. Stoll[Sto17] shows a way to get around computing generators of the Mordell–Weil group by replacing the

Jacobian with a 푝–Selmer set.

Yet other results have attempted to relax the restriction that rank 퐽(Q) ≤ 푔 − 1. These methods, including elliptic curve Chabauty as developed in [FW99] and [Bru03], usually apply descent in some way.

1.3.1 Kim’s nonabelian Chabauty

Kim has also proposed a vast generalization of Chabauty’s method, which replaces and with a tower of global and local Selmer varieties. These Selmer 퐽(Q) 퐽(Q푝) varieties, constructed using the pro-unipotent étale fundamental group of 푋 carry

26 the structure of algebraic varieties over . The first level of the tower (essentially) Q푝 specializes to Chabauty’s original method.

As with classical Chabauty, the Chabauty-Kim method proves 푋(Q) is finite when the dimension of the global Selmer variety (corresponding to dim 퐽(Q)) is less than the dimension of the local Selmer variety (corresponding to ). As one goes dim 퐽(Q푝) up the tower, the dimensions of both varieties grow. When the dimension of the local Selmer variety grows faster than the dimension of the global Selmer variety

(asymptotically), Kim’s method shows that 푋(퐾) is finite. Kim’s method can also

be used to show that the set of 풪퐾,푆 points on an affine curve is finite under similar conditions. Kim’s method has been applied to give new proofs of finiteness of integral points

for several classes of curves. [Kim05] uses this approach to show that 1 is P (풪Q,푆) finite. [CK10] uses multivariable Iwasawa theory to prove that 푋(Q) is finite in the case where 푋 is a curve of genus at least 2 whose Jacobian is isogenous to a product of abelian varieties with complex multiplication. [EH17] builds on this work to show

that 푋(Q) is finite whenever 푋 is a geometrically Galois cover of P1 with solvable Galois group.

A variant of Kim’s method using adelic points instead of 푝-adic points was proved in [Ove16] to cut out exactly the set of points which survive torsors for finite étale nilpotent group schemes of odd order. Very recently, Dogra proved an unlikely intersection result for iterated integrals and used it to extend the results of finiteness results of [Kim05], [CK10], and [EH17]

to give nonabelian Chabauty proofs of the finiteness of 풳(풪퐾,푆) for curves 풳 of the same forms, but over any number field 퐾 and for any finite set of places 푆. (See [Dog19] for details.) Restricting the method to the second step of the tower (this is called quadratic

Chabauty), the Chabauty-Kim method can be applied to bound the set 푋(Q) in several cases even when rank 퐽(Q) = dim 퐽. For instance, as a relatively easy to state example of a much more general result, there is

Theorem 1.3.3 ([BD18a, special case of Corollary 1.2]). Let 푋 be a smooth projective

27 hyperelliptic curve of genus 푔 with good reduction at 3 and potential good reduction at

all primes. Let 퐽 be the Jacobian of 푋, let 푟 = rank 퐽(Q), and let 휌 be the rank of the Néron–Severi group of 퐽. Suppose that 푟 = 푔 and 휌 > 1. Then

3 2 #푋(Q) < 24푔 + 228푔 + 120푔 + 72 .

In addition to general bounds, it is often possible to use quadratic Chabauty in practice to compute an explicit subset of which contains . For instance, 푋(Q푝) 푋(Q) the method was used to complete the classification of Q-points on the modular curves + + 푋s(ℓ) := 푋(ℓ)/퐶s(ℓ) , where 퐶s(ℓ) is the normalizer of a split Cartan subgroup of GL .[BDS+17] showed 2(Fℓ)

+ Theorem 1.3.4 ([BDS 17]). #푋s(13)(Q) = 7 .

The curve 푋s(13)(Q) has genus 3 and absolutely simple Jacobian of rank 3; this made it inaccessible to all methods used to tackle other levels.

1.3.2 The Mordell–Weil sieve

No summary of improvements to Chabauty’s method would be complete without mentioning the Mordell-Weil sieve. Although, strictly speaking, it is not a part of Chabauty’s method, the Mordell–Weil sieve allows one to combine information from

reducing modulo many different primes to show that 푋(Q) must map into certain cosets of subgroups of 퐽(Q). This often allows one to rule out the existence of rational points inside of some residue discs on Using the Mordell–Weil sieve makes it 푋Q푝 . much more likely that the subset of computed via Chabauty will be equal to 푋(Q푝) the set 푋(Q). We recommend [Sik15] as a reference.

28 1.4 History of the 푆–unit equation

From one perspective, much of this thesis is devoted to studying a classical object of over a century of research — the set of solutions to the 푆-unit equation, i.e., the set

× × ∼ 1 푈퐾,푆 := {(푥, 푦) ∈ 풪퐾,푆 × 풪퐾,푆 : 푥 + 푦 = 1} = (P r {0, 1, ∞})(풪퐾,푆) .

Building on earlier work of Siegel and Mahler on 푆–integral points on curves of positive genus, [Lan60] shows that 푈퐾,푆 is finite. These proofs were ineffective. In other words, they gave no information about the following questions:

1. Can 1 be bounded uniformly in terms of and ? (P r {0, 1, ∞})(풪퐾,푆) [퐾 : Q] #푆

2. Given 퐾 and 푆, is it possible to compute the set 푈퐾,푆 ? If so, can one compute

푈퐾,푆 efficiently?

Regarding the first question: For many years, the best known (general) upper

bounds on #푈퐾,푆 have built on work of Baker [Bak67] and Yu [Yu89] on the theory of linear forms in logarithms and linear forms in 푝-adic logarithms. For instance, in [Eve84], it is proved that

∞ 푑+2·#푆+2·#푀 2#푆+3푟1+4푟2 #푈퐾,푆 ≤ 3 · 7 퐾 = 3 · 7 .

This upper bound is far from the largest known lower bounds. [EST88] show that

there exists a positive constant 퐶 and sets 푆 with #푆 arbitrarily large satisfying 1/2 . In the context where the field varies instead of , 푈Q,푆 > exp(퐶(푠/ log 푠) ) 퐾 푆 [Gra96] shows that 2 for cyclotomic fields. 푈퐾,∅ ≥ 퐶 · [퐾 : Q] Regarding the second question: Many authors have produced effective bounds for the heights of elements of 푈퐾,푆 by combining data from linear forms in logarithms at both the infinite and the finite places of 퐾. The current state of the art can be found in [Győ19]. It is also possible to prove effective bounds on the height of elements

of 푈퐾,푆 without using the theory of linear forms in logarithms [MP13]. In principle,

these bounds allow one to compute 푈퐾,푆 by exhaustively checking all elements of

29 × up to a given height. In practice, it is possible to do much better by using 풪퐾,푆 a combination of sieving methods and LLL-based techniques for lattice reduction

[Sma99]. An efficient implementation is now available in Sage[AKM+18].

At this point, bounds on #푈퐾,푆 and methods for computing 푈퐾,푆 are well-understood. What is the value in continuing to study this set? We propose two answers.

First, algorithms for computing 푈퐾,푆 involve a mixture of techniques. It seems

likely that new ideas for studying 푈퐾,푆 could lead to more efficient computations. For instance, it may be possible to use our ideas to improve the efficiency of the sieving

step of [AKM+18].

Being able to compute 푈퐾,푆 efficiently has several applications. For instance, it is a key step in computing all elliptic curves over 퐾 with good reduction outside 푆 [Š63] (as explained in [Sil09, proof of theorem 6.1],) all genus 2 curves over 퐾 with good reduction outside 2 [MS93], or all Fermat curves with good reduction outside 2 and 3 [BKSW19]. There are also applications to the study of recurrence sequences, decomposable form equations, and much more. Second, the unit equation has recently been an important testing ground for strate- gies for new proofs of Faltings’ Theorem. The first triumph of Kim’s nonabelian Chabauty program was a new proof that is finite [Kim05]. This work has since 푈Q,푆 been turned into an explicit algorithm for bounding inside of the vanishing set 푈Q,푆 of 푝-adic power series, at least when 푆 is small (and also for number fields of small degree) [DCW15]. Further development of Kim’s approach has led to new bounds on

푋(Q) for 푋 a hyperelliptic curve with rank 퐽(Q) = 푔 and Néron–Severi rank greater than 1 [BD18b, BD17], as well as the explicit computation of the Q-points on ‘cursed’ + modular curve 푋푛푠(13) with rank and genus equal to 3 [BDS 17]. Before they completed their new 푝-adic proof of Faltings’ theorem, Lawrence and

Venkatesh also reproved the finiteness of 푈퐾,푆 by their method [LV18]. It served an important motivational role, and continues to be useful as a source of intuition regarding their method. Our (wildly optimistic) dream is that the methods we develop in this thesis, possibly adapted to the setting of Kim’s nonabelian Chabauty, could someday be part

30 of a new, computationally-oriented proof of Faltings’ theorem. This day seems far off and may well never arrive. Nevertheless, we hope this dream will serve as sufficient

motivation for our study of the 푆-unit equation.

1.5 Outline of this thesis

The remainder of this thesis is organized as follows: Chapter2 sets the notation that we use throughout the paper and discusses some background material. It may be best viewed as a reference section to be consulted as needed. It focuses on three topics: generalized Jacobians, descent, and Newton polygons. Section 2.2 sets up the theory necessary to describe an embedding of the punctured curve 풞 into a semi-abelian scheme over 풪퐾,푆. Section 2.3 reviews the theory of descent and gives explicit examples of descent sets for punctured genus 0 curves. Chapter3 reviews the classical theory of Chabauty’s method. While most expo- sitions of Chabauty’s method consider Q-points on projective curves, we work in the full generality of 풪퐾,푆-points on integral models of curves. Although this complicates the presentation somewhat, we hope that it will prepare the reader to understand the method of Chabauty for restrictions of scalars of curves. In Chapter4, we present explain the RoS Chabauty method. Our treatment differs from Siksek’s [Sik13] in two main ways. Firse, we develop the theory for the more general setting of integral points on curves. Second, while Siksek always considers the points of and , we often consider the -points of these Z푝 Res퐾/Q 푋 Res퐾/Q 퐽 퐾p varieties. This allows us to simplify formulas for regular differentials substantially, at the cost of working over larger fields. In this chapter, we also explain howbase change obstructions to RoS Chabauty arise. In Chapter5, we study the power of classical and RoS Chabauty together with descent for bounding 1 . We show that classical Chabauty and (P r {0, 1, ∞})(풪퐾,푆) descent by genus 0 covers suffices to prove finiteness ofthe 푆-unit equation only in very special cases. In contrast, when 퐾 has does not contain a CM subfield, we show

31 that all known obstacles to RoS Chabauty for P1 r {0, 1, ∞} can be avoided using descent. This should be viewed as evidence that RoS Chabauty and descent may suffice to prove finiteness ofthe 푆-unit equation. In Chapter6, we give several extended examples where we use RoS Chabauty to give a new proof that 1 is finite. We cover the cases where is (P r {0, 1, ∞})(풪퐾 ) 퐾 a quadratic field, a mixed cubic field, a totally complex quartic field, a mixed quartic field with a real quadratic subfield. We also show that there are no solutions tothe unit equation when [퐾 : Q] is not divisible 3 and 3 splits completely in 퐾. The last two cases involve some particularly interesting applications of the method.

32 Chapter 2

Preliminaries

This chapter provides notation and a variety of background material which will be used later in this thesis. The reader may find it most helpful to skip this section on a first reading and to consult it as necessary.

2.1 Notation and conventions

In this section, we collect notations. There will be several forward references justifying our notation.

Throughout this thesis, we assume that 푋 is a nice curve of genus 푔 over a number field 퐾, i.e., 푋 is smooth, proper, and geometrically integral. Given an effective divisor m on 푋 which is defined over 퐾, we let 퐶 be the open subscheme of 푋 given

by 퐶 = 푋 r supp(m). We assume m has multiplicity at most 1 at any point of 푋. We also assume that ⎧ ⎪3, if 푔 = 0, ⎪ ⎨⎪ deg m ≥ 1, if 푔 = 1, ⎪ ⎪ ⎩⎪0, if 푔 ≥ 2 ,

so that 퐶 is a hyperbolic curve.

Let 퐽퐶 := GJac(퐶, m) be the generalized Jacobian of 푋 with modulus m. (See

Section 2.2.) The group scheme 퐽퐶 is a semi-abelian variety over 퐾. This means that

33 퐽퐶 is the extension of an abelian variety 퐴 by a torus 푇 . In other words, 퐽퐶 is the middle term in an exact sequence

0 → 푇 → 퐽퐶 → 퐴 → 0 .

In this sequence, the abelian variety 퐴 is the Jacobian of 푋 and the torus 푇 depends on the modulus m. Set 훾 = max(0, deg m − 1). Then, dim 푇 = 훾 and dim 퐴 = 푔, so

dim 퐽퐶 = 푔 + 훾.

We write 푟1 = 푟1(퐾) for the number of real embeddings of 퐾 and 푟2 = 푟2(퐾) pairs of complex conjugate embeddings of . We write for 퐾 푑 = [퐾 : Q] = 푟1 + 2푟2 the degree of 퐾 over Q. We say that a number field 퐾 is CM if 퐾 is a totally complex, degree 2 extension of a totally real field.

Let 푆′ be a finite set of finite places of Q, and let 푆 be the set of places of 퐾 lying over ′. We write for the ring of -integers of and write . Note 푆 풪퐾,푆 푆 퐾 Z푆′ := 풪Q,푆′ that is the integral closure of in and is free of rank as a -module, so 풪퐾,푆 Z푆′ 퐾 푑 Z푆′ that there is a restriction of scalars functor which takes -schemes to -schemes. 풪퐾,푆 Z푆′ In our convention, 푆 does not contain the infinite places. We warn the reader that many authors use different conventions. We use this notation because the infinite and finite places will behave differently in many of our arguments.

Chabauty’s method is a 푝-adic method. As such we will need to work with com- pletions of 퐾. Let p ∈/ 푆 be a prime of 퐾. We write 퐾p for the completion of 퐾 at p and 풪p for the ring of integers of 퐾p.

′ Given a scheme 풵 over a base 푇 and scheme 푇 /푇 , we let 풵푇 ′ denote the base change of 풵 to 푇 ′. When 푇 ′ = Spec 퐿 for a ring 퐿, we abuse notation and write

풵퐿 := 풵Spec 퐿.

In this thesis, we will be interested in the set of integral points on certain 풪퐾,푆- models of 퐶. We now define the models which are acceptable for our method.

Definition 2.1.1. Suppose that 푋 is a smooth, proper, geometrically integral curve over 퐾 and let 풳 be a proper regular model for 푋 over 풪퐾,푆. Then the closure

34 of supp(m) in 풳 is a horizontal divisor Γ of 풳. Write 풞 = 풳 r Γ. Then, 풞 is an

풪퐾,푆-model for 퐶.

We call a 풪퐾,푆-model 풞 of this form a clearance hole model for 퐶.

If, moreover, 풞 is smooth, separated, and of finite-type over 풪퐾,푆 and the fibers

of 풞 over the closed points of 풪퐾,푆 are smooth and geometrically connected, we say that 풞 is a sound clearance hole model for 퐶.

Also, we say a scheme 풞 over 풪퐾,푆 is a (sound) clearance hole curve if 풞 is a

(sound) clearance hole model for its generic fiber 풞퐾 .

Remark 2.1.2. A clearance hole is a hole through an object which is large enough to enable the threads of a screw or bolt to pass through but not the head of the screw or bolt. In other words, it is a hole which passes all the way through an object and is exactly as large as it needs to be, but no larger. The word sound is intended in the sense of ‘structurally sound’.

Remark 2.1.3. Given a clearance hole model 풞 for 퐶, it is always possible to enlarge 푆 to get a sound clearance hole model. Since we assume that 퐶 is a hyperbolic curve, given any clearance hole model 풞 for 퐶, the set 풞(풪퐾,푆) is finite by Faltings’ theorem and Siegel’s theorem.

Given a sound clearance hold model 풞 for 퐶 over 풪퐾,푆, let 풥풞 be the connected component of the identity of the lft-Néron model of 퐽퐶 over 풪퐾,푆. We call 풥풞 the

generalized Jacobian of 풞. Note however, that 풥풞 depends only on 퐶 and 풪퐾,푆.

Since 풞 is smooth and has geometrically connected fibers, given 푃 ∈ 풞(풪퐾,푆),

the Abel-Jacobi map 푗 : 퐶 → 퐽퐶 based at 푃 extends to a map 푗 : 풞 → 풥풞. (See Section 2.2 for more on this construction.)

Remark 2.1.4. When 풞 = 1 Γ, such a morphism can be realized explicitly P풪퐾,푆 r without enlarging 푆, even in the absence of an 푆-integral point. For instance, if 풞 = 1 {0, 1, ∞}, we can take the morphism defined by P풪퐾,푆 r

1 {0, 1, ∞} → × P풪퐾,푆 r G푚,풪퐾,푆 G푚,풪퐾,푆 푥 ↦→ (푥, 푥 − 1).

35 See Section 2.2.5 for the general construction.

Finally, given a scheme 풵 over 풪퐾,푆, we always use dim 풵 to refer to the relative dimension of 풵 over 풪퐾,푆. In all situations we consider, this will be equivalent to the dimension of the generic fiber 풵퐾 as a 퐾-scheme.

2.2 Generalized Jacobians

In this section we discuss the basic properties of generalized Jacobians needed for the rest of this thesis.

2.2.1 Generalized Jacobians over Algebraically Closed Fields

In this section, our exposition mostly follows that of [Ser88].

To start, let 푋 be a nice (smooth, projective, and geometrically integral) curve defined over an algebraically closed field 퐾. Let 퐽 be the Jacobian of 푋. The abelian variety 퐽 plays two main roles with respect to 푋.

1. The Jacobian 퐽 is the Picard variety of 푋; i.e., it parameterizes degree 0 divisors on 푋 up to linear equivalence.

2. The Jacobian 퐽 is the Albanese variety of 푋; i.e., given a point 푃 ∈ 푋(퐾), the Abel-Jacobi morphism 푗 : 푋 → 퐽 taking 푃 to the identity on 퐽 satisfies the following universal property: For any abelian variety 퐴 and any morphism 푓 : 푋 → 퐴 which takes 푃 to the identity on 퐴, there is a unique morphism of group schemes 휑 : 퐽 → 퐴 such that 푓 = 휑 ∘ 푗.

Generalized Jacobians have analogues of these properties. They have a similar relationship with the Jacobian as a ray class group has with the class group of a number field. As when defining ray class groups, we need a notion of modulus.

A modulus m on 푋 is a formal nonnegative integer linear combination of points of 푋(퐾), i.e., an effective divisor on 푋. Any modulus can be expressed a integer linear

36 combination ∑︁ m = 푛푃 푃 푃 ∈푋(퐾)

where 푛푃 = 0 for all but finitely many 푃 ∈ 푋(퐾) and 푛푃 ≥ 0 always. The support of m is the finite set

supp(m) := {푃 ∈ 푋(퐾): 푛푃 ̸= 0} .

If 휙 ∈ 퐾(푋) is a rational function, we write 휙 ≡ 1 (mod m) if 휙 − 1 has a zero of order at least 푛푃 for each 푃 ∈ supp(m).

The generalized Jacobian 퐽m of 푋 with modulus m exists and is characterized by the universal property in Theorem 2.2.3.

Before we state the theorem, we define a property of functions on 푋 equipped with a modulus m.

Definition 2.2.1. Let 푓 : 푋 99K 퐺 be a rational map from 푋 to a commutative group ∑︀ scheme over 퐾. Given a divisor 퐷 = 푛푃 푃 on 푋, let 푓(퐷) denote the element of ∑︀ 퐺 given by 푛푃 푓(푃 ) (writing the group operation additively.) If both

1. 푓 is regular away from supp(m), and

2. 푓(퐷) = 0 whenever 퐷 = (휙) is the divisor of a function with 휙 ≡ 1 mod m ,

say that 푓 is m-good.

Definition 2.2.2. A rational map 휙 : 퐻 → 퐺 of commutative algebraic groups is an affine homomorphism if 휙 is a composition of a homomorphism and a translation.

Theorem 2.2.3 ([Ser88, Theorem I.2]). For every modulus m, there exists a commutative algebraic group and a rational map such that the following 퐽m 푗m : 푋 99K 퐽m property holds:

For every commutative algebraic group 퐺 and every m-good rational map 푓 : 푋 99K , there is a unique affine homomorphism such that . 퐺 휃 : 퐽m 99K 퐺 푓 = 휃 ∘ 푗m

As with the usual Abel–Jacobi morphism, 푗m is unique up to the choice of a base point 푃 ∈ 푋(퐾) mapping to the identity element of 퐽(퐾).

37 We will abuse notation slightly by referring to 푗m as the Abel–Jacobi morphism, in keeping with standard terminology for Jacobians. Much like the usual Abel–Jacobi morphism, the restriction of to is a locally closed embedding. 푗m 푋 r supp(m)

푛푃 The structure of 퐽m is well understood. For each 푃 , let 푅푃 := 퐾[푡]/(푡 ). Then,

퐽m fits into the following exact sequence:

⎛ ⎞ ∏︁ 0 → Res / → 퐽 → 퐽 → 0 . (2.2.4) ⎝ 푅푃 /퐾 G푚⎠ G푚 m 푃 ∈supp(m)

We are most concerned with the case where 푛푃 = 1 for all 푃 ∈ supp(m), i.e. the case where m (viewed as a subscheme of 푋 in the natural way) is étale over Spec(퐾). In this situation, for all , and so is an extension of an 퐺푃 = G푚 푃 ∈ supp(m) 퐽m abelian variety by a torus, i.e. a semiabelian variety.

Recall that 훾 = max(0, deg(m) − 1). By 2.2.4, the toric part of 퐽m has dimension 훾 and the abelian variety quotient has dimension 푔 = genus(푋). To ease notation throughout the rest of the paper, we make the following definition:

Definition 2.2.5. Suppose that 푋/퐾 is a smooth, proper, and geometrically integral curve. Let be a modulus with for all . Set . m 푛푃 = 1 푃 ∈ supp(m) 퐶 := 푋 r m

The generalized Jacobian of 퐶, which we denote by 퐽m or 퐽퐶 or GJac(퐶), is the generalized Jacobian of 푋 with modulus m. Similarly, we denote the Jacobian of 푋 by Jac(푋).

2.2.2 Generalized Jacobians over (Number) Fields

Let 푋 be a nice curve over a number field 퐾 and let m be a reduced effective divisor on 푋. Then, 퐶 := 푋 r supp(m) is a smooth, geometrically integral curve over 퐾. Fix an algebraic closure 퐾 of 퐾. Much like the usual Jacobian, generalized Jacobians descend. This means that is

38 there is a commutative group scheme GJac(퐶) defined over 퐾 such that

GJac(퐶)퐾 = GJac(퐶퐾 ) .

If moreover 퐶(퐾) ̸= ∅, fixing any 푃 ∈ 퐶(퐾) determines an Abel-Jacobi morphism which descends to a locally closed embedding 푗 : 퐶퐾 → GJac(퐶)퐾

푗 : 퐶 → GJac(퐶) ,

defined over 퐾 and such that 푗(푃 ) is the identity element of GJac(퐶). We comment briefly on the structure of GJac(퐶). Recall that m is identified with

the reduced zero-dimensional subscheme 푋 r 퐶. Then, m = Spec(퐿) for some étale algebra 퐿/퐾. Then, refining (2.2.4), GJac(퐶) fits into the exact sequence

0 → (Res퐿/퐾 G푚)/G푚 → GJac(퐶) → Jac(푋) → 0 .

We discuss the structure of in further detail in Section 2.2.5. Res퐿/퐾 G푚

2.2.3 Néron Models

Before we define the generalized Jacobian for relative curves, we first recall some facts about Néron models of semiabelian varieties. Our exposition is mostly drawn from [BLR90]. We first recall the definition of a Néron model:

Definition 2.2.6. [BLR90, Definition 1.2.1.] Let 푆 be a Dedekind scheme with ring of rational functions 퐾.

Let 푋퐾 be a smooth 퐾-scheme. A Néron model of 푋퐾 (resp. a Néron lft-model) is an 푆-model of 푋 which is smooth, separated, and of finite type (resp. locally of finite type,) and which satisfies the following universal property, called theNéron mapping property:

For each smooth 푆-scheme 푌 and each 퐾-morphism 푢퐾 : 푌퐾 → 푋퐾 there is a

39 unique 푆-morphism 푢 : 푌 → 푋 extending 푢퐾 .

By the Néron mapping property, a Néron lft-model 퐺 of an algebraic group 퐺퐾 over 퐾 is a group scheme over 푆 (if it exists). Note, however, that 퐺 need not be proper over 푆, even if 퐺퐾 is proper over 퐾. We require the following existence result:

Theorem 2.2.7 ([BLR90, page 310]). Let 푆 be an excellent Dedekind scheme with ring of rational functions 퐾 of characteristic 0. Let 퐺퐾 be a smooth, commutative, connected algebraic group over 퐾. The following conditions are equivalent:

1. 퐺퐾 admits a Néron lft-model over 푆.

2. contains no subgroup of type . 퐺퐾 G푎

In particular, if 퐺퐾 is a semi-abelian variety, then 퐺퐾 admits a Néron lft-model over 푆. The Néron lft-model is of finite type if and only if 퐺퐾 is an abelian variety.

Remark 2.2.8. The fibers of a Néron lft-model 퐺 of a smooth, commutative, connected algebraic group need not be connected. We let 퐺∘ denote the (fiber-wise) connected component of the identity section. The subscheme 퐺∘ is also a commutative group scheme.

Example 2.2.9. Suppose that is an elliptic curve. Let be the minimal 퐸/Q푝 ℰ/Z푝 proper regular model of 퐸. Then the Néron model of 퐸 is the open subscheme of ℰ obtained by removing any singular points on the special fiber of ℰ.

Example 2.2.10. We describe the lft-Néron model over of . 퐺 Spec Z G푚,Q We construct by gluing an infinite disjoint union of copies of , indexed by 퐺 G푚,Z a prime and an integer. Write 푛 for the copy indexed by and . This is just 푝 G푚,Z 푝 푛 notation, but it will be suggestive of the gluing. We have:

⋃︁ ⋃︁ 푛 G푚,Z ∪ 푝 G푚,Z . 푝 prime 푛∈Z

To get , we glue to 푛 over via multiplication by 푛. 퐺 G푚,Z 푝 G푚,Z Spec Z r {푝} 푝 The generic fiber of the resulting scheme is clearly . The special fiber over G푚,Q is an disjoint union of copies of indexed by . Writing 푛 for the copy 푝 G푚,F푝 Z 푝 G푚,F푝

40 indexed by and writing for the multiplication on , the group law on the 푛 푚 G푚,F푝 special fiber is given by

⋃︁ 푛 ⋃︁ 푛 ⋃︁ 푛 푀 : 푝 G푚,F푝 × 푝 G푚,F푝 → 푝 G푚,F푝 , 푛∈Z 푛∈Z 푛∈Z ∐︁ 푖 푗 푖+푗 = 푚 : 푝 G푚,F푝 × 푝 G푚,F푝 → 푝 G푚,F푝 . (푖,푗)∈Z2

The inversion maps and co-units are defined similarly.

The -points of are clearly in bijection with ×. Indeed, an element Z 퐺 G푚,Q(Q) = Q of 퐺(Z) is an element of 푥 ∈ Q× together with the 푝-adic valuation of 푥 and the congruence class modulo 푝 of 푝−푣푝(푥)푥 for each prime 푝.

Moreover, the connected component of the identity is

∘ 퐺 = G푚,Z .

One can check that the Néron property holds and that this construction generalizes to number fields without much further work. See[BLR90, Example 10.1.5] for details.

Taking Néron models ‘commutes’ with Weil restriction of scalars as follows.

Proposition 2.2.11. let 푇 ′ → 푇 be a finite flat extension of Dedekind schemes with rings of rational functions 퐿 and 퐾, respectively. Suppose that 퐺퐿 is a smooth group scheme over 퐿 with Neron lft-model 퐺 over 푇 ′.

The Weil restriction of scalars Res푇 ′/푇 퐺 is an lft-Néron model for Res퐿/퐾 퐺퐿.

Proof. The Weil restriction of scalars Res푇 ′/푇 퐺 is a smooth, separated, locally of finite type model for Res퐿/퐾 퐺퐿, so it suffices to check the Néron mapping property. Moreover, the universal property of restriction of scalars (twice) and the Néron

41 property for 퐺 imply that for any 푇 -scheme 푇 ′′,

′′ ′′ ′ Hom푇 (푇 , Res푇 ′/푇 퐺) = Hom푇 ′ (푇 ×푇 푇 , 퐺)

′′ ′ = HomSpec(퐿)((푇 ×푇 푇 )퐿, 퐺퐿)

′′ = HomSpec(퐿)(푇퐾 ×Spec 퐾 Spec(퐿), 퐺퐿)

′′ = HomSpec(퐾)(푇퐾 , Res퐿/퐾 퐺퐿) .

Hence, Res푇 ′/푇 퐺 is a Néron model for Res퐿/퐾 퐺퐿.

∘ ∘ Remark 2.2.12. We claim that (Res푇 ′/푇 퐺) = (Res푇 ′/푇 퐺 ). Indeed, since the universal property implies that the fibers of the restriction of scalars are the restriction of scalars of the fibers, so it suffices to check this fiber-wise. Now, thefibersof 퐺

∘ are smooth, so [CGP15, Proposition A.5.9] shows that the fibers of (Res푇 ′/푇 퐺 ) are connected. So

∘ ∘ ∘ ∘ (Res푇 ′/푇 퐺) = (Res푇 ′/푇 퐺 ) = Res푇 ′/푇 퐺 , as claimed.

As an immediate application, note that if 퐺 is the Néron model over 풪퐾,푆 of the group scheme and ′′ is the integral closure of in , then Res퐿/퐾 G푚,퐾 풪퐿푖,푆 풪퐾,푆 퐿

퐺∘ = Res . 풪퐿,푆′′ /풪퐾,푆 G푚,풪퐿,푆′′

Néron models also behave well under étale base change.

2.2.4 Generalized Jacobians over Rings of 푆-Integers

In this section, we assume that over is a sound clearance hole model 풞 = 풳 r Γ 풪퐾,푆 (see Definition 2.1.1) for the 퐾-curve 퐶 = 푋 r m. Note in particular that m is étale over Spec 퐾 while Γ is flat, but not necessarily

étale over Spec 풪퐾,푆.

Recall that 퐽퐶 is the generalized Jacobian of 퐶. Suppose that the basepoint of the Abel-Jacobi morphism 푗 : 퐶 → 퐽m extends to a point of 풞(풪퐾,푆). Since m is étale

42 over 퐾, the commutative group scheme 퐽m is a semiabelian variety. In particular, 퐽m admits a Néron lft-model over . Let ∘ be the (fiber-wise) connected 풥m 풪퐾,푆 풥풞 := 풥m

component of the identity of 풥m.

We call 풥풞 the generalized Jacobian of 풳 with modulus Γ, or the generalized Jacobian of . Note, however that depends only on and . 풞 := 풳 r Γ 풥풞 퐶 풪퐾,푆 We briefly justify our terminology.

By the Néron propety, the Abel-Jacobi morphism 푗 : 퐶 ˓→ 퐽m extends to a

morphism 푗 : 풞 ˓→ 풥m. Since 풞 is a sound clearance hole model, the fibers of 풞 are geometrically connected. Hence, the image will lie in the connected component of the identity. So in fact we have an Abel-Jacobi morphism

∘ 푗 : 풞 ˓→ 풥m = 풥풞 .

For example, if 풞 = 1 {0, 2, ∞}, then 퐽 ∼ × , so 풥 ∼ × . PZ r m = G푚,Q G푚,Q 풞 = G푚,Z G푚,Z If we fix the base point 1 ∈ 풞(Z), the Abel-Jacobi morphism is given by

1 푗 : PZ r {0, 2, ∞} → G푚,Z × G푚,Z , 푥 ↦→ (푥, 2 − 푥) .

2.2.5 Generalized Jacobians of Genus Zero Curves

In the examples in this article, we will focus primarily on the case where 풞 is a punctured curve of genus zero. In preparation, we recall some facts about the structure of the generalized Jacobian of such a curve. Set

풞 = 1 {(푥 : 푦): 푓(푥, 푦) = 0} P풪퐾,푆 r

for some homogeneous square-free polynomial 푓 ∈ 풪퐾,푆[푥, 푦] of content 1.

In this case, the generalized Jacobian of 풞 can be understood explicitly in terms

43 of norm tori. Write ∏︀푐 where each is irreducible over . Set 푓 = 푖=1 푓푖 푓푖 풪퐾,푆

⎧ ⎨⎪the integral closure of 풪퐾,푆 in 퐾[푥]/푓(푥, 1) if 푓푖 ̸= 푦 , 푅푖 := ⎩⎪풪퐾,푆 if 푓푖 = 푦 .

Let 퐿푖 be the fraction field of 푅푖 and let 푆푖 be the set of places of 퐿푖 which lie over some prime in 푆.

Each 푅푖 is module-finite over 풪퐾,푆 and we have

(︃ 푐 )︃ ∼ ∏︁ (2.2.13) 풥풞 = Res푅푖,풪퐾,푆 G푚,푅푖 /Δ(G푚,풪퐾,푆 ) , 푖=1 where indicates a diagonally embedded copy of . If for Δ(G푚,풪퐾,푆 ) G푚,풪퐾,푆 푓푖 = 푦 some 푖, so that 풞 is a punctured copy of A1, the formula can be simplified by leaving out the 푖th component and the quotient.

Independent of the existence of a rational point on 풞, there is an Abel-Jacobi map

푗 : 풞 → 풥풞 defined as follows. Over an algebraic closure, 푓푖 factors as a product of conjugates of (훼푖푥 − 훽푖푦) for 훼푖 and 훽푖 integral. Then, we have

푗 : 풞 → 풥풞

(푥 : 푦) ↦→ (훼1푥 − 훽1푦, . . . , 훼푐푥 − 훽푐푦) ,

where we can naturally interpret 훼1푥 − 훽1푦 as an element of the restriction of scalars.

For example, if 푓(푥, 푦) = 푥(푥 − 푦)푦, so that 풞 = 1 {0, 1, ∞}, then P풪퐾,푆 r

∼ 풥풞 = G푚,풪퐾,푆 × G푚,풪퐾,푆 . and

푗 : 풞 → 풥풞 푥 ↦→ (푥, 푥 − 1) ,

44 where we have set 푦 = 1 and forgotten the last coordinate because we are using the simplified form.

The expression (2.2.13) makes it easy to compute the dimension and rank of . Since is finite index in and has the same rank as , we have 풥풞(풪퐾,푆) 푅푖 풪퐿푖,푆푖 풪퐿푖,푆푖

푐 푐 ∑︁ ∑︁ dim 풥풞 = deg(푓) − 1 = deg(푓푖) − 1 = [퐿푖 : 퐾] − [퐾 : 퐾] 푖=1 푖=1 and

푐 ∑︁ × × rank 풥풞(풪퐾,푆) = rank 푅푖 − rank 풪퐾,푆 푖=1 푐 ∑︁ = rank 풪× − rank 풪× (2.2.14) 퐿푖,푆푖 퐾,푆 푖=1 푐 ∑︁ = [푟1(퐿푖) + 푟2(퐿푖) + #푆푖 − 1] − [푟1(퐾) + 푟2(퐾) + #푆 − 1] . 푖=1

We can also express the rank in terms of the action of the absolute Galois group of 퐾 on the set of punctures of our genus zero curve. We state the general result, noting that for the purpose of proving finiteness of integral points, we may assume the curve is a punctured 1, or else it automatically has no points. P 풪퐾,푆

Lemma 2.2.15. Let be a minimal proper regular model of 1 .. Let 풳 /풪퐾,푆 P퐾 퐺 =

Gal(퐾/퐾) be the absolute Galois group of 퐾 and for a place p, let 퐺p denote the decomposition group of p. Let Γ be a horizontal divisor on 풳 . Set

풞 = 풳 r Γ .

The curve 풞 is a sound clearance hole model for its generic fiber 퐶 := 풞퐾 . Write 퐺∖Γ(퐾) for the set of orbits for the action of 퐺 on Γ(퐾).

Then, the generic fiber 퐽퐶 of 풥풞 is a torus and

dim 풥풞 = dim 퐽퐶 = #Γ(퐾) − 1,

45 and 풥풞(풪퐾,푆) is an abelian group of rank

∑︁ rank 풥풞(풪퐾,푆) = [#(퐺p∖Γ(퐾)) − 1] − [#(퐺∖Γ(퐾)) − 1] .

p∈푆∪Σ∞

We omit the proof, which is a fairly straightforward computation in the character theory of tori. All of the ideas needed can be found, for example, Theorem 8.7.2 of Cohomology of Number Fields by Neukirch, Schmidt, and Wingberg [NSW08], or in Chapter 6 of Eisenträger’s Ph.D. Thesis [Eis03].

2.3 Descent

When the pair (풞, 풪퐾,푆) does not satisfy the classical Chabauty inequality (3.2.6) or the RoS Chabauty inequality (4.1.5), we must use another approach to prove that

풞(풪퐾,푆) is finite. One possibility is to replace (풞, 풪퐾,푆) with a finite set of covers of

풞 such that each point in 풞(풪퐾,푆) is the image of an integral point on some curve in the set. We define

Definition 2.3.1. A descent set for (풞, 풪퐾,푆) is a finite set D of clearance hole curves

풟 over 풪퐾,푆 (see Definition 2.1.1) equipped with morphisms 푓풟 : 풟 → 풞 such that

⋃︁ 풞(풪퐾,푆) ⊆ 푓풟(풟(풪퐾,푆)) . 풟∈D

If, moreover, the generic fibers 풟퐾 of all of the curves 풟 are punctured genus 0 curves, we say that D is a genus 0 descent set.

Showing that 풟(풪퐾,푆) is finite for each 풟 ∈ D proves that 풞(풪퐾,푆) is also finite.

Remark 2.3.2. If D is a descent set of (풞, 풪퐾,푆) and for some 풟 ∈ D, we have that E is a descent set for , then is also a descent set for . (풟, 풪퐾,푆) (D r {풟}) ∪ E (풞, 풪퐾,푆) This iterated descent approach can be used to construct complicated descent sets from relatively simple building blocks.

In this remainder of this section, we give a brief overview of the theory of Galois descent, proving that descent sets exist and giving explicit examples.

46 Our exposition mostly follows [Poo17, Sections 6.5 and 8.4]. We recommend [Sko01] as a further reference.

2.3.1 Torsors

In order to explain the theory of integral descent, we first recall the notion of a torsor under a group scheme over a base scheme.

Definition 2.3.3 ([Poo17, Definition 6.5.1]). Let 퐺 → 풮 be an fppf group scheme. A (right) 퐺-torsor over 풮 is an fppf 풮-scheme 푍 equipped with a right 퐺-action

푍 ×풮 퐺 → 푍 such that one of the following equivalent conditions holds:

′ 1. There exists an fppf base change 풮 → 풮 such that 푍풮′ is isomorphic to 퐺풮′ ′ as 풮 -schemes with 퐺풮′ action. (Here the action of 퐺풮′ on itself is by right translation.)

2. The morphism

푍 ×풮 퐺 → 푍 ×풮 푍 (푥, 푔) ↦→ (푥, 푥 · 푔)

is an isomorphism.

The first condition says that 푍 is a twist of 퐺 (as a scheme with 퐺-action). As such, it should not be a surprise that torsors are parameterized by 퐻1 in a suitable cohomology theory. For the remainder of this section and the following section, we make the convention 1 ˇ 1 . We have 퐻 := 퐻푓푝푝푓

Theorem 2.3.4 ([Poo17, Theorem 6.5.10]). Let 퐺 be an fppf group scheme over a locally noetherian scheme 푆. Suppose that 퐺 → 풮 is an affine morphism. Letting 퐻1

47 denote Čech cohomology on the fppf site, we have

{퐺-torsors}/isomorphism ∼= 퐻1(풮, 퐺). as pointed sets. (The neutral element on the left-hand side is given by the class of the trivial torsor 퐺 with right 퐺-action.)

We are primarily concerned with cases where 퐺 is finite over 풮, so the assumption that 퐺 → 풮 is affine is not an inconvenience. Following [Poo17, Section 6.5.6] we assume that 퐺 → 푆 is affine from now on. Write 퐻 = 퐺 viewed as a scheme with a left-action on 퐺. Given a right 퐺-torsor 푇 corresponding to 휏 ∈ 퐻1(푆, 퐺), there is a left-action of the twisted group scheme 풢휏 on 푇 . So, in fact, 푇 is a 퐻휏 -퐺-bitorsor. It is also possible to give 푇 the structure of a 퐺-퐻휏 -bitorsor by inverting the group action. For instance, one can define the map 퐺 × 푇 → 푇 by 푔 · 푡 = 푡푔−1. The resulting 퐺-퐻휏 -bitorsor is denoted 푇 −1. Now, given two right 퐺-torsors 푍 and 푇 over 푆 represented in 퐻1(풮, 퐺) by 휁 and 퐺 −1 −1 휏, respectively, let 푍 × 푇 be the quotient of 푍 ×풮 푇 by the 퐺-action (푥, 푡) · 푔 = 퐺 (푥푔, 푔−1푡). The scheme 푍 × 푇 −1 is a 퐺휁 -퐺휏 -bitorsor over 풮 and is affine over 풮.

2.3.2 Constructing descent sets

Suppose now that 풮 is finite-type and separated over 풪퐾,푆. Let 풢 be a finite flat group scheme over 풪퐾,푆, and suppose that 푓 : 푍 → 풮 is a 풢풮 -torsor. For any 풢 1 torsor 푇 over 풪퐾,푆 corresponding to 휏 ∈ 퐻 (풪퐾,푆, 풢), the twist

풢 휏 풮 −1 푍 := 푍 × (푇풮 )

휏 휏 휏 is a right (풢 )풮 -torsor over 풮, with structure map 푓 : 푍 → 풮. Now, slightly generalizing [Poo17, Section 8.4.1], we can define an evaluation map

1 1 풮(풪퐾,푆) × 퐻 (풮, 풢) → 퐻 (풪퐾,푆, 풢) (푥, 휁) ↦→ 휁(푥) := 푥*(휁)

48 by pulling back cohomology classes. Define

풮(풪퐾,푆)휏 := {푥 ∈ 풮(풪퐾,푆): 휁(푥) = 휏} .

Clearly, ⋃︁ 풮(풪퐾,푆) = 풮(풪퐾,푆)휏 . 1 휏∈퐻 (풪퐾,푆 ,풢) By a slight generalization of [Poo17, Theorem 8.4.1],

휏 휏 푓 (푍 (풪퐾,푆)) = 풮(풪퐾,푆)휏 , so in fact,

⋃︁ 휏 휏 풮(풪퐾,푆) = 푓 (푍 (풪퐾,푆)) . 1 휏∈퐻 (풪퐾,푆 ,풢)

1 By [Poo17, Exercise 8.4], 퐻 (풪퐾,푆, 풢) is finite, so in fact, we have

Proposition 2.3.5. The set

{︀ 휏 휏 1 }︀ 풵 = 푓 : 푍 → 풮 : 휏 ∈ 퐻 (풪퐾,푆, 풢)

is a descent set for (풮, 풪퐾,푆).

2.3.3 Explicit examples of descent sets

In certain cases, we can construct descent sets explicitly.

Suppose that 풜 is a separated group scheme of finite type over 풪퐾,푆 and we have a morphism 푗 : 풞 ˓→ 풜 such that the restriction of 푗 to the generic fiber is finite.

Suppose that ℬ is another separated group scheme of finite type over 풪퐾,푆 equipped with a finite, flat map 휑 : ℬ → 풜 of group schemes such that 휑(ℬ(풪퐾,푆)) has finite index in 풜(풪퐾,푆). Suppose also that 풜(풪퐾,푆) is finitely generated. For example, this holds if 풜 is the connected component of the identity of the Néron model of a semiabelian variety.)

Choose (right) coset representatives 푃1, . . . , 푃푛 for 휑(ℬ(풪퐾,푆)) in 풜(풪퐾,푆). For

49 each 푖 ∈ {1, . . . , 푛}, there is a morphism of 풪퐾,푆-schemes

휑푖 : ℬ → 풜 ,

푄 ↦→ 휑(푄) · 푃푖 .

Then,

푛 푛 ∐︁ ∐︁ 풜(풪퐾,푆) = 휑(ℬ(풪퐾,푆)) · 푃푖 = 휑푖(ℬ(풪퐾,푆)) . (2.3.6) 푖=1 푖=1

Let 풟푖 be the pullback of 풞 by 휑푖. From the pullback diagram

푗풟 풟푖 ℬ

휑푖 휑푖 푗 풞 풜

and (2.3.6), we see that 푄 ∈ 풞(풪퐾,푆) belongs to 휑푖(풟푖(풪퐾,푆)) if and only if 푄 ∈

ℬ(풪퐾,푆) · 푃푖. It follows that

푛 ∐︁ 풞(풪퐾,푆) = 휑푖(풟푖(풪퐾,푆)) . 푖=1

In particular, {풟푖 : 푖 ∈ {1, . . . , 푛}} will be a descent set for (풞, 풪퐾,푆) so long as the

풟푖 are sound clearance hold curves over 풪퐾,푆.

In this construction, each 풟푖 is a torsor over 풞 for the finite group scheme ker(휑푖 : ℬ → 풜). In particular, this construction is an example of descent.

Example 2.3.7. If is 1 ′ then there is a natural inclusion 풞/풪퐾,푆 P r ({0, ∞} ∪ Γ ), . Taking to be the th power map from to itself, this procedure 풞 ˓→ G푚,풪퐾,푆 휑 푛 G푚 gives a descent set for indexed by elements of × ×푛 D (풞, 풪퐾,푆) 풪퐾,푆/풪퐾,푆. Every covering curve in D is a punctured P1 for which we can write down explicit equations. If we choose a polynomial 푓 such that

Γ′ = {푥 : 푓(푥) = 0} ,

50 and let range over × × 푛, then consists of the curves 훼 풪퐾,푆/(풪퐾,푆) D

1 푛 1 푓훼 : P r {0, ∞} ∪ {푥 : 푓(훼푥 ) = 0} → P r {0, ∞} ∪ {푥 : 푓(푥) = 0} 푥 ↦→ 훼푥푛

Remark 2.3.8. We show that Example 2.3.7 combined with the iterated descent described in Remark 2.3.2 is essentially the only way to produce descent sets consisting

of punctured genus 0 curves which are torsors from group schemes over the base. Let 휑 : 1 → 1 be a Galois cover. I.e., suppose that the induced map on P퐾 P퐾 function fields identifies 휑*(퐾(푡)) as a subgroup of 퐾(푠) such that the extension is Galois.

Let 퐵 be the set of 퐾-points on the base over which of 휑 is branched. The Riemann-Hurwitz theorem implies that

∑︁ 1 2 deg(휑) − 2 = deg(휑) · #퐵 − #{푄 ∈ P (퐾): 휑(푄) = 푃 } . 푃 ∈퐵

For any 푃 ∈ 퐵, we have #{푄 ∈ P1(퐾): 휑(푄) = 푃 } ≤ 푑/2, so 1 < #퐵 < 4. If #퐵 = 2, then 휑 : P1 → P1 is a isomorphic to the 푛th power map. If , then and 1 1 is an -cover. Using the structure of #퐵 = 3 deg(휑) = 6 휑 : P → P 푆3

푆3 as a solvable group, 휑 can be constructed as the composition of two Galois covers which are ramified at exactly 2 points.

51 52 Chapter 3

The Method of Chabauty-Coleman-Skolem

In this section, we recall the 푝-adic method of Chabauty and Coleman for bounding the number of 푆-integral points on a sound clearance hole curve 풞/풪퐾,푆. (See Defini- tion 2.1.1.) Although Chabauty’s method is typically described in the special case of

퐾-rational points on smooth, proper, and geometrically integral curves, we present the general case with an eye towards computing 푆-integral points on punctured genus zero curves. In a way, this makes a return to the original inspiration for Chabauty’s method — an analogous strategy (due to Skolem) for solving Thue equations, i.e. computing the set

2 {(푎, 푏) ∈ Z : 푓(푎, 푏) = 푐} for 푓 ∈ Z[푥, 푦] a homogeneous polynomial of degree 푑 ≥ 3. Our exposition mostly follows that of [MP12], which we enthusiastically recommend.

3.1 Chabauty’s result

We begin with an overview of Chabauty’s idea adapted to our more general setting. Our setup is as in Section 2.1.

Our goal is to compute the set 풞(풪퐾,푆) of 푆-integral points on the punctured

53 curve 풞.

Fix a prime p ∈/ 푆. Let 퐾p be the completion of 퐾 at p and let 풪p denote the ring of integers of 퐾p. Then there is a commutative diagram:

풞(풪퐾,푆) 풞(풪p)

풥 (풪퐾,푆) 풥 (풪p) .

In particular, inside of 풥 (풪p), we have the inclusion

풞(풪퐾,푆) ⊆ 풥 (풪퐾,푆) ∩ 풞(풪p) .

Now, 풞 has relative dimension one over 풪퐾,푆, so if we can find locally 푝-adic analytic functions on 풥 (풪p) which vanish identically on 풥 (풪퐾,푆) but do not vanish identically on 풞(풪p), we will be able to prove that the intersection, and therefore

풞(풪퐾,푆), is finite. Let denote the relative dimension of over . Then, is a - dim 풥 풥 풪퐾,푆 풥(풪p) Q푝 analytic manifold (in the naïve sense of Serre [Ser65] or [Ser06, Part 2, Section 3]) of dimension . (dim 풥) · [퐾p : Q푝]

Also, 풥 (풪퐾,푆) is a finitely generated group of some rank 푟. As we will see, its closure in is a -adic Lie subgroup of some dimension . If 풥 (풪퐾,푆) 풥 (풪퐾p ) 푝 휌 ≤ 푟 , then will be a proper -adic analytic Lie 휌 < (dim 풥 ) · [퐾p : Q푝] 풥 (풪퐾,푆) $ 풥(퐾p) 푝 subgroup. Since 풞 is an irreducible curve, if 풞(풪p) is not contained in 풥(풪퐾,푆), the intersection

풥 (풪퐾,푆) ∩ 풞(풪p) will be finite. Indeed, when and is an Abel-Jacobi embedding of a proper 퐾 = Q푝 푗 : 풞 → 풥 curve into its Jacobian, this is what Chabauty proved.

Theorem 3.1.1 ([Cha41], as stated in [MP12, Theorem 4.4]). Let 푋 be a smooth, projective, geometrically integral curve of genus 푔 ≥ 2 over Q. Let 퐽 be the Jacobian of 푋. Let 푝 be a prime and let 휌 = dim 퐽(Q) ≤ rank 퐽(Q). Suppose that 휌 < 푔. Then, is finite. It follows that is finite as well. 푋(Q푝) ∩ 퐽(Q) 푋(Q)

54 Chabauty’s original result was not effective. Coleman later used his theory of 푝- adic integration to show how to explicitly compute equations vanishing on 풥 (풪퐾,푆). The remainder of this section is an overview of the ideas needed for Coleman’s approach, followed by some more recent further improvements.

3.2 The 푝-adic logarithm map

In this section, we assume that 퐾p is a 푝-adic field with ring of integers 풪p. Let 풥 be a semi-abelian variety over 풪p of relative dimension 푛. The set has the structure of a -Lie group of dimension . Let 풥 (퐾p) Q푝 푛 · [퐾p : Q푝] be the identity element of and write for the Lie algebra of , 푒 퐽(퐾p) Lie 풥퐾p 풥 (퐾p) i.e. the tangent space of 풥 (퐾p) at 푒.

Since 풥 is of finite type, 풥 (풪p) has the structure of a profinite group. The profinite topology agrees with the topology from 풥 (퐾p).

In particular, for any 푥 ∈ 풥 (풪p), there is some increasing sequence of positive

푛푖 integers 푛푖 such that lim푖→∞ 푥 = 푒.[Bou98, III.7.6., Proposition 10] says that there is a unique 푝-adic analytic homomorphism

log : 풥 (풪p) → Lie 풥퐾p , called the 푝-adic logarithm, which is injective with analytic inverse after restricting the domain and codomain to suitable open subgroups.

The 푝-adic logarithm map can be understood explicitly as an antiderivative, as we now explain.

Let 0 1 denote the vector space of translation-invariant global differ- 퐻 (풥퐾p , Ω )inv entials on . We have 풥퐾p 0 1 dim퐾p 퐻 (풥퐾p , Ω )inv = 푛 .

Let 0 1 and choose local coordinates on at . In a 휔 ∈ 퐻 (풥퐾p , Ω )inv 푡1, . . . , 푡푛 풥퐾p 푒

55 neighborhood of 푒, the 1-form 휔 can be expanded as a power series

푔+훾 ∑︁ 휔|푈 = 퐹푖(푡1, . . . , 푡푔+훾)푑푡푖 . 푖=1

Integrating the power series formally, and restricting 푈 further if necessary gives a map

휂 : 푈 → 퐾p , ∫︁ 푃 푃 → 휔 . 푒

The map 휂 extends uniquely to a homomorphism 휂 : 풥 (풪p) → 퐾p. We abuse notation slightly and write ∫︀ 푃 for , even when is not in the radius 푒 휔 휂 푃 of convergence of the 푝-adic power series. Now, formal integration is linear in the differential form, so in fact, this defines a homomorphism:

0 1 퐻 (풥퐾p , Ω )inv → Hom(풥 (풪p), 퐾p) .

Equivalently, we have a morphism of 푝-adic Lie groups

0 1 ∨ ∼ log : 풥 (풪p) → (퐻 (풥퐾p , Ω )inv) = Lie 풥퐾p ,

which can be identified with the 푝-adic logarithm map defined earlier.

This map can be written explicitly as follows: Let {휔1, . . . , 휔푛} be a basis for 0 1 . Taking the dual basis determines an isomorphism ∼ 푛. In 퐻 (풥퐾p , Ω )inv Lie 풥퐾p = 퐾p this setup, the logarithm map can be written explicitly as

푛 log : 풥 (풪p) → 퐾p , (︂∫︁ 푃 ∫︁ 푃 )︂ 푃 ↦→ 휔1,..., 휔푛 . 푒 푒

Since the logarithm map is a homomorphism and 푛 is torsion-free, we see that 퐾p

56 the torsion subgroup of is contained in . In fact, these sets are equal. 풥 (풪퐾p ) ker(log) The 푝-adic logarithm map is a local homeomorphism, with inverse defined on a neighborhood of 0 by a formal exponential map. As a consequence, we have:

Proposition 3.2.1. Suppose that 퐺 ⊂ 풥 (풪p) is a rank 푟 finitely-generated subgroup.

The closure 퐺 of 퐺 in 풥 (퐾p) is a 푝-adic Lie subgroup of dimension 휌 ≤ 푟.

Proof. Since log is a homomorphism and ker(log) consists of torsion elements, log 퐺 ⊂ is a finitely-generated subgroup of rank . Then, is a -adic Lie 퐾p 푟 log 퐺 = Z푝 · log 퐺 푝 subgroup of 푛. Moreover, 퐾p

휌 = rankZ푝 Z푝 · log 퐺 ≤ rank 퐺 = 푟 .

Since log is locally a homeomorphism, log 퐺 = log 퐺 and so 퐺 is a 푝-adic Lie subgroup

of 퐽(퐾p) of dimension 휌 ≤ 푟.

Remark 3.2.2. Let be as in Proposition 3.2.1. The set has a natural -structure, 퐺 퐺 Z푝

but in general will not carry an 풪p-structure. If we want to construct locally 퐾p- analytic functions which vanish on 퐺, we can do the following. Let ′ . Clearly ′ . 휌 = dim풪p 풪p · log 퐺 휌 ≤ 휌 ′ ′ When 휌 < 푛, we construct 푛 − 휌 functions on 풥 (풪p) which vanish on 퐺.

1. Fix a set of generators {푃1, . . . , 푃푟} for a finite index subgroup of 퐺.

2. Compute the images log(푃1),..., log(푃푟) and put them in the rows of a matrix ′ 푀. The resulting matrix has rank 휌 over 퐾p.

3. Let ⃗푎1 = (푎1,1, . . . , 푎푛,1), . . . ,⃗푎푛−휌′ span the kernel of 푀.

4. Set ′ ∑︀푛 . The functions 휔푗 = 푖=1 푎푖,푗휔푖

휂푗 : 풥 (풪p) → 퐾p (3.2.3) ∫︁ 푃 ′ 푃 ↦→ 휔푗 휖

57 ′ for 푗 = 1, . . . , 푛 − 휌 vanish whenever 푃 ∈ {푃1, . . . , 푃푟}. Since the 휔푖 are

translation-invariant, by additivity, the 휂푗 vanish on all of 퐺. We typically hope to apply this construction to the finitely generated subgroup

풥 (풪퐾,푆). Remark 3.2.4. Unfortunately, the functions constructed in Remark 3.2.2 are somewhat inefficient. As a Lie group, their common vanishing locus has dimension Q푝 ′ , which could be much larger than . 휌 · [퐾p : Q푝] 휌 = dim 퐺 Of course this problem does not exist when . If we replaced with 퐾p = Q푝 풥퐾p before taking integrals, this problem would go away. Later on, we take Res퐾p/Q푝 풥퐾p this restriction of scalars at the level of number fields, which also allows us to use

information from other primes above 푝.

Suppose now that we are in the situation where 푗 : 풞 → 풥 is an Abel-Jacobi embedding of a curve into generalized Jacobian (in the sense of Section 2.2). Take

′ 퐺 = 풥 (풪퐾,푆) in Remark 3.2.2. Although is is possible that 휌 < 푟, in the typical case, we expect that 푟 = 휌′. For instance, under Leopoldt’s Conjecture, 푟 = 휌′ whenever

푔풞 = 0.

In particular, we know that Chabauty’s method will prove that 풞(풪퐾,푆) is finite when 푟 ≤ dim 풥퐾 − 1. To ease our future discussion of when Chabauty’s method applies, we make the following definition.

Definition 3.2.5. Let 풞/풪퐾,푆 be a sound clearance hole curve. Let 풥 be the generalized Jacobian of 풞.

We say that the pair (풞, 풪퐾,푆) satisfies the classical Chabauty inequality if

rank 풥 (풪퐾,푆) ≤ dim 풥퐾 − 1 . (3.2.6)

3.3 Coleman’s approach

In this section, we discuss Coleman’s approach to computing the 푝-adic analytic functions constructed in Remark 3.2.2 and bounding the number of points of 풞(풪퐾,푆).

58 The key idea is to replace integration on the Jacobian (which is hard to compute) with integration on the curve (which is easier to compute).

The morphism 푗 : 풞 → 풥 induces a homomorphism of vector spaces

* 0 1 0 1 푗 : 퐻 (풥퐾p , Ω ) → 퐻 (풞퐾p , Ω ) .

Remark 3.3.1. If is the generalized Jacobian of , the morphism * induces an 풥퐾p 풞퐾p 푗 isomorphism of 퐾p-vector spaces

* 0 1 0 1 푗 : 퐻 (풥퐾p , Ω )inv . → 퐻 (풞퐾p , Ω (log)), where Ω1(log) denotes the sheaf of meromorphic 1-forms with poles of order at most at the punctures of and no other poles. (See [Ser88, Section V.2.10]) 1 풞퐾p

Suppose that and reduce to the same point of . Then, for any 푃1 푃2 풞(Fp) 휔 ∈ 0 1 , we have 퐻 (풥퐾p , Ω )inv . ∫︁ [푃1−푃2] ∫︁ 푃2 휔 = 푗*휔 , 푒 푃1 where the second integral is given by formal integration of power series in terms of a local parameter on 풞.

More precisely, fix and let be such that the restriction to the 푄 ∈ 풞(Fp) 푡 ∈ 풪풞,푄 special fiber 푡| is a uniformizer at 푄. The map 풞Fp

∼ 푡 : {푃 ∈ 풞(풪p): 푃 (mod p) = 푄} → p · 풪p

from the residue disc around 푃 to p풪p is a bijection. Moreover, if 휔 ̸= 0, then up to scaling by an element of ×, the -form * can be written in the form for 퐾p 1 푗 휔 푓(푡)푑푡 some . We may assume that some coefficient of lies in ×. Let be the 푓 ∈ 풪p[[푡]] 푓 풪p 퐹 formal antiderivative of 푓. Then

∫︁ 푃2 ∫︁ 푡(푃2) * 푗 휔 = 푓(푡) 푑푡 = 퐹 (푡(푃2)) − 퐹 (푡(푃1)) . 푃1 푡(푃1) 59 If we take 휔 ̸= 0 such that ∫︁ 푃 휔 = 0 푒 for all 푃 ∈ 풥 (풪퐾,푆) as in Remark 3.2.2, and fix a basepoint 푃0 in each residue disc

on 풞, we can bound #풞(풪퐾,푆) by using Newton polygons to bound the number of

zeros of 퐹 (푡(푃 )) − 퐹 (푡(푃0)) on each residue disc based on the 푝-adic valuations of the coefficients of 퐹 . With this approach, Coleman proved

Theorem 3.3.2 ([Col85, Corollary 4a]). Let 푋 be a smooth, projective, geometrically integral curve of genus 푔 ≥ 2 over 퐾. Let 퐽 be the Jacobian of 푋. Let p be a prime of 퐾 which is unramified over Q, lying above 푝 > 2푔. Let 휌 = dim 퐽(Q) ≤ rank 퐽(Q). Suppose that 휌 < 푔. Then,

#푋(퐾) ≤ #푋(Fp) + (2푔 − 2) .

Coleman also showed how to bound #푋(퐾) when 푝 ≤ 2푔, although the bounds are not as good.

Remark 3.3.3. In Remark 3.2.2, we constructed at least dim 풥퐾 − 휌 functions simultaneously vanishing on 풥 (풪p). If one considers all of these functions instead of a single function, [Sto06, Corollary 6.7] shows that Coleman’s bound can be improved to . #푋(Fp) + 2휌

The same ideas used by Coleman and Stoll apply to bound 풪퐾,푆 points on a sound clearance hole curve 풞/풪퐾,푆, although the bounds produced by the method may be different.

60 Chapter 4

Chabauty for Restrictions of Scalars

This section aims to give an overview of Chabauty’s method for restrictions of scalars of curves (or RoS Chabauty, for short). Our description differs from Siksek [Sik13] in two main ways. First, with an eye towards testing the theory on the unit equation, we develop the method for 푆-integral points on models of curves which may not be complete. Second, after taking restrictions of scalars, we consider points valued in the

Galois closure of the field that we start with. This simplifies the multivariate 푝-adic power series that appear when applying the method.

4.1 Introduction to RoS Chabauty

The classical Chabauty inequality (3.2.6) is somewhat inefficient in two ways:

1. As we saw in Remark 3.2.4, the vanishing locus of the Chabauty functions can be much larger than , especially when is large. 풥 (풪퐾,푆) [퐾p : Q푝]

2. If 푝 splits in 퐾, the classical Chabauty-Coleman approach uses information only at one of the primes p lying over 푝.

We can solve both of these problems and weaken the hypothesis of classical Chabauty somewhat using restriction of scalars, as follows:

61 Set

풱 := Res 풞 and 풜 := Res 풥 . 풪퐾,푆 /Z푆′ 풪퐾,푆 /Z푆′

The map 푗 induces a generically finite map 푗 : 풱 ˓→ 풜. Similarly to classical Chabauty, there is a commutative diagram:

풞(풪퐾,푆) 풱(Z푆′ ) 풱(Z푝) (4.1.1)

풥 (풪퐾,푆) 풜(Z푆′ ) 풜(Z푝).

We observe:

1. Although 풜 might not be semi-abelian, there is a logarithm map

푑·dim 풥퐾 log: 풜(Z푝) → Q푝

much like in the case of classical Chabauty, even if 푝 is ramified in 퐾. In particular, is a -adic Lie group of dimension over . 풜(Z푝) 푝 푑 · dim(풥 ) Z푝

2. The closure of in the -adic topology is a -adic Lie group 풜(Z푆′ ) ⊂ 풜(Z푝) 푝 푝 of dimension over for some . In contrast to classical Chabauty (as 휌 Z푝 휌 ≤ 푟 described in Section3 and in particular Remark 3.2.4,) the set is finite- 풜(Z푆′ ) index in some subgroup cut out by locally analytic 푝-adic functions obtained by

composing with linear functionals 푑·dim 풥퐾 . log Q푝 → Q푝

3. Inside of 풥 (풪p), we have the inclusion

(4.1.2) 풞(풪퐾,푆) = 풱(Z푆′ ) ⊆ 풱(Z푝) ∩ 풜(Z푆′ ).

If , or equivalently if codim(풱, 풜) + codim(풜(Z푆′ ), 풜(Z푝)) ≥ dim 풜

휌 ≤ 푑(dim 풥풞 − 1), (4.1.3)

62 then if the intersection is ‘sufficiently generic’ (e.g. transverse) the right-hand sideof

(4.1.2) will consist of isolated points, which would imply that 풞(풪퐾,푆) is finite. By analogy with the classical case, we make the following definition.:

Definition 4.1.4. Let 풞/풪퐾,푆 be a sound clearance hole curve (see Definition 2.1.1) and let 풥 be the generalized Jacobian of 풞 in the sense of Section 2.2.

The pair (풞, 풪퐾,푆) satisfies the RoS Chabauty inequality if

푟 ≤ 푑(dim 풥퐾 − 1). (4.1.5)

Since the generic fiber of 풜 is a semiabelian variety, we can still understand the logarithm map in terms of integration on 풜. In particular, the vector space

{︂ ∫︁ 푃 }︂ ′ 0 1 for all ′ (4.1.6) Ω := 휔 ∈ 퐻 (풜Q푝 , Ω )inv : 휔 = 0 푃 ∈ 풜(Z푆 ) , 0 has dimension at least 푑 · dim 풥퐾 − 푟, and

{︂ ∫︁ 푃 }︂ for all ′ 풥(풪퐾,푆) = 풜(Z푆′ ) ⊆ 푃 ∈ 풜(Z푝): 휔 = 0 휔 ∈ Ω . 0

Analogously to the classical Chabauty case, when are the images 푃1, 푃2 ∈ 풜(Z푆′ ) of points of and is an invariant differential, we can compute the integral 풱(Z푆′ ) 휔

∫︁ 푃2 휔 푃1 on by taking a pullback. More precisely, fix and let in the local 풱 푄 ∈ 풱(F푝) 푡1, . . . , 푡푑 ring 풪풱,푄 be generators for the maximal ideal m풱,푄 of 풪풱,푄. The functions 푡1, . . . , 푡푑 induce a map

∼ 푑 (푡1, . . . , 푡푑): {푃 ∈ 풱(풪p): 푃 (mod 푝) = 푄 → (푝Z푝) , which is a bijection from the residue disc around to 푑. Much like in the 푃 (푝Z푝) classical case, after possibly changing scaling by an element of ×, the -form * Q푝 1 푗 휔

63 can be written in the form ∑︀푑 for some 푗=1 푓푗(푡1, . . . , 푡푑)푑푡푗 푓1, . . . , 푓푑 ∈ Z푝[[푡1, . . . , 푡푑]] where some has a coefficient in ×. 푓푗 Z푝 Since invariant 1-forms on 풜 are closed and pullbacks of closed forms are closed, there is some such that 퐹휔 ∈ Q푝[[푡1, . . . , 푡푑]] 푑퐹휔 = 휔

Then, for 푃1 and 푃2 in the same residue disc as 푃 , we have

∫︁ 푃2 * 푗 (휔) = 퐹휔((푡1, . . . , 푡푑)(푃2)) − 퐹휔((푡1, . . . , 푡푑)(푃1)) . 푃1

Notably, the functions 퐹휔 are no longer expressible as (locally) 푝-adic analytic

functions in a single variable. Instead, on sufficiently small balls, the 퐹휔 will be given by 푑-variable power series which converge on a neighborhood of the balls. Let be a completion of and let be its ring of integers. C푝 Q푝 풪C푝 Given linearly independent one-forms and , such that the converge 푑 휔 훼 ∈ Q 퐹휔 on a neighborhood of 푃 + 푝훼풪푑 , the common vanishing locus of these power series C푝 will have finitely many isolated points in the disc 푃 + 푝훼풪푑 . C푝 However, if we are unlucky (or in a bad situation), the common vanishing locus of these functions may not consist exclusively of isolated points. Unfortunately, such bad situations can occur for a geometric reason. Before continuing with the development of RoS Chabauty, we discuss two geomet-

ric reasons why the common vanishing locus of the 퐹휔 for the curve 풞/풪퐾,푆 may not consist of isolated points.

One such geometric obstruction occurs when 풞 is the base change of a curve 풟 for which the vanishing locus of the in is infinite (e.g., because does 퐹휔 (Res 풟)(Z푝) 풟 not satisfy the RoS Chabuaty inequality.)

A second geometric obstruction occurs when there is a morphism 푓 : 풞 → 풟 such

that the vanishing locus of the 퐹휔 for 풟 does not consist of isolated points and the * Prym scheme 풫 (the group scheme quotient 풫 := 풥풞/푓 (풥풟)) has the property that the 푝-adic closure

(Res 풫)(풪 ) ⊂ (Res 풫)( ) 풪퐾,푆 /Z푆′ 퐾,푆 풪퐾,푆 /Z푆′ Z푝

64 is finite-index. We discuss these obstructions, which we call base change obstructions and full Prym obstructions in the next two sections. In all known cases where does not consist of finitely many isolated 풱(Z푝)∩풜(Z푆′ ) points, it can be explained by iterated application of these obstructions.

4.2 Base change obstructions

Suppose that 푘 ⊂ 퐾 is a subfield and that 푆푘 is the set of primes of 푘 lying under primes in . Suppose also that is a sound clearance hole curve which becomes 푆 풟/풪푘,푆푘

isomorphic to 풞 after base change to 풪퐾,푆, i.e., a curve such that

∼ 풟풪퐾,푆 = 풞.

We also have ∼ compatibly with the Abel-Jacobi maps. (풥풟)풪퐾,푆 = 풥풞

Letting 풲 := (Res풪 / 풟) and ℬ := (Res풪 / 풥풟), we have the following 푘,푆푘 Z푆′ 푘,푆푘 Z푆′ commutative diagram:

풲(Z푆′ ) ℬ(Z푆′ )

풱(Z푆′ ) 풜(Z푆′ )

풲(Z푝) ℬ(Z푝)

풱(Z푝) 풜(Z푝) Clearly, inside of . If 풲(Z푝) ∩ ℬ(Z푆′ ) ⊂ 풱(Z푝) ∩ 풜(Z푆′ ) 풜(Z푝)

dim ℬ(Z푆′ ) > [푘 : Q] · (dim 풥풟 − 1), then will (typically) have positive dimension as a -adic analytic 풲(Z푝) ∩ ℬ(Z푆′ ) 푝 variety in the sense of [Ser65], so will have positive dimension as a 풱(Z푝) ∩ 풜(Z푆′ ) 푝-adic analytic variety as well. In particular, both sets will be infinite.

65 Based on this observation, we say

Definition 4.2.1. A base change obstruction to RoS Chabauty for (풞, 풪퐾,푆) is a pair where is a subfield of , is the set of primes of lying under primes (풟, 풪푘,푆푘 ) 푘 퐾 푆푘 푘 in , is a sound clearance hole curve over , and ∼ such that 푆 풟 풪푘,푆푘 풟풪퐾,푆 = 풞

푗((Res풪 / 풟)( 푝)) ∩ (Res풪 / 풥풟)( 푆′ ) 푘,푆푘 Z푆′ Z 푘,푆푘 Z푆′ Z

inside (Res풪 / 풥풟)( 푝) is infinite. 푘,푆푘 Z푆′ Z A strong base change obstruction to RoS Chabauty is a base change obstruction such that

(4.2.2) rank 풥풟(풪푘,푆푘 ) > [푘 : Q] · (dim 풥풟 − 1) .

This is the inequality that one would expect to lead to the above intersection being infinite (although this implication does not always hold.) If nosuch 풟 exists, we say that (풞, 풪퐾,푆) has no strong base change obstruction to RoS Chabauty.

Strong base change obstructions do show up in practice.

Example 4.2.3. If 풞 = P1 r {0, 1, ∞}, 퐾 is a CM sextic field, and 푘 is the totally real cubic subfield of , then ∼ and 퐾 풥 = G푚 × G푚

× 2 · rank 풪퐾 = 4 ≤ 6 = [퐾 : Q] · 1 , so (4.1.5) holds. Also

× 2 · rank 풪푘 = 4 > 3 = [푘 : Q] · 1 , so (4.2.2) also holds. It follows that if the intersection in Definition 4.2.1 is infinite, then ( 1 {0, 1, ∞}, 풪 ) is a strong base change obstruction to RoS Chabauty for P풪푘 r 푘 ( 1 {0, 1, ∞}, 풪 ). This is the case in typical examples. P풪퐾 r 퐾

66 Example 4.2.4. If 풞/Q is a smooth projective curve of genus 푔 and 퐾 is a number field with

rank 풥풞(Q) ≥ 푔,

but

rank 풥풞(퐾) ≤ [퐾 : Q] · (푔 − 1), then (풞, Q) is typically a strong base change obstruction to RoS Chabauty for (풞, 퐾).

4.3 Full Prym obstructions

In [Dog19], Dogra shows that there is another geometric reason that the intersection why RoS Chabauty sometimes fails to prove finiteness of rational points. In this section, we give a heuristic-based explanation of the situation that [Dog19] describes.

Suppose that 풞 → 풟 is a finite morphism of sound clearance hole curves over

풪퐾,푆. Suppose that the following running assumptions (for this section) hold:

(i) The intersection 푍 := (Res 풟)( )∩(Res 풥 )( ′ ) is infinite. In 풪퐾,푆 /Z푆′ Z푝 풪퐾,푆 /Z푆′ 풟 Z푆

particular, this means that RoS Chabauty fails to prove that 풟(풪퐾,푆) is finite.

(ii) . rank 풥풞(풪퐾,푆) − rank 풥풟(풪퐾,푆) ≥ [퐾 : Q] · (dim 풥풞 − dim 풥풟)

Under these conditions, we will now explain why the intersection

(Res 풞)( ) ∩ (Res 풥 )( ′ ) 풪퐾,푆 /Z푆′ Z푝 풪퐾,푆 /Z푆′ 풞 Z푆

is likely to be infinite, so that RoS Chabauty will fail to prove that 풞(풪퐾,푆) is finite.

* Note that there are pullback and pushforward maps 푓 : 풥풞 → 풥풟 and 푓* : 풥풟 →

풥풞. Let 풫 := ker(푓*) and let 퐺 := coker(풥풞(풪퐾,푆) → 풥풟(풪퐾,푆)). We have the exact sequence

0 → 풫(풪퐾,푆) → 풥풞(풪퐾,푆) → 풥풟(풪퐾,푆) → 퐺 → 0

67 which we can rewrite as

0 → (Res 풫)( ′ ) → (Res 풥 )( ′ ) → (Res 풥 )( ′ ) → 퐺 → 0 . 풪퐾,푆 /Z푆′ Z푆 풪퐾,푆 /Z푆′ 풞 Z푆 풪퐾,푆 /Z푆′ 풟 Z푆

Now imagine that the following hypotheses hold:

1. 퐺 = {1}, i.e., 푓* : 풥풞(풪퐾,푆) → 풥풟(풪퐾,푆) is surjective,

2. (Res 풫)( ′ ) is dense in (Res 풫)( ), 풪퐾,푆 /Z푆′ Z푆 풪퐾,푆 /Z푆′ Z푝

3. is surjective. 푓 : 풞(풪퐾,푆 ⊗ Z푝) → 풟(풪퐾,푆 ⊗ Z푝)

From hypotheses 1 and 2, for each 푃 ∈ 푍 ⊂ (Res 풥 )( ′ ), the 푝-adic 풪퐾,푆 /Z푆′ 풟 Z푆 −1 closure (Res 풥 )( ′ ) would contain (푓 ) (푃 ). 풪퐾,푆 /Z푆′ 풞 Z푆 *

−1 푓 (푍) ⊂ (Res 풞)( ) ∩ (Res 풥 )( ′ ) . 풪퐾,푆 /Z푆′ Z푝 풪퐾,푆 /Z푆′ 풞 Z푆

Under hypothesis 3, 푓 −1(푃 ) ⊂ Res ( ) is non-empty for each 푃 ∈ 푍, so 풪퐾,푆 /Z푆′ Z푝 푓 −1(푍) is infinite as well. Hence, these conditions would imply that (Res 풞)( )∩ 풪퐾,푆 /Z푆′ Z푝

(Res 풥 )( ′ ) is infinite. 풪퐾,푆 /Z푆′ 풞 Z푆

We illustrate this situation with the further approximation 풥풞 ≈ 풥풟 ×풫 in Figure 4-1.

In practice, these hypotheses 1, 2, and 3 are unlikely to hold. However, as we now discuss, each is likely to hold up to finite index/on an open subset of 푍 under running assumptions (i) and (ii).

* Since 푓*∘푓 is the multiplication by deg(푓) map and 풥풟(풪퐾,푆) is finitely generated, 퐺 is a finite group, so 1 is true up to finite index. For 2, note that

rank(Res 풫)( ′ ) = rank 풥 (풪 ) − rank 풥 (풪 ) 풪퐾,푆 /Z푆′ Z푆 풞 퐾,푆 풟 퐾,푆

68 Figure 4-1: An approximate graphical illustration of a full Prym obstruction. In the notation of this section, the curve downstairs is . 푍 = (Res 풥풟)(풪퐾,푆) ∩ (Res 풟)(Z푝) Upstairs, contains a finite index subgroup of the preimage −1 (Res(풫 × 풥풟))(Z푝) 휋 (푍) of the curve downstairs. The image of intersects −1 in the curve (Res 풞)(Z푝) 휋 (푍) upstairs, and contains this curve. (휑 ∘ 푗)(Res 풞)(Z푝) ∩ (Res 풫 × 풥풟)(풪퐾,푆)

69 and

dim(Res 풫) = [퐾 : ] · dim 풫 = [퐾 : ] · (dim 풥 − dim 풥 ) . 풪퐾,푆 /Z푆′ Q Q 풞 풟

Running assumption (ii) says that

rank(Res 풫)( ′ ) ≥ dim(Res 풫), 풪퐾,푆 /Z푆′ Z푆 풪퐾,푆 /Z푆′

so in generic situations, (Res 풫)( ′ ) will be finite index in (Res 풫)( ). 풪퐾,푆 /Z푆′ Z푆 풪퐾,푆 /Z푆′ Z푝 Hence, 2 is likely to hold up to finite index. Finally, 3 is a ultimately a statement about some collection of polynomials having roots over a local field with finite residue field. Suppose that . 푃 ∈ 풞(풪퐾,푆 ⊗ Z푝) By the implicit function theorem, after a small 푝-adic perturbation, we may assume that is étale at the points given by . Then, by Krasner’s lemma, for any 푓 Q푝 푃 , there is an open neighborhood of such 푃 ∈ 풞(풪퐾,푆 ⊗ Z푝) 푈 푓(푃 ) ∈ 풟(풪퐾,푆 ⊗ Z푝) that . In particular, if then 푈 ⊂ 푓(풞(풪퐾,푆 ⊗ Z푝)) 풞(풪퐾,푆 ⊗ Z푝) ̸= ∅ 푓(풞(풪퐾,푆 ⊗ Z푝)) contains a nonempty open subset of 풟(풪퐾,푆 ⊗ Z푝). Putting these together, under running assumptions (i) and (ii), we see that heuris- tically,

(Res 풞)( ) ∩ (Res 풥 )( ′ ) 풪퐾,푆 /Z푆′ Z푝 풪퐾,푆 /Z푆′ 풞 Z푆 is likely to be infinite whenever it is nonempty, even if hypotheses 1, 2, and3are not satisfied on the nose. This seems especially likely in the case that 풞(풪퐾,푆) is nonempty, which is often the case when trying to apply variants of the Chabauty– Coleman method in practice. To capture situations where this sort of obstruction could exist, we make the following definition:

Definition 4.3.1. A full Prym obstruction to RoS Chabauty for (풞, 풪퐾,푆) is a mor-

phism 풞 → 풟 to a sound clearance hole curve 풟/풪퐾,푆 such that both

(i)

푗((Res 풟)( )) ∩ (Res 풥 )( ′ ) 풪퐾,푆 /Z푆′ Z푝 풪퐾,푆 /Z푆′ 풟 Z푆

70 inside (Res 풥 )( ) is infinite and 풪퐾,푆 /Z푆′ 풟 Z푝

(ii)

rank 풥풞(풪퐾,푆) − rank 풥풟(풪퐾,푆) ≥ [퐾 : Q] · (dim 풥풞 − dim 풥풟) .

Example 4.3.2. Full Prym obstructions (that are not simultaneously base change √︀ obstructions) do exist. Following [Dog19], take 푏 = 11/27 and 퐾 = Q(푏), to be the smooth projective genus 3 hyperelliptic curve 퐶 over 퐾 with affine model

2916 · 푏 + 484 −128304 · 푏 + 168112 푦2 = 푥8 + 푥6 + 푥4 297 8019 214057729 · 푏 − 35529472 −10784721024 · 푏 + 874208708 + 푥2 + . 23181643 64304361

One can compute that

rank 풥퐶 (퐾) = 4 < 6 = [퐾 : Q] · dim 풥퐶 , so the RoS Chabauty inequality holds for 퐶. One can also check that 퐶 is not a base change of any curve defined over Q, so 퐶 does not have a base change obstruction.

On the other hand, 퐶 covers the base change 퐷퐾 of the smooth, projective, genus 2 hyperelliptic curve 퐷/Q with affine model

(︂ 11)︂ (︂ 27)︂ 푦2 = 푥4 − 푥2 − . 27 11

We claim that 퐶 → 퐷퐾 is a full Prym obstruction to RoS Chabuaty for 퐶. Now, and both have rank . In particular, 풥퐷(Q) 풥퐷(퐾) 2

rank 풥퐷(Q) = 2 > 1 = dim 풥퐷 − 1 ,

so it is plausible that 퐷 is a base change obstruction to RoS Chabauty for 퐷퐾 . With a bit more work, one can show that this indeed the case. In particular, (i) of definition 4.3.1 is satisfied.

Moreover, the Prym variety of 퐶 → 퐷퐾 is an elliptic curve 퐸 defined over 퐾.

71 A computation shows that rank 퐸(퐾) = 2 and the 푝-adic closure 퐸(퐾) is finite- index in . Hence, (Res퐾/Q 퐸)(Q푝)

rank 풥퐶 (퐾) − rank 풥퐷(퐾) = 2 ≥ 2 = [퐾 : Q](dim 풥퐶 − dim 풥퐷) ,

so (ii) of definition 4.3.1 is also satisfied. Thus, there is a full Prym obstruction

to RoS Chabauty for 퐶. The preceding discussion suggests that the intersection of and is likely to be infinite. (Res퐾/Q 퐶)(Q푝) 풥퐶 (퐾)

4.4 Simplifying regular differentials via base change

′ [Sik13] presents a sufficient condition to determine if the functions 퐹휔 for 휔 ∈ Ω constructed in equation 4.1.6 have at most a single zero in a residue disc. The condition that [Sik13] presents is very computational and seems difficult to strengthen or to analyze in general. One difficulty is that the 퐹휔, when expressed as power series in local parameters on a disc, will typically have ‘few’ nonzero coefficients. For both computation and abstract reasoning, it would be nicer to express the 퐹휔 in terms of sparse power series.

In this section, we describe a different approach, which allows us to do this. Our plan is to expand the 퐹휔 as power series in local parameters which are defined only over a field extension of . This will be useful in Chapter6. Q푝 Suppose that is unramified in . Let be a finite extension of which is 푝 퐾/Q 퐿P Q푝 a splitting field for . Then, . 퐾/Q푝 # Hom(퐾, 퐿P) = [퐾 : Q] Let denote the valuation ring of . Then, 풪퐿P 퐿P

∼ ∏︁ 풪퐾,푆 ⊗ 풪퐿P = 풪퐿P , 휏

where ranges over 휏 Hom(풪퐾,푆, 풪퐿P ) = Hom(퐾, 퐿).

Now, suppose that 풵 is a quasi-projective 풪퐾,푆-scheme. For 휏 as above, let 풵휏

72 over be the base change. Then, 풪퐿P

∼ ∏︁ (Res 풵)풪 = 풵휏 . 풪퐾,푆 /Z푆′ 퐿P 휏

The composition

∏︁ 풵(풪 ) = (Res 풵)( ′ ) ˓→ (Res 풵)(풪 ) = 풵 (풪 ) 퐾,푆 풪퐾,푆 /Z푆′ Z푆 풪퐾,푆 /Z푆′ 퐿P 휏 P 휏 is given by 푃 ↦→ (휏(푃 ))휏 .

Applying this for 풵 = 풞 and 풵 = 풥, we obtain a variant of the RoS Chabauty diagram (4.1.1). We have

′ ∏︀ 풞(풪퐾,푆) 풱(Z푆 ) 풱(풪퐿P ) 휏 풞휏 (풪P) (4.4.1)

′ ∏︀ 풥 (풪퐾,푆) 풜(Z푆 ) 풜(풪퐿P ) 휏 풥휏 (풪P).

The advantage of replacing the restrictions of scalars 풱 and 풜 by their base change to is that the spaces of regular -forms can be related directly to the -forms on 풪퐿P 1 1 풥 and 풞. Specifically,

(︃ )︃ 0 1 ∼ 0 ∏︁ 1 ∼ ∏︁ 0 1 ∼ ∏︁ 0 1 퐻 (풜퐿P , Ω )inv = 퐻 풥휏 , Ω = 퐻 (풥휏 , Ω )inv = (퐻 (풥 , Ω )inv ⊗휏 퐿P) 퐾 휏 inv 휏 휏

and similarly for 0 1 . For in 0 1 ∏︀ 0 1 퐻 (풱퐿P , Ω )inv 휔 = (휔휏 ) 퐻 (풜퐿P , Ω )inv = 휏 (퐻 (풥 , Ω )inv⊗휏 퐾 and in ∏︀ , we have 퐿P) 푃 = (푃휏 ) 풜(풪퐿P ) = 휏 풥휏 (풪퐿P )

∫︁ 푃 ∑︁ ∫︁ 푃휏 휔 = 휔휏 . 0 휏 0

Similarly, -adic integrals on decompose as a sum of integrals on the mul- 푝 풱퐿P tiplicands . If we choose local parameters near in 풞휏 푡1, . . . , 푡푑 푃 = (푃휏 ) 풱(풪퐿P ) = ∏︀ which are local parameters on the multiplicands, then for in the 휏 풞휏 (풪퐿P ) 푃1, 푃2

73 same (sufficiently small) residue disc, there exist functions 퐹휔,푖 ∈ 퐿P[[푡푖]] such that

푑 ∫︁ 푃2 * ∑︁ 푗 휔 = 퐹휔,푖(푡푖(푃1)) − 퐹휔,푖(푡푖(푃2)) . 푃1 푖=1

In particular,

−1 ∏︁ 푗 (풥(풪퐾,푆)) ⊂ 풞휏 (풪퐿P ) 휏 is contained in the vanishing locus of a collection of power series of this form, which we can compute explicitly by linear algebra analogous to that in Remark 3.2.2. Such power series are said to be pure because the coefficients of any mixed terms are zero. In particular, these power series are sparse.

This sparse representation comes at some cost – computations over 퐿P are significantly more expensive than computations over and the -valued vanishing set Q푝 풪퐿P of our equations may be larger than the set computed when working over . This Z푝 latter cost can be mitigated by keeping track of the action of . We expect Gal(퐿P/Q푝) that the savings allowed by the sparse power series will compensate for the former cost in practice. And as we show in Chapter6, this sparse representation is certainly helpful when trying to reason about RoS Chabauty for many base changes of a curve simultaneously.

74 Chapter 5

RoS Chabauty and Genus 0 Descent: General Theory

In this section, we combine Chabauty’s method and RoS Chabauty with the method

of descent, which reduces the problem of computing 풞(풪퐾,푆) to computing 풟(풪퐾,푆) for a finite set of curves which map to 풞. We will be particularly interested in the case where the 풞 and 풟 are all sound clearance hole curves such that the generic fiber is a punctured genus zero curve, and the 풟 are 풢-torsors over 풞 for some finite Galois group 풢. We prove that in most cases, classical Chabauty’s method together with Ga- lois descent by genus 0 covers cannot succeed in proving that ( 1 {0, 1, ∞})(풪 ) P풪퐾,푆 r 퐾,푆 is finite. In contrast, so long as 퐾 does not contain a CM subfield, RoS Chabauty applied to a suitable set of genus zero covers of ( 1 {0, 1, ∞})(풪 ) has no P풪퐾,푆 r 퐾,푆

strong base change obstruction. For each cover in this set, there exists an 훼 ∈ 풪퐾,푆 and and a morphism to 풟′ := 1 {푥 ∈ 퐾 퐾 : 훼푥푞 − 1 = 0}. The 풟′ for 훼 not 훼 P풪퐾,푆 r r 훼 a 푞th power have no obstruction to RoS Chabauty coming solely from a combination of strong base change and full Prym obstructions.

The review of generalized Jacobians of genus zero curves in Section 2.2.5 and the review of descent in Section 2.3 (particularly the classification of genus 0 descent sets in Section 2.3.3) may be helpful as background.

75 5.1 Classical Chabauty and genus 0 descent.

Suppose that is a sound, genus clearance hole curve over . Taking 풞 = 풳 rΓ 0 풪퐾,푆 풥 to be the generalized Jacobian of 풞, the classical Chabauty inequality (3.2.6) becomes

rank 풥 (풪퐾,푆) ≤ (#Γ(퐾) − 1) − 1 = #Γ(퐾) − 2 . (5.1.1) in this case. We use Lemma 2.2.15 to study when (5.1.1) holds (or fails to hold) for

풞. As a warm up, note that the classical Chabauty inequality is satisfied for 1 P풪퐾,푆 r if and only if × , or equivalently and either or {0, 1, ∞} rank 풪퐾,푆 < 1 푆 = ∅ 퐾 = Q 퐾 is an imaginary quadratic field. More generally, we note the following:

Proposition 5.1.2. Let 퐾 be a number field with absolute Galois group 퐺 = Gal(퐾/퐾). Suppose that 풞 = 1 Γ for some horizontal divisor Γ is a sound clearance hole P풪퐾,푆 r curve. The classical Chabauty-Coleman inequality (5.1.1) for 풞 fails if any of the following conditions hold:

1. 푟2(퐾) ≥ 1 and 푟1(퐾) + 푟2(퐾) + #푆 ≥ 2,

2. 푟1(퐾) ≥ 3 and 푟1(퐾) + #푆 ≥ 4,

3. 푟1(퐾) = 2 and 푟1(퐾) + #푆 ≥ 4 and #(퐺∖Γ(퐾)) ≥ 2,

4. and cannot be written as a disjoint union of -orbits ′ 푟1(퐾) = 3 Γ(퐾) 퐺 {푃푖, 푃푖 }

which remain 퐺p orbits for each real place p. (E.g. The condition on Γ(퐾) is automatic if #Γ(퐾) is odd.)

Of course, the classical Chabauty inequality (5.1.1) may also fail under many other conditions, especially when 푆 is large. We sketch the proof.

Proof. We use Lemma 2.2.15 to express rank 풥풞(풪퐾,푆) in terms of the action of Gal(퐾/퐾) on Γ(퐾). The statement follows quickly from three observations:

76 1. If p ∈ Σ∞ is a complex place 퐺p = {1}, so #(퐺p∖Γ(퐾)) − 1 = #Γ(퐾) − 1. ⌈︁ ⌉︁ 2. If is a real place , so #Γ(퐾) , with p ∈ Σ∞ 퐺p = Z/2Z #(퐺p∖Γ(퐾)) − 1 ≥ 2 − 1

equality if and only if Γ(퐾) decomposes into complex conjugate pairs over 퐾p.

3. . [#(퐺∖Γ(퐾)) − 1] ≤ minp∈푆∪Σ∞ [#(퐺p∖Γ(퐾)) − 1]

For example, if 푟2(퐾) ≥ 1 and 푟1(퐾) + 푟2(퐾) + #푆 ≥ 2 , let ∞1 be an infinite

place and let p ̸= ∞1 be another place with #(퐺p∖Γ(퐾)) minimal. By Lemma 2.2.15,

rank 풥풞(풪퐾,푆) ≥ [#(퐺∞1 ∖Γ(퐾)) − 1] + [#(퐺p∖Γ(퐾)) − 1] − [#(퐺∖Γ(퐾)) − 1]

≥ [#(퐺∞1 ∖Γ(퐾)) − 1] + [#(퐺p∖Γ(퐾)) − 1] − [#(퐺p∖Γ(퐾)) − 1] = #Γ(퐾) − 1 .

The other cases are similar, hinging on the fact that the −[#(퐺∖Γ(퐾)) − 1] term in Lemma 2.2.15 can cancel at most the minimal positive term.

consisting of genus zero covers of 1 {0, 1, ∞}. Under Leopoldt’s Conjecture, this P풪퐾,푆 r implies that the combination of classical Chabauty and descent by genus zero covers is insufficient to prove that

( 1 {0, 1, ∞})(풪 ) P풪퐾,푆 r 퐾,푆 is finite.

Remark 5.1.4. If 퐾 = Q the results of the next section show that (Z/푞Z)-descent and classical Chabauty suffice to prove finiteness of 1 for any set (P r {0, 1, ∞})(Z푆′ ) 푆′ of primes. Of course, when 퐾 is imaginary quadratic, classical Chabauty without descent trivially suffices to prove that 1 is finite. (P r {0, 1, ∞})(풪퐾 )

77 In the remaining cases, where 퐾 is a totally real quadratic or cubic field, one

can check that (Z/푞Z)-descent and classical Chabauty is not sufficient to prove the desired finiteness result. It may be possible to prove a similar finiteness result inthese cases using an iterated cyclic descent, or a descent-like procedure using covers which are not torsors over the base curve, but this would require a more careful analysis.

5.2 RoS Chabauty and genus 0 descent.

For the genus 0 sound clearance hole curve 풞 = 1 Γ, the condition for RoS P풪퐾,푆 r Chabauty to apply becomes

(︀ )︀ rank 풥풞(풪퐾,푆) ≤ 푑 #Γ(퐾) − 2 . (5.2.1)

The main results of this section are the following two theorems.

rank 풥풟(풪퐾,푆) ≤ 푑(#Γ풟(퐾) − 2). (5.2.3) holds for all 풟 ∈ D. II. Under the further assumption that 퐾 does not contain a CM field, D can be chosen so that there is no base change obstruction to RoS Chabauty for (풟, 풪퐾,푆) for any 풟 ∈ D.

Theorem 5.2.4. Let 퐾 be a number field which does not contain a CM subfield and let 푞 be a prime. Fix an 훼 ∈ 퐾 which is not a 푞th power in 퐾 and let

풞 := 1 {푥 ∈ 퐾 : 푥푞 − 훼 = 0} . P풪퐾,푆 r

78 For 푞 sufficiently large (depending only on 퐾 and 푆), there is no full Prym obstruction to RoS Chabauty for 풞. Moreover, if 풞 is the base change of some curve 풟, there is no full Prym obstruction to RoS Chabauty for 풟. Also, there is no base change obstruction to RoS Chabauty for 풞.

Remark 5.2.5. We saw in Section 5.1 that descent by genus 0 covers and classical Chabauty are very unlikely to prove finiteness of ( 1 {0, 1, ∞})(풪 ), except P풪퐾,푆 r 퐾,푆 in a few special cases with 퐾 of low degree. In Remark 6.1.1, we will see that RoS Chabauty alone cannot prove finiteness of

( 1 {0, 1, ∞})(풪 ) unless every subfield 푘 ⊆ 퐾 has at most two real places. P풪퐾,푆 r 퐾,푆 In contrast, Theorems 5.2.2 and 5.2.4 suggest that descent and RoS Chabauty

together are likely to be sufficient to prove that ( 1 {0, 1, ∞})(풪 ) is finite, P풪퐾,푆 r 퐾,푆 so long as 퐾 does not contain a CM subfield. Of course, there are many other ways to prove that this set is finite; the real value of Theorems 5.2.2 and 5.2.4 is not to prove this finiteness, but to provide evidence that Chabauty’s method could be used to prove finiteness of integral points on a curve over a wide range of number fields.

Before we begin the proof, we note that if 1 Γ and 1 Γ are isomorphic, P풪퐾,푆 r 1 P풪퐾,푆 r 2 then there is some fractional linear transformation 푎푥+푏 with 휑 : 푥 ↦→ 푐푥+푑 푎, 푏, 푐, 푑 ∈ 퐾 such that 휑(Γ1(퐾)) = Γ2(퐾). To determine when RoS Chabauty applies, we will need to understand the number of real and complex embeddings of the fields generated by

the Galois orbits in Γ1 and Γ2, as well as the splitting of the primes in 푆 in these fields. The following lemmas will be helpful in the proof of Theorem 5.2.2 and Theo- rem 5.2.4.

Lemma 5.2.6. Let 푞 be a prime. Suppose that 퐾 is a number field with [퐾(휁푞): 퐾] = 푞 − 1. Let 푎, 푏, 푐, 푑 ∈ 퐾 be such that 푎푑 − 푏푐 ̸= 0. Define a map 휑 : 퐾 → 퐾 by 푎푥+푏 . Suppose that is a subfield such that . Let 휑(푥) = 푐푥+푑 푘 ⊂ 퐾 [푘(휑(휁푞)) : 푘] = 푞 − 1

푟1(푘) be the number of real embeddings of 푘.

a) If [퐾 : Q] is odd, then,

푟1(푘(휑(휁푞))) ≤ 2푟1(푘) .

79 b) If 퐾 does not contain a CM subfield, then

푟1(푘(휑(휁푞))) ≤ [푘 : Q](푞 − 1) − (푞 − 3) .

For the proof, it will be convenient to have the following lemma.

Lemma 5.2.7. Suppose that 퐾 is a number field and that 훾 is algebraic over 퐾. Let be such that . Define a map by 푎푥+푏 . 푎, 푏, 푐, 푑 ∈ 퐾 푎푑 − 푏푐 ̸= 0 휑 : 퐾 → 퐾 휑(푥) = 푐푥+푑 Suppose that 푘 ⊂ 퐾 is a subfield such that [푘(휑(훾)) : 푘] = [퐾(훾): 퐾].

a) Fix an embedding and some . Suppose that for all embeddings 휄 : 퐾 → C 푟 ∈ R>0 ′ extending , the image ′ lies on the circle 휄 : 퐾(푎) ˓→ C 휄 휄 (훾) 푈푟 := {푥 ∈ C : |푥| = . If is finite, then there are at most real embeddings 푟} 휄(퐾)∩푈푟 2 푘(휑휄(훾)) ˓→ R

extending 휄|푘.

b) For each fix some . Suppose that for all embeddings 휄 : 퐾 → C 푟휄 ∈ R>0 ′ extending , the image ′ lies on the circle 휄 : 퐾(푎) ˓→ C 휄 휄 (훾) 푈푟휄 := {푥 ∈ . If for all is finite, then there C : |푥| = 푟휄} 퐾bad := {푥 ∈ 퐾 : 휄(푥) ∈ 푈푟휄 휄} exists some some such that there are at most real embeddings 휄 2 푘(휑휄(훾)) ˓→ R

extending 휄|푘.

Proof of Lemma 5.2.7. Let 1 1 be the fractional linear transforma- 휑휄 : P (C) → P (C) tion induced by 휑 and 휄.

(a) Suppose for the sake of contradiction that three of the conjugates of 휑휄(훾) (extending 휄(퐾)) lie in R. Since fractional linear transformations take circles/lines to circles/lines, 1 . For any , we have −1 . But −1 휑휄(푈푟) = P (R) 푥 ∈ Q 휑휄 (푥) ∈ 푈푟 ∩ 휄(퐾) 휑휄

is injective and 푈푟 ∩ 휄(퐾) is finite, so this is clearly impossible. So, is real in at most of the embeddings extending . Since 휑휄(훾) 2 퐾(휑휄(훾)) ˓→ C 휄 , there are at most real embeddings of [푘(휑(훾)) : 푘] = [퐾(훾): 퐾] 2 푘(휑휄(훾)) ˓→ C

extending 푘|휄.

(b) Suppose for the sake of contradiction that three of the conjugates of 휑휄(훾) (extending 휄(퐾)) lie in R in every embedding 휄 : 퐾 ˓→ C. Then, for any 푥 ∈ Q,

80 −1 −1 we have 휑 (푥) ∈ 퐾bad. But 휑 is injective and 퐾bad is finite, so this is clearly impossible.

So, there is some 휄 such that 휑휄(훾) is real in at most 2 of the embeddings extending . Since , there are at most 퐾(휑휄(훾)) ˓→ C 휄 [푘(휑(훾)) : 푘] = [퐾(훾): 퐾] real embeddings of extending . 2 푘(휑휄(훾)) ˓→ C 푘|휄

Proof of Lemma 5.2.6. Let 휄 : 퐾 → C be a complex embedding of 퐾. Let | · | be the corresponding complex absolute value. Let 푈 := {푥 ∈ C : |푥| = 1}. Now, 휄 and 휑 induce a fractional linear transformation 1 1 . Any such map takes 휑휄 : P (C) → P (C) lines and circles to lines and circles.

(a) Suppose [퐾 : Q] is odd. We claim that

푈 ∩ 휄(퐾) = {±1} .

Suppose for the sake of contradiction that 푥 ∈ 휄(퐾) with |푥| = 1 and 푥 ̸= ±1, then the complex conjugate 푥 = 푥−1 ∈ 휄(퐾) as well. In particular, the minimal polynomial

of 푥 over 휄(퐾) ∩ R is 푧2 − (푥 + 푥−1)푧 + 1. So, 푥 has degree 2 over 휄(퐾) ∩ R, whence 퐾 has even degree. But 퐾 has odd degree, which gives the desired contradiction.

Then, Lemma 5.2.7 part (a) applies with 푟 = 1 for each embedding 휄. So,

푟1(푘(휑(휁푞))) ≤ 2푟1(푘) ,

as desired.

(b) Now we consider the case that 퐾 does not contain a CM-subfield.

Suppose that 푥 ∈ 퐾 (푥 ̸= ±1) lies on the unit circle in every complex embedding.

Then, Q(푥) is a degree 2, totally complex extension of the totally real field Q(푥+푥−1). But then Q(푥) is a CM subfield of 퐾, which is a contradiction.

Then, Lemma 5.2.7 part (b) applies with 푟휄 = 1 for all 휄. For some 휄, the are at

most 2 real places of 푘(휑휄(휁푞)) extending 휄|푘. For each other place, there are at most

81 푞 − 1 real places extending 휄|푘. So,

푟1(푘(휑(휁푞))) ≤ (푟1(푘) + 푟2(푘) − 1)(푞 − 1) + 2 ,

≤ [퐾 : Q](푞 − 1) − (푞 − 3) ,

as desired.

√ Lemma 5.2.8. Suppose that 퐾 is a number field and 훼 ∈ 퐾 with [퐾( 푞 훼): 퐾] = 푞. Let be such that . Define a map by 푎푥+푏 . 푎, 푏, 푐, 푑 ∈ 퐾 푎푑 − 푏푐 ̸= 0 휑 : 퐾 → 퐾 휑(푥) = 푐푥+푑

Suppose that 푘 ⊂ 퐾 is a subfield such that [푘(휑푀 (훼)) : 푘] = 푞. Let 푟1(푘) be the number of real embeddings of 푘.

1. If [퐾 : Q] is odd, then, √ 푞 푟1(푘(휑( 훼))) ≤ 2푟1(푘) .

2. If 퐾 does not contain a CM subfield, then

√ 푞 푟1(푘(휑( 훼))) ≤ [푘 : Q]푞 − (푞 − 2) .

Proof of Lemma 5.2.8. The proof of Lemma 5.2.8, is very similar to the proof of Lemma 5.2.6.

(a) Suppose is odd and fix . Set √︀푞 . For any [퐾 : Q] 휄 : 퐾 ˓→ C 푟휄 = |휄(훼)| √ embedding extending 휄, we have | 푞 훼| = 푟. If there is some 훽 ∈ 휄(퐾) such that |휄(훽)| = 푟, then |휄(훽푞/훼)| = 1. Since 훼 is not a 푞th power, we have 훽푞/훼 ̸= ±1. So, the existence of such a 훽 implies the existence of an element on the unit circle in that embedding. From part (a) of the proof of Lemma 5.2.6, we see that 훽푞/훼 has even degree, which is a contradiction. Then, part (a) of Lemma 5.2.7 implies that there are at most two real embeddings √ 푞 of 푘(휑( 훼)) extending 휄|푘. Since 휄 was arbitrary, we have

√ 푞 푟1(푘(휑( 훼))) ≤ 2푟1(푘) .

82 (b) If there were some 훽 such that |휄(훽)| = √︀푞 |휄(훼)| for all 휄, then 휄(훽푞/훼) would lie on the unit circle in all embeddings 휄 : 퐾 ˓→ C. As in the proof of part (b) of the proof of Lemma 5.2.6, this would imply that 퐾 contains a CM field, so it cannot be the case.

√︀푞 Apply part (b) of Lemma 5.2.7 with 푟휄 = |휄(훼)|. For some 휄, the are at most 2 √ 푞 real places of 푘(휑휄( 훼)) extending 휄|푘. For each other place, there are at most 푞 − 1 real places extending 휄|푘. So,

√ 푞 푘(휑( 훼)) ≤ (푟1(푘) + 푟2(푘) − 1)푞 + 2 ,

≤ [퐾 : Q]푞 − (푞 − 2) , as desired.

Proof of Theorem 5.2.2. We construct an explicit set D consisting of covers of twists of the 푞th power map for any sufficiently large prime 푞∈ / 푆′. We determine how large 푞 needs to be later in the proof. Choose a set of coset representatives for × ×푞 with . 푈 풪퐾,푆/풪퐾,푆 1 ∈ 푈 Let 풟 be the curve 1 {푥(푥푞 − 푢−1) = 0}. If we take Γ to be the divisor 푢 A풪퐾,푆 r 푢 (defined over 퐾) given by

푞 Γ푢 = {0, ∞} ∪ {푥 ∈ 퐾 : 푢푥 − 1 = 0} ,

then 풟 = 1 Γ . The curve 풟 maps to 1 {0, 1, ∞} via 푢 P풪퐾,푆 r 푢 푢 P풪퐾,푆 r

1 {푥(푥푞 − 푢−1) = 0} → 1 {0, 1, ∞} A풪퐾,푆 r P풪퐾,푆 r 푥 ↦→ 푢푥푞 .

Let D be the set {풟푢 : 푢 ∈ 푈}. For each 푢 ∈ 푈, the image of 풟푢(풪퐾,푆) is in 푢풪×푞 ∩( 1 {0, 1, ∞})(풪 ). As a result, is a descent set for 1 {0, 1, ∞}. 퐾,푆 P풪퐾,푆 r 퐾,푆 D P풪퐾,푆 r

Let 풥푢 be the generalized Jacobian of 풟푢. For each 푢 ∈ 푈, the generic fiber of 풟푢 is 1 with a reduced divisor of degree removed. Hence, . P 푞 + 2 dim 풥푢 = 푞 + 1

83 Bounding rank 풥푢 will require more work. We break into two cases based on whether or not 푢 = 1.

Case 1: 푢 = 1

Part I: (풟1, 풪퐾,푆) satisfies the RoS Chabauty inequality.

Let 퐺 be the absolute Galois group Gal(퐾/퐾) and let 퐺p be the decomposition group at the place p. We use Lemma 2.2.15 to compute the rank of 풥1 in terms of the action of the 퐺p on Γ1(퐾). By construction,

#(퐺∖Γ1(퐾)) = 4 .

Moreover,

⎧ ⎪푞 + 2 if p is a complex place of 퐾 , ⎪ ⎨⎪ 푞−1 #(퐺p∖Γ1(퐾)) = 3 + if p is a real place of 퐾 , ⎪ 2 ⎪ ⎪ primes of above if is a finite place of ⎩⎪3 + #{ P 퐾(휁푞) p} p 퐾 .

While an exact formula for the number of primes of 퐾(휁푞) above p is complicated, we can bound it as follows when p does not lie over 푞.

Let 휅p be the residue field of 퐾p. Let 푎p = [휅p(휁푞): 휅p]. Then, 푎p is the order of in ×, so #휅p (Z/푞Z) ⌈︂ln(푞 − 1)⌉︂ 푎p ≥ . ln(#휅p) Also, primes of above 푞 − 1 푞 − 1 #{ P 퐾[휁푝] p} = ≤ ⌈︁ ⌉︁ . 푎p ln(푞−1) ln(#휅p) Thus, given any 휀 > 0, we may choose 푞 sufficiently large (depending on 휀, 퐾, and 푆) so that

∑︁ #{primes P of 퐾[휁푝] above p} ≤ 휀(푞 − 1) . (5.2.9) p∈푆

84 For such 푞, Lemma 2.2.15 implies that

(︂푞 + 3)︂ rank 풥 (풪 ) ≤ 푟 (퐾) + (푞 + 1)푟 (퐾) + 2#푆 + 휀(푞 − 1) − 3 1 퐾,푆 2 1 2 (︂푟 (퐾) + 2푟 (퐾) )︂ = 1 2 + 휀 푞 + 푂(1) , 2 where the 푂(1) term depends only on 퐾 and 푆. In comparison,

[퐾 : Q](dim 풥1 − 1) = (푟1(퐾) + 2푟2(퐾))푞 .

Hence, if 푞 is sufficiently large,

rank 풥1(풪퐾,푆) ≤ [퐾 : Q](dim 풥1 − 1) .

In other words, (풟1, 풪퐾,푆) satisfies the RoS Chabauty inequality (4.1.5).

Part II: (풟1, 풪퐾,푆) has no base change obstruction. We claim that if 퐾 does not contain a CM subfield then for sufficiently large 푞 the pair (풟1, 풪퐾,푆) has no base change obstruction to RoS Chabauty.

Let 푘 be a subfield of 퐾. Let 푆푘 be the set of primes lying under primes in 푆.

Let 퐻 = Gal(푘/푘) be the absolute Galois group of 푘. Given a prime l of 풪푘, let 퐻l be the decomposition group at l. Note that if p is a prime of 풪퐾 lying over l, we can identify 퐺p with a subgroup of 퐻l.

Suppose that 풟′/풪 is a curve with 풟′ ∼ 풟 . We know that the generic 푘,푆푘 풪퐾,푆 = 1 fiber ′ of ′ is a punctured genus curve. A priori, the projective closure of ′ 풟푘 풟 0 풟푘 could be a twisted form of 1. However, the set of punctures gives a reduced divisor, P푘 which has degree . Since this degree is odd, the projective closure of ′ must 푞 + 2 풟푘 be 1. P푘

′ 1 In particular, there is some 푘-rational divisor so that 풟 ∼= Γ ′ . (Note that 푘 P푘 r 풟푘 although there may be room to choose the integral structure of 풟′, we have that

′ is independent of the choice.) rank 풥풟 (풪푘,푆푘 )

85 푎푥+푏 ′ Let 휑(푥) = define the isomorphism (풟 ) ∼= (풟 ) . Then, Γ ′ (푘) = (휑({0, ∞}∪ 푐푥+푑 1 퐾 퐾 풟푘 {푥 : 푥푞 − 1 = 0}).

Since 퐾 does not contain a CM subfield, a slight generalization of (the proof of)

Lemma 5.2.6 implies that there is some embedding 휄 : 푘 ˓→ such that #(Γ ′ (푘) ∩ C 풟푘 R) ≤ 4.

For the corresponding place l ∈ Σ , consider the action of 퐻 on Γ ′ (푘). Let 푀 푘,∞ l 풟푘 be the number of 퐻l orbits of size 1 and let 푁 be the number of 퐻l orbits of size 2. We have 푀 + 2푁 = 푞 + 2 and 푀 ≤ 4, so 푀 + 푁 ≤ (푞 + 6)/2.

For all other infinite places of 푘, we use the trivial bound #(퐻l∖Γ풟′ (푘)) ≤

#Γ풟′ (푘) = 푞 + 2. So,

∑︁ 푞 + 6 #(퐻 ∖Γ ′ (푘)) ≤ (푟 (푘) + 푟 (푘) − 1)(푞 + 2) + l 풟 1 2 2 l∈Σ푘,∞ (︂ 1)︂ ≤ [푘 : ] − 푞 + 푂(1) . Q 2

Since 퐾 has finitely many subfields 푘, the 푂(1) term depends only on 퐾.

For any place l of 푘 lying under a place p of 퐾, we have

#(퐻l∖Γ풟′ (푘)) ≤ #(퐺p∖Γ풟′ (퐾)) = #(퐺p∖Γ1(퐾)) .

Applying (5.2.9), and accounting for the 3 additional rational points {0, 1, ∞} we have

∑︁ ∑︁ ∑︁ #(퐻l∖Γ풟′ (푘)) ≤ #(퐺p∖Γ풟′ (퐾)) = #(퐺p∖Γ1(퐾)) ≤ 휀(푞 − 1) + 3#푆 .

l∈푆푘 p∈푆 p∈푆

By Lemma 2.2.15, we have

(︂ 1 )︂ rank 풥 ′ (풪 ) [푘 : ] − + 휀 푞 + 푂(1) . 풟 푘,푆푘 Q 2

Choosing 휀 = 1/4 and 푞 sufficiently large (independent of the choice of 푘 ⊂ 퐾)

86 we may arrange that

rank 풥풟(풪푘,푆푘 ) ≤ [푘 : Q] · 푞 = [푘 : Q](dim 풥풟 − 1) .

Case 2: ×푞 푢 ̸∈ 풪퐾,푆

Part I: (풟푢, 풪퐾,푆) satisfies the RoS Chabauty inequality. The proof in this case is very similar to the previous case. There is a slight difference in the treatment of the finite places.

⎧ ⎪푞 + 2 if p is a complex place of 퐾 , ⎪ ⎨⎪ 푞−1 #(퐺p∖Γ푢(퐾)) = 3 + if p is a real place of 퐾 , ⎪ 2 ⎪ √ ⎪ 푞 ⎩⎪2 + #{primes P of 퐾[ 푢] above p} if p is a finite place of 퐾 .

With 푎p defined as in the previous case, for a finite place p of 퐾 not lying over 푞,

⎧ ×푞 √ ⎨⎪1, if 푢∈ / 휅p , #{primes P of 퐾[ 푞 푢] above p} =

⎩⎪1 + (푞 − 1)/푎p, otherwise .

For 휀 as in Part 1, Lemma 2.2.15 gives

(︂푞 + 3)︂ rank 풥 (풪 ) ≤ 푟 (퐾) + (푞 + 1)푟 (퐾) + 2#푆 + 휀(푞 − 1) − 2 . 푢 퐾,푆 2 1 2

Again, in comparison,

[퐾 : Q](dim 풥푢 − 1) = (푟1(퐾) + 2푟2(퐾))푞 .

Taking 푞 large enough, we have

rank 풥푢(풪퐾,푆) ≤ [퐾 : Q](dim 풥푢 − 1) ,

so that (풟푢, 풪퐾,푆) satisfies the RoS Chabauty inequality (4.1.5).

87 Part II: (풟푢, 풪퐾,푆) has no base change obstruction.

The proof that 풟푢 has no base change obstruction so long as 퐾 does not contain a CM

subfield is essentially identical to the proof for 풟1, except that it a mild generalization of Lemma 5.2.8 in place of the mild generalization of Lemma 5.2.6. Again, for any

풟′/풪 with 풟′ ∼ 풟 , we find by Lemma 2.2.15 that 푘,푆푘 풪퐾,푆 = 푢

(︂ 1 )︂ rank 풥 ′ (풪 ) [푘 : ] − + 휀 푞 + 푂(1) . (5.2.10) 풟 푘,푆푘 Q 2

Choosing 휀 = 1/4 and 푞 sufficiently large we may arrange that

rank 풥풟(풪푘,푆푘 ) ≤ [푘 : Q] · 푞 = [푘 : Q](dim 풥풟 − 1) .

Remark 5.2.11. Retain the notation of Theorem 5.2.2 and its proof. In contrast to

the situation where 퐾 does not contain a CM field, if 퐾 is a CM field, we will show

that there is a base change obstruction to RoS Chabauty for (풟1, 풪퐾,푆) for any large 푞, at least when #푆 ≥ 3.

Suppose 퐾 is CM, with maximal totally real subfield 푘. Let 훼 ∈ 풪퐾 be such that 퐾 = 푘(훼). Let 훼 be the Galois conjugate of 훼 under the Gal(퐾/푘) action. Then,

훼 is the complex conjugate of 훼 for every embedding 퐾 ˓→ C. Define the fractional linear transformation:

1 1 푓 : P → P , 훼푥 − 훼 푥 ↦→ . 푥 − 1

If 푎 ∈ C r R, then 푎푥 − 푎 푥 ↦→ 푥 − 1 maps to . Applying this to and (under {푧 ∈ C : |푧| = 1} R ∪ {∞} 푎 = 훼 푥 = 휁푞 any embedding ) shows that is totally real. Thus, the minimal 퐾(휁푞) ˓→ C 푓(휁푞) polynomial of 푓(휁푞) over 퐾 has coefficients in 푘.

88 푞 Also, 푓(1) = ∞ and 푓({0, ∞}) = {훼, 훼}, so Γ풟 := 푓({푥 : 푥 − 1 = 0} ∪ {0, ∞}) is defined over 푘. Moreover, for each infinite place l of 푘, we have

#(퐻l∖Γ풟(푘)) = 푞 + 1 .

The set (퐻∖Γ풟(푘)) consists of the orbits {∞}, {훼, 훼}, and at least one other orbit, so #(퐻∖Γ풟(푘)) ≥ 3. Thus, for any finite place l of 푘,

#(퐻l∖Γ풟(푘)) ≥ #(퐻∖Γ풟(푘)) ≥ 3 .

Set 1 and let be its Jacobian. When , we can absorb the 풟 = 풪 Γ풟 풥풟 푆푘 ̸= ∅ P 푘,푆푘 r

#(퐻∖Γ풟(푘)) − 1 term of Lemma 2.2.15 into one of the finite places to get implies that

rank 풥풟(풪푘,푆푘 ) ≥ [푘 : Q]푞 + 2#푆푘 − 2

> [푘 : Q](dim 풥풟 − 1) + 2#푆푘 − 3 .

In particular, if , then so is a base change obstruction #푆퐾 ≥ 3 #푆푘 ≥ 2 (풟, 풪푘,푆푘 ) to RoS Chabauty for (풟1, 풪퐾,푆). Of course, if 퐾 is not CM, but contains a CM field, we may apply the same argument to the CM subfield to construct a base change obstruction.

Before proving Theorem 5.2.4, we state and prove one more technical lemma, which says roughly that full Prym obstructions to RoS Chabauty for a genus 0 sound clearance hole curve 풞 over 풪퐾,푆 only occur when the map ‘forgets’ a puncture or when the rank of the Jacobian is close to (more precisely, within approximately × rank 풪퐾,푆 of) the bound from the RoS Chabauty inequality.

Lemma 5.2.12. Let 1 and let 1 be sound clearance hole 풞1 := P r Γ1 풞2 := P r Γ2 curves over 풪퐾,푆. Let 풥1 be the generalized Jacobian of 풞1 and suppose that

rank 풥1(풪퐾,푆) + rank G푚(풪퐾,푆) + 1 < [퐾 : Q] · #Γ1(퐾) .

89 a) Suppose that Γ2(퐾) consists of a single Galois orbit and that we are given a morphism such that the induced morphism 1 1 maps 휙 : 풞1 → 풞2 휑 : P → P

Γ1(퐾) to Γ2(퐾) surjectively. Then, 휙 is not a full Prym obstruction to RoS

Chabauty for 풞1.

b) Suppose that Γ1 consists of a single Galois orbit. Then, there is no full Prym

obstruction to RoS Chabauty for 풞1.

Proof. (a) Let 푛 = #(퐺∖Γ1(퐾)). Let 훿 = #Γ1(퐾)/#Γ2(퐾). Let 풥2 be the generalized Jacobian of 풞 . Then, adding rank (풪 ) = −1 + ∑︀ 1 to the formula 2 G푚 퐾,푆 p∈푆∪Σ∞ from Lemma 2.2.15 gives

∑︁ (5.2.13) rank 풥1(풪퐾,푆) + rank G푚(풪퐾,푆) + 푛 = #(퐺p∖Γ1(퐾)) . p∈푆∪Σ∞

Similarly,

∑︁ rank 풥2(풪퐾,푆) + rank G푚(풪퐾,푆) + 1 = #(퐺p∖Γ2(퐾)) . p∈푆∪Σ∞

Now, for each p ∈ 푆 ∪ Σ∞, we have #(퐺p∖Γ1(퐾)) ≤ 훿#(퐺p∖Γ2(퐾)), so

훿 − 1 rank 풥 (풪 ) − rank 풥 (풪 ) ≤ (rank 풥 (풪 ) + rank (풪 ) + 1) . 1 퐾,푆 2 퐾,푆 훿 1 퐾,푆 G푚 퐾,푆

On the other hand,

훿 − 1 dim 풥 − dim 풥 = (#Γ (퐾) − 1) − (#Γ (퐾) − 1) = #Γ (퐾) . 1 2 1 2 훿 1

Combining (5.2.13) with the previous two sentences shows that

rank 풥1(풪퐾,푆) − rank 풥2(풪퐾,푆) < dim 풥1 − dim 풥2 .

Thus, 풞1 → 풞2 is not a large Prym obstruction.

Part (b) follows immediately from part (a), since if Γ1 consists of a single Galois

orbit, 휑(Γ1) must also consist of a single Galois orbit.

90 Theorem 5.2.4 is essentially a corollary of Lemma 5.2.12 and the rank bounds from the proof of Theorem 5.2.2.

Proof of Theorem 5.2.4. Suppose that 푘 is a subfield of 퐾 and 풟 is a sound clearance hole curve over such that ∼ . Since differs from the curves considered 풪푘,푆푘 풟풪퐾,푆 = 풞 풟 in (5.2.10) only by a finite set of punctures, the ranks will be the same uptoan 푂(1) term (depending on 퐾 and 푆, but not on 푞.) We get

(︂ 1 )︂ rank 풥 (풪 ) + rank (풪 ) + 1 ≤ [푘 : ] − + 휀 · 푞 + 푂(1) . 풟 푘,푆푘 G푚 푘,푆푘 Q 2

Taking 휀 = 1/4 and 푞 sufficiently large, this is strictly less that [푘 : Q] · 푞 = [푘 : . Applying Lemma 5.2.12 completes the proof. Q] · #Γ풟(퐾)

91 92 Chapter 6

1 RoS Chabauty and P r {0, 1, ∞}: Computations

In this chapter, we study the application of restriction of scalars and Chabauty’s method to compute solutions to the unit equation (without using descent). Most of the section is an extended series of examples of RoS Chabauty for P1 r{0, 1, ∞}. The final subsection uses a slight variant of this setup to prove thatif 3 splits completely in and then 1 . 퐾 3 - [퐾 : Q] (P r {0, 1, ∞})(풪퐾 ) = ∅

6.1 RoS Chabauty and P1 r {0, 1, ∞}: Generalities

We make explicit the theory of RoS Chabauty from Sections 4.1 and 4.4 in the context of P1 r {0, 1, ∞}. Our notation is as in Section 4.4. From here on, we drop subscripts signifying the base ring to ease notation if we think it is unlikely to cause confusion.

Remark 6.1.1. If 풞 = 1 {0, 1, ∞} then 풥 ∼ × . Let 푟 and P풪퐾,푆 r 풞 = G푚,풪퐾,푆 G푚,풪퐾,푆 1

푟2 be the number of real and complex embeddings of 퐾, so 푑 = 푟1 + 2푟2. We have

dim 풥풞 = 2 and

× rank 풥 (풪퐾,푆) = 2 rank 풪퐾,푆 = 2(푟1 + 푟2 + #푆 − 1) .

93 In this case, the RoS Chabauty inequality (4.1.5) becomes:

2(푟1 + 푟2 + #푆 − 1) ≤ (푟1 + 2푟2)(2 − 1) , or equivalently

푟1 + 2#푆 ≤ 2 .

Hence, 1 satisfies the RoS Chabauty inequality if and only ifone (P r {0, 1, ∞}, 풪퐾,푆) of the following holds:

∙ 푟1 = 0 and #푆 ≤ 1,

∙ 푟1 = 1 and #푆 = 0,

∙ 푟1 = 2 and #푆 = 0.

If , then 1 violates the RoS Chabauty inequality and #푆 > 0 (P r {0, 1, ∞}, Z푆′ ) therefore is a base change obstruction to RoS Chabauty for 1 (P r {0, 1, ∞}, 풪퐾,푆). As a result, we do not lose much generality if we restrict to the case 푆 = ∅. We will do so for the rest of this section to ease notation.

Throughout, we fix the Abel-Jacobi map:

1 푗 : P r {0, 1, ∞} ˓→ G푚 × G푚 푥 ↦→ (푥, 푥 − 1) .

The outside of the diagram (4.4.1) becomes

1 1 푑 (P r {0, 1, ∞})(풪퐾 ) (P r {0, 1, ∞})(풪퐿P ) (6.1.2)

푑 푑 G푚(풪퐾 ) × G푚(풪퐾 ) G푚(풪퐿P ) × G푚(풪퐿P ) .

Let 휎1, . . . , 휎푑 : 퐾 → 퐿P be the embeddings of 퐾 into 퐿P. The bottom horizontal map in (6.1.2) is given by

(푥, 푦) ↦→ ((휎1(푥), . . . , 휎푑(푥)), (휎1(푦), . . . , 휎푑(푦))) ,

94 while the rightmost vertical map in (6.1.2) is given by

(푢1, . . . , 푢푑) ↦→ ((푢1, . . . , 푢푑), (푢1 − 1, . . . , 푢푑 − 1)) .

Let be the local coordinates on 푑 푑 . We have 푥1, . . . , 푥푑, 푦1, . . . , 푦푑 G푚 × G푚

{︂ }︂ 0 푑 푑 1 푑푥1 푑푥푑 푑푦1 푑푦푑 퐻 (G푚 × G푚, Ω )inv = Span ,..., , ,..., . 푥1 푥푑 푦1 푦푑

In these coordinates, the 푝-adic integration theory has a particularly simple interpre- tation: For 푖 ∈ {0, 1} set

(푖) (푖) (푖) (푖) 푄푖 = ((푥1 , . . . , 푥푑 ), (푦1 , . . . , 푦푑 )) .

Let log denote the 푝-adic logarithm. Then,

푑 푑 (︃ (1) (1) )︃ ∫︁ 푄1 (︂ )︂ ∑︁ 푑푥푗 푑푦푗 ∑︁ 푥푗 푦푗 푎 + 푏 = 푎 log + 푏 log . 푗 푥 푗 푦 푗 (0) 푗 (0) 푄0 푗=1 푗 푗 푗=1 푥푗 푦푗

Since we will only ever evaluate at points where the (푖) and (푖) are units, we do not 푥푗 푦푗 need to specify the value of log 푝.

Now, we must gather information about the space of 푎푗 and 푏푗 such that for any

′ ′ ′ ′ 푑 푑 푄 = ((푥1, . . . , 푥푑), (푦1, . . . , 푦푑)) ∈ G푚(풪퐾 ) × G푚(풪퐾 ) ⊂ G푚(풪퐿P ) × G푚(풪퐿P ) , we have ∫︁ 푄 푑 (︂ )︂ 푑 ∑︁ 푑푥푗 푑푦푗 ∑︁ 푎 + 푏 = (︀푎 log 푥′ + 푏 log 푦′ = 0)︀ . 푗 푥 푗 푦 푗 푗 푗 푗 (1,1) 푗=1 푗 푗 푗=1

Let 푑 be the subspace of such that A ⊂ 퐿P (푎1, . . . , 푎푑)

푑 ∑︁ 푎푗 log 휎푗(푥) = 0 푗=1

for all . Let . Then with equality if 푥 ∈ G푚(풪퐾 ) 푟 = rank G푚(풪퐾 ) dim A ≥ 푑 − 푟

95 Leopoldt’s conjecture holds.

For any , we have ∏︀푑 In particular, 푥 ∈ 풪퐾 푗=1 휎(푥) ∈ G푚(Z) = {±1}.

푑 ∑︁ log 휎푗(푥) = 0 , 푗=1

so (1,..., 1) ∈ A regardless of 퐾. Let 푑 푑 be the common vanishing locus of the (at least) 푍 ⊆ G푚(풪퐿P ) × G푚(풪퐿P ) 2(푑 − 푟)-dimensional vector space of functions:

{︃ 푑 }︃ {︃ 푑 }︃ ∑︁ ∑︁ ℱ := 푎푗 log 푥푗 :(푎1, . . . , 푎푑) ∈ A ⊕ 푎푗 log 푦푗 :(푎1, . . . , 푎푑) ∈ A . 푗=1 푗=1 (6.1.3)

By construction, we have G푚(풪퐾 ) × G푚(풪퐾 ) ⊆ 푍. Roughly speaking, our strategy for proving finiteness of 1 is (P r {0, 1, ∞})(풪퐾 ) to prove finiteness of the intersection

1 푑 푑 푑 푍 ∩ (P r {0, 1, ∞}) (풪퐿P ) ⊂ G푚(풪퐿P ) × G푚(풪퐿P ) , which contains the image of 1 . (P r {0, 1, ∞})(풪퐾 ) Since the image of 1 푑 is cut out by the equations (P r {0, 1, ∞}) (풪퐿P )

풢 := {푥푗 − 푦푗 = 1 : 푗 ∈ [1, . . . , 푑]} , (6.1.4)

this intersection is the common vanishing locus of at least 2푑 − 2푟 + 푑 functions on the -dimensional 푑 푑. On sufficiently small closed compact 2푑 G푚(풪퐿P ) × G푚(풪퐿P ) neighborhoods 푈 all of these functions can be written simultaneously as multivariate

푝-adic power series with 퐿P-coefficients which converge on an open neighborhood of 푈. In other words, the 푝-adic power series are overconvergent on 푈. When 2푟+푑 ≤ 2푑, we may hope that this intersection, the common vanishing locus of the functions in

ℱ and 풢, consists of a finite set of isolated points.

96 In subsection 6.2.4, we need a slight variant of this high-level strategy. To explain our approach, we first recall a few facts from the theory of rigid analytic geometry as they apply to our setup.

Let be the completion of the algebraic closure of . Given any as above, C푝 Q푝 푈 we may assume that there exist such that is (isomorphic to) 푎1, . . . , 푎2푑 ∈ R 푈

2푑 for all { (푡1, . . . , 푡2푑) ∈ 퐿P : |푡푖| ≤ 푎푖 푖 ∈ {1,..., 2푑}} .

Choosing linearly independent functions 퐹1, . . . , 퐹2푑−2푟 ∈ ℱ and 퐺1, . . . , 퐺푑 ∈ 풢 and expressing them as overconvergent power series in the local coordinates for 푈, we may assume that they converge for all with on . 푡푖 ∈ C푝 |푡푖| < 푎푖 + 휖 푈

Let ∑︀2푑 for 2푑 After rescaling the local coordinates so ‖(푢1, . . . , 푢2푑)‖ = 푖=1 푢푖 푢 ∈ Z≥0. that all of the are , the set of common zeros -valued zeros of these power series 푎푖 0 C푝 in is the of the affinoid algebra 풪C푝 MaxSpec

{︃ }︃ ∑︁ 푢 as 퐴 := 푏푢푡 ∈ C푝[[푡1, . . . , 푡2푑]]: |푏푢| → ∞ ‖푢‖ → ∞ / ⟨퐹1, . . . , 퐹2푑−2푟, 퐺1, . . . , 퐺푑⟩ , 푢

This gives the common vanishing set of our power series (on each disc) the structure of an (affinoid) rigid analytic space. The ring 퐴 is noetherian and irreducible components of MaxSpec(퐴) correspond to the finitely many minimal prime ideals of 퐴. (See for instance [Con99], where this theory is developed in significantly greater generality.)

Since any isolated common zeros of 퐹1, . . . , 퐹2푑−2푟, 퐺1, . . . , 퐺푑 are irreducible components of MaxSpec(퐴), this implies that our power series have finitely many common zeros which are isolated among their -valued common zeros. 풪C푝 In the next section, we will use tangent space computations to prove in several particular cases that if is in the image of 1 , then is iso- 푃 ∈ 푈 (P r {0, 1, ∞})(풪퐾 ) 푃 lated among -valued common zeros of the and . It then follows immediately 풪C푝 퐹푖 퐺푖 from the discussion above that 1 is finite. (P r {0, 1, ∞})(풪퐾 )

97 6.2 Finiteness of the Chabauty set: Examples

In the upcoming subsections, 퐾 will be a quadratic field, mixed cubic field, ora complex quartic field. In the final subsection, 퐾 will be a mixed quartic field with a totally real quadratic subfield. We always take 푝 to be a prime which is unramified in and let be a finite extension of which is a splitting field for . 퐾 퐿P Q푝 퐾 Following the strategy and notation of section 6.1, we will show that with finitely many exceptions, for 1 the intersection of the tangent spaces 푃 ∈ (P r {0, 1, ∞})(풪퐾 ) of the 퐹1, . . . , 퐹2푑−2푟, 퐺1, . . . , 퐺푑 at 푗(푃 ) is zero-dimensional. Hence, 푗(푃 ) is isolated among -valued common zeros of the and . By the discussion at the end of 풪C푝 퐹푖 퐺푖 section 6.1, this implies that 1 is finite. (P r {0, 1, ∞})(풪퐾 )

6.2.1 Real quadratic fields

Proposition 6.2.1. Let 퐾 be a real quadratic field. Let

푗 : 1 {0, 1, ∞} ˓→ × P풪퐾 r G푚,풪퐾 G푚,풪퐾 푥 ↦→ (푥, 푥 − 1) .

Also use 푗 to refer to the corresponding map on the restriction of scalars to Z. Then, inside , the intersection (Res풪퐾 /Z(G푚 × G푚))(Z푝)

1 푗((Res풪퐾 /Z P r {0, 1, ∞})(Z푝)) ∩ (Res풪퐾 /Z(G푚 × G푚))(Z) consists of finitely many isolated points. It follows that 1 . #(P r {0, 1, ∞})(풪퐾 ) < ∞

Proof. Suppose that is a real quadratic field. In this case, × and 퐾 rank 풪퐾 = 1 [퐾 : Q] = 2, so A is 1-dimensional. Thus, A is spanned by (1, 1).

98 We may take 퐹1, 퐹2, 퐺1, 퐺2 to be the equations

퐹1 : log 푥1 + log 푥2 = 0 ,

퐹2 : log 푦1 + log 푦2 = 0 ,

퐺1 : 푥1 − 푦1 − 1 = 0 ,

퐺2 : 푥2 − 푦2 − 1 = 0 .

After accounting for finitely many exceptions, we show that if solves 푥 ∈ G푚(풪퐿P ) these equations then the intersection of the tangent spaces of the zero sets of 퐹1, 퐹2, 퐺1, and 퐺2 at 푥 is zero-dimensional. Equivalently, we will show that the matrix of derivatives at 푥 = (푥1, 푥2, 푦1, 푦2)

⎛ ⎞ 1 1 푥 푥 0 0 ⎜ 1 2 ⎟ ⎜ 0 0 1 1 ⎟ ⎜ 푦1 푦2 ⎟ 푀푥 := ⎜ ⎟ ⎜ ⎟ ⎜ 1 0 −1 0 ⎟ ⎝ ⎠ 0 1 0 −1 has rank 4. After some elementary column/row operations and substituting 푦푗 =

푥푗 − 1, the condition rank 푀푥 = 4 is equivalent to

⎛ ⎞ 푥1 − 1 푥2 − 1 rank ⎝ ⎠ = 2 . 푥1 푥2

This holds unless 푥1 = 푥2. In that case, log 푥1 = log 푥2 = 0, which only holds for finitely many . Hence, the set of points where the rank drops is finite. 푥1, 푥2 ∈ 풪퐿P

On residue discs where 퐹1, 퐹2, 퐺1 and 퐺2 can be expressed as overconvergent power series, this implies that all common zeros are isolated (even among the -valued 풪C푝 zeros.) In particular, the set of -valued common zeros of is finite, which 풪퐿P 퐹1, 퐹2, 퐺1, 퐺2 proves the claim.

In fact, since the equations used in the proof of Proposition 6.2.1 do not depend

99 on the choice of field, the proof is uniform in the field in the following sense. Wehave:

Corollary 6.2.2.

⋃︁ 1 (P r {0, 1, ∞})(풪퐾 ) < ∞. 퐾:[퐾:Q]=2

In fact, the only solutions to the unit equation in real quadratic fields are

√ √ √ √ √ √ 1 + 5 1 − 5 3 + 5 −1 − 5 3 − 5 −1 + 5 , , , , , and . 2 2 2 2 2 2

As an amusing digression, we note a quick geometric proof of this result.

2 1 ∼ Sym (P r {0, 1, ∞}) = G푚 × G푚 , {푥, 푦} ↦→ (푥푦, (푥 − 1)(푦 − 1)) .

Since and each element corresponds to an unordered pair of #(G푚 × G푚)(Z) = 4 solutions, there are exactly 8 solutions to the unit equation in quadratic fields. In [DCW15], Dan-Cohen and Wewers develop an explicit version Kim’s of nonabelian Chabauty to study points on 1 . Their computation cuts (P r {0, 1, ∞})(풪퐾 ) out solutions using single-variable polylogarithms instead of sums of logarithms in different variables. In the case where 퐾 is a quadratic field, their method is also uniform in the field, giving another proof that these are the only solutions totheunit equation in quadratic fields where 11 splits.

6.2.2 Complex cubic fields

Proposition 6.2.3. Let 퐾 be a complex cubic field. Let

푗 : 1 {0, 1, ∞} ˓→ × P풪퐾 r G푚,풪퐾 G푚,풪퐾 푥 ↦→ (푥, 푥 − 1) .

100 Also use 푗 to refer to the corresponding map on the restriction of scalars to Z. Then, inside , the intersection (Res풪퐾 /Z(G푚 × G푚))(Z푝)

1 푗((Res풪퐾 /Z P r {0, 1, ∞})(Z푝)) ∩ (Res풪퐾 /Z(G푚 × G푚))(Z) consists of finitely many isolated points. It follows that 1 . #(P r {0, 1, ∞})(풪퐾 ) < ∞

Proof. Suppose now that is a complex cubic field. In this case, × and 퐾 rank 풪퐾 = 1 [퐾 : Q] = 3, so A is 2-dimensional. Thus, A is spanned by (1, 1, 1) and one other vector, call it (푎1, 푎2, 푎3).

We wish to show that the any common zero in (풪× )6 to the equations 퐿P

퐹1 : log 푥1 + log 푥2 + log 푥3 = 0 ,

퐹2 : 푎1 log 푥1 + 푎2 log 푥2 + 푎3 log 푥3 = 0 ,

퐹3 : log 푦1 + log 푦2 + log 푦3 = 0 ,

퐹4 : 푎1 log 푦1 + 푎2 log 푦2 + 푎3 log 푦3 = 0 ,

퐺1 : 푥1 − 푦1 − 1 = 0 ,

퐺2 : 푥2 − 푦2 − 1 = 0 ,

퐺3 : 푥3 − 푦3 − 1 = 0 ,

consists of points which are isolated inside the set of -valued solutions. Let be 풪C푝 푋 the set -valued common solutions of the and . 풪퐿P 퐹푖 퐺푗

To do this, we will check that with finitely many exceptions, the intersection of the

tangent spaces of the equations 퐹1, 퐹2, 퐹3, 퐹4, 퐺1, 퐺2, 퐺3 at 푥 ∈ 푋 is zero-dimensional.

To do this, we will show that the matrix of derivatives at 푥 = (푥1, 푥2, 푥3, 푦1, 푦2, 푦3) ∈

101 푋, namely ⎛ 1 1 1 ⎞ 푥 푥 푥 0 0 0 ⎜ 1 2 3 ⎟ ⎜ 푎1 푎2 푎3 0 0 0 ⎟ ⎜ 푥1 푥2 푥3 ⎟ ⎜ ⎟ ⎜ 0 0 0 1 1 1 ⎟ ⎜ 푦1 푦2 푦3 ⎟ ⎜ ⎟ 푎 푎 푎 푀푥 := ⎜ 0 0 0 1 1 1 ⎟ , ⎜ 푦1 푦2 푦3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 0 0 −1 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 0 0 −1 0 ⎟ ⎝ ⎠ 0 0 1 0 0 1

has rank 6 except for a finite subset of 푋 consisting of isolated points. We may assume without loss of generality that 푎1 = 0. Substituting 푦푗 = 푥푗 − 1 and applying

elementary row and column operations, the condition rank 푀푥 ̸= 6 is equivalent to

⎛ ⎞ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ 푥1 푥2 푥3 ⎟ rank ⎜ ⎟ = 3 . ⎜ ⎟ ⎜ 0 푎2 푎3 ⎟ ⎝ ⎠ 0 푎2푥2 푎3푥3

If this fails, then (0, 푎2, 푎3) is proportional to (0, 푎2푥2, 푎3푥3). If these vectors are proportional, then either (i) 푎2 = 0, (ii) 푎3 = 0, or (iii) 푥2 = 푥3.

If we are in case (i), we may replace (0, 0, 푎3) with (−푎3, −푎3, 0) and (0, 0, 푎3푥3)

with (−푎3푥1, −푎3푥2, 0) and reindex the variables to reduce to case (iii). We can do the

same in case (ii). Making a similar transformation, we may assume that 푎2 + 푎3 ̸= 0.

So, when rank 푀푥 ̸= 6, we must have 푥2 = 푥3 and 푦2 = 푦3.

When we plug this into the 퐹푖 and 퐺푖, we get (푎2 +푎3) log 푥2 = 0. So, if rank 푀푥 ̸=

6, then log 푥2 = 0. It follows that if rank 푀푥 ̸= 6,

log 푥1 = log 푥2 = log 푥3 = log 푦1 = log 푦2 = log 푦3 = 0 .

The solutions to this equation are isolated in (풪× )6, so we have shown that the C푝 dimension of the the intersection of the tangent spaces of the 퐹푖 and 퐺푖 is zero- dimensional for 푥 ∈ 푋 except perhaps at a finite set of isolated points. We have seen

102 that the points of are all isolated among the solutions to the and in their 푋 C푝 퐹푖 퐺푖 residue disc. Therefore, 푋 is finite. The conclusion follows immediately.

6.2.3 Totally complex quartic fields

Let be a totally complex quartic field. In this case, × and , 퐾 rank 풪퐾 = 1 [퐾 : Q] = 4

so A is 3-dimensional. Thus, A is spanned by 푣1 := (1, 1, 1, 1) and two other vectors, which we may assume have the form 푣2 := (0, 1, 푎3, 푎4) and 푣3 := (0, 0, 1, 푏4) after possibly reindexing. Let 푣(푖) denote the 푖th component of the vector 푣.

As in the previous cases, for 푖 ∈ {1, 2, 3} and 푗 ∈ {1, 2, 3, 4} we can set

(1) (2) (3) (4) 퐹푖 : 푣푖 log 푥1 + 푣푖 log 푥2 + 푣푖 log 푥3 + 푣푖 log 푥4 , (1) (2) (3) (4) 퐹푖+3 : 푣푖 log 푦1 + 푣푖 log 푦2 + 푣푖 log 푦3 + 푣푖 log 푦4 ,

퐺푗 : 푥푗 − 푦푗 − 1 = 0 .

Let 푋 be the set of solutions to the 퐹 and 퐺 in (풪× )8. As in previous cases, 푖 푗 퐿P we claim that each is isolated among the -valued solutions to the and 푥 ∈ 푋 풪C푝 퐹푖

퐺푗 in any residue disc about 푥.

We check that at 푥 ∈ 푋, the intersection of the tangent spaces of the 퐹푖 and 퐺푗 is zero-dimensional except possibly on a finite subset of 푋 consisting of isolated points. This is equivalent to checking that the matrix

⎛ ⎞ 1 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ 푥1 푥2 푥3 푥4 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 푎3 푎4 ⎟ 푀푥 := ⎜ ⎟ ⎜ ⎟ ⎜ 0 푥2 푎3푥3 푎4푥4 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 1 푏4 ⎟ ⎝ ⎠ 0 0 푥3 푏4푥4

has rank 4. If this fails, then either (i) 푥3 = 푥4 or (ii) 푏4 = 0. If 푥3 = 푥4 and

103 rank 푀푥 < 4, we must have

⎛ ⎞ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ 푥1 푥2 푥4 ⎟ rank ⎜ ⎟ < 3 . ⎜ ⎟ ⎜ 0 1 (푎3 + 푎4)/2 ⎟ ⎝ ⎠ 0 푥2 푥4(푎3 + 푎4)/2

If 푏4 = 0 and rank 푀푥 < 4, we must have

⎛ ⎞ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ 푥1 푥2 푥4 ⎟ rank ⎜ ⎟ < 3 . ⎜ ⎟ ⎜ 0 1 푎4 ⎟ ⎝ ⎠ 0 푥2 푎4푥4

In both cases, the proof that the rank drops at points 푥 in a subset of 푋 consisting of isolated points is essentially the same as in the cubic field case, so we omit it. We conclude:

Proposition 6.2.4. If 퐾 is a totally complex quartic field, then the intersection

1 푑 푑 푑 (P r {0, 1, ∞})(풪퐿P ) ∩ G푚(풪퐾 ) × G푚(풪퐾 ) ⊂ G푚(풪퐿P ) × G푚(풪퐿P ) consists of finitely many isolated points. Thus,

1 #(P r {0, 1, ∞})(풪퐾 ) < ∞

6.2.4 Mixed quartic fields with a totally real subfield

Suppose now that 퐾 is a mixed quartic field, i.e. that 푟1(퐾) = 2 and 푟2(퐾) = 1. In this case, × and , so is -dimensional. Thus, is spanned rank 풪퐾 = 2 [퐾 : Q] = 4 A 2 A by (1, 1, 1, 1) and one other vector, call it (푎1, 푎2, 푎3, 푎4). We claim:

Proposition 6.2.5. If 퐾 is a mixed quartic field with a totally real quadratic sub-

104 field, then for any , the image 1 in 푑 푑 is 푝 푗((P r {0, 1, ∞})(풪퐾 )) G푚(풪퐿P ) × G푚(풪퐿P ) contained in the (finite) subset of 풪× -valued common solutions to the equations 퐿P

퐹1 : log 푥1 + log 푥2 + log 푥3 + log 푥4 = 0 ,

퐹2 : 푎1 log 푥1 + 푎2 log 푥2 + 푎3 log 푥3 + 푎4 log 푥4 = 0 ,

퐹3 : log 푦1 + log 푦2 + log 푦3 + log 푦4 = 0 ,

퐹4 : 푎1 log 푦1 + 푎2 log 푦2 + 푎3 log 푦3 + 푎4 log 푦4 = 0 ,

퐺푗 : 푥푗 − 푦푗 − 1 = 0 for 푗 ∈ {1, 2, 3, 4} , which are isolated among the 풪× -valued common solutions. C푝 In particular,

1 #(P r {0, 1, ∞})(풪퐾 ) < ∞ .

Let 푘 be the totally real subfield of 퐾. The field 푘 has two embeddings 휏1, 휏2 ˓→ 퐿P.

Say that 휎1 and 휎2 extend 휏1 and that 휎3 and 휎4 extend 휏2. (︀ )︀ (︀ )︀ In our chosen coordinates, we can identify Res풪 / 푚 inside of Res풪 / 푚 푘 ZG 퐿P 푘 ZG 퐿P as the space 푥1 = 푥2, 푥3 = 푥4.

If we restrict 퐹1 and 퐹2 to this subspace, then they must vanish on the points corresponding to elements of ×. Taking parameters and , the space of functions 풪푘 푥1 푥3 of the form 푏1 log 푥1 + 푏3 log 푥3 on

(︀ )︀ Res풪 / 푚 (퐿P) 푘 ZG 퐿P which vanish on the image of is one-dimensional vector space spanned by G푚(풪푘) log 푥1 + log 푥3. This implies that 푎1 + 푎2 = 푎3 + 푎4. So, we may assume that

푎1 = 0, 푎2 = 1, 푎3 = 푏, 푎4 = 1 − 푏 .

Substituting for the 푦푖 using the 퐺푖, if we followed the pattern from the previous subsections we would now show that the set 푋 ⊂ (풪× ∩ (풪× − 1))4 of common 퐿P 퐿P

105 solutions of the equations

log 푥1 + log 푥2 + log 푥3 + log 푥4 = 0 ,

log 푥2 + 푏 log 푥3 + (1 − 푏) log 푥4 = 0 ,

log(1 − 푥1) + log(1 − 푥2) + log(1 − 푥3) + log(1 − 푥4) = 0 ,

log(1 − 푥2) + 푏 log(1 − 푥3) + (1 − 푏) log(1 − 푥4) = 0 . consists of isolated points inside the -valued solutions. Instead, we will show that if 풪C푝 푥 ∈ 푋 is not an isolated inside the 풪 -valued solutions then 푥∈ / ( 1 {0, 1, ∞})(풪 ). C푝 P풪퐾 r 퐾

For 푥 = (푥1, . . . , 푥4) ∈ 푋, let

⎛ ⎞ 1 1 1 1 푥 푥 푥 푥 ⎜ 1 2 3 4 ⎟ ⎜ 0 1 푏 1−푏 ⎟ ⎜ 푥2 푥3 푥4 ⎟ 푀푥 := ⎜ ⎟ . ⎜ 1 1 1 1 ⎟ ⎜ 푥 −1 푥 −1 푥 −1 푥 −1 ⎟ ⎝ 1 2 3 4 ⎠ 0 1 푏 1−푏 푥2−1 푥3−1 푥4−1

If rank(푀푥) < 4, then

⎛ ⎞ 1 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ 푥1 푥2 푥3 푥4 ⎟ det ⎜ ⎟ = 0 . ⎜ ⎟ ⎜ 0 1 푏 1 − 푏 ⎟ ⎝ ⎠ 0 푥2 푏푥3 (1 − 푏)푥4

So, rank(푀푥) < 4 if and only if

2 (푥1 − 푥3)(푥2 − 푥4) (0 − 푏)(1 − (1 − 푏)) 푏 = = 2 . (푥2 − 푥3)(푥1 − 푥4) (1 − 푏)(0 − (1 − 푏)) (1 − 푏)

In particular, if some point 1 is not isolated then must 푥 ∈ (P r {0, 1, ∞})(풪퐾 ) ⊂ 푋 푏 be algebraic.

We will prove by contradiction that 푏 cannot be algebraic. The key tool is a 푝-adic analogue due to Brumer [Bru67] of Baker’s Theorem on the linear independence of logarithms of algebraic numbers.

106 Theorem 6.2.6 (Brumer, [Bru67]). Let 훼1, . . . , 훼푛 be elements of the completion of the algebraic closure of which are algebraic over the rationals and whose - Q푝 Q 푝 adic logarithms are linearly independent over Q. These logarithms are then linearly independent over the algebraic closure of in . Q Q Q푝

Choose generators for a finite index subgroup of × with ×. 푢1, 푢2 풪퐾 푢1 ∈ 풪푘

If 푏 is algebraic, the contrapositive of Brumer’s theorem implies that there are integers ′ ′ ′ , not all zero, such that 푐2, 푐3, 푐4 ∈ Z

′ ′ ′ 푐2 log 휎2(푢2) + 푐3 log 휎3(푢2) + 푐4 log 휎4(푢2) = 0 ,

or equivalently that there are , not all zero, such that 푐2, 푐3, 푐4 ∈ Z

푐2 푐3 푐4 휎2(푢2) 휎3(푢2) 휎4(푢2) = 1 .

Since 푘 is a quadratic field, 휏1(푘) = 휏2(푘) ⊂ 퐿P as subsets of 퐿P. Moreover, since 퐾 is

a quadratic extension of 푘, we have 휎1(퐾) = 휎2(퐾) ⊂ 퐿P and 휎3(퐾) = 휎4(퐾) ⊂ 퐿P

as subsets of 퐿P. On the other hand, 휎2(퐾) ∩ 휎3(퐾) = 휏1(푘) as subsets of 퐿P.

푐2 푐3 푐4 푐2 Then, if 휎2(푢2) 휎3(푢2) 휎4(푢2) = 1, we must have 휎2(푢2) ∈ 휏1(푘) ⊂ 퐿P. Hence,

there are nonzero such that 푐·푐2 푑. But this is only possible 푐, 푑 ∈ Z 휎2(푢2) = 휎2(푢1)

if 푐2 = 0, since 푢1 and 푢2 generate a rank 2 group and are therefore multiplicatively independent.

Hence, we have

푐3 푐4 휎3(푢2) 휎4(푢2) = 1 .

By the same argument applied to in place of , there are non-zero 푢1푢2 푢2 푑3, 푑4 ∈ Z such that

푑3 푑4 휎3(푢2푢1) 휎4(푢2푢1) = 1 . (6.2.7)

107 So, we have

푐3푑3 푐4푑3 휎3(푢2) 휎4(푢2) = 1 ,

푐3푑3 푐3푑4 휎3(푢2푢1) 휎4(푢2푢1) = 1 .

푐4푑3−푐3푑4 푐3푑3 푐3푑4 This implies that 휎4(푢2) = 휎3(푢1) 휎4(푢1) ∈ 휏2(푘). So, 푐4푑3 = 푐3푑4. Then,

푑3 푑4 푐3 (휎3(푢1) 휎4(푢1) ) = 1 , so 푑3 푑4 푑3+푑4 is a root of unity. But if 푑3+푑4 is a root of unity, 휎3(푢1) 휎4(푢1) = 휏2(푢1) 푢1

푑3 = −푑4, since 푢1 generates the free part of 풪푘.

푑3 푑3 푑3 Then, 6.2.7 becomes 휎3(푢2) = 휎4(푢2) , which implies that 휎3(푢2) ∈ 휏2(푘).

Since 푑3 and 푑4 are non-zero and 푢2 ∈/ 푘, this is a contradiction. This proves that 푏 cannot be algebraic. As a consequence, if some point is not isolated in the solutions to our 푥 ∈ 푋 풪C푝 equations on its residue disc, then 1 . 푥∈ / (P r {0, 1, ∞})(풪퐾 ) This completes the proof.

6.3 Fields where 3 splits completely

Theorem 6.3.1. Suppose that [퐾 : Q] is not divisible by 3 and that 3 splits completely in . Then there is no pair × such that . Equivalently, 퐾 푥, 푦 ∈ 풪퐾 푥 + 푦 = 1

1 (P r {0, 1, ∞})(풪퐾 ) = ∅.

Remark 6.3.2. Before we begin the proof, we note that the corresponding result for

2 is trivial. In fact, if there is any prime p above 2 such that 퐾p is a totally ramified extension of and then there are no × such that . In that Z2 p ∈/ 푆 푥, 푦 ∈ 풪퐾,푆 푥 + 푦 = 1 setup, the proof is very short; there is a local obstruction at p. Solutions to the unit equation cannot be congruent to 0 or 1 modulo p. On the other hand, if 퐾p is totally ramified over , then every element of is either congruent to or modulo . Z2 풪퐾,푆 0 1 p

108 Proof. Although it is disguised in our argument, the main idea of the proof is to use a variant of Chabauty’s method to prove that for any unit × , the intersection 푢 ∈ 풪퐾

1 푛 × (Res풪퐾 /Z P r {0, 1, ∞})(Z3) ∩ {푢 : 푛 ∈ Z} × 풪퐾 inside is empty. (Res풪퐾 /Z(G푚 × G푚))(Z3) Suppose that splits completely in and choose any × . The splitting 3 퐾 푢 ∈ 풪퐾 gives different maps . Let be the images of in under these 푑 풪퐾 → Z3 푢1, . . . , 푢푑 푢 Z3 maps. Since 푢 is a unit,

푑 ∏︁ × 푢푖 = Nm퐾/Q 푢 ∈ Z = {±1} . 푖=1

Now, suppose further that −푢 is a solution to the unit equation, so that there exists some × with . Since and are units, we must have 푣 ∈ 풪퐾 −푢 − 푣 = 1 푢 푣 for all . Moreover, we have 푢푖 ∈ 1 + 3Z3 푖 ∈ {1, . . . , 푑}

푑 ∏︁ 푑 (1 + 푢푖) = Nm퐾/Q −푣 = (−1) . 푖=1

In particular, this says that 푛 = 1 is a solution to the 푝-adic analytic equation

푛 푛 푑 푓(푛) := (1 + 푢1 ) ··· (1 + 푢푑 ) − (−1) = 0 .

Now, ∏︀푑 by the norm equation, so 푖=1 푢푖 = 1

푑 ∏︁ −푛 푑 푓(−푛) = (1 + 푢푖 ) − (−1) 푖=1 푑 푑 ∏︁ −푛 ∏︁ 푛 푑 = 푢푖 (1 + 푢푖 ) − (−1) 푖=1 푖=1 푑 ∏︁ 푛 푑 = (1 + 푢푖 ) − (−1) 푖=1 = 푓(푛) ,

109 so 푓 is an even function. In particular, if we expand 푓 as a 푝-adic power series, only even-degree terms will have non-zero coefficients. Now, we can rewrite 푓 as

푑 푑 ∏︁ 푓(푛) = −(−1) + (1 + exp(푛 log 푢푖)),. 푖=1

Since log 푢푖 has 3-adic valuation ≥ 1 and exp converges so long as the 3-adic valuation of is , this power series converges on an open neighborhood of (푛 log 푢푖) > 1/2 풪C3 ⊂ . C3 Expanding 푓 as a power series,

푑 ∏︁ 푛2 푛3 푓(푛) = −(−1)푑 + (2 + 푛 log 푢 + (log 푢 )2 + (log 푢 )3 + ··· ) . 푖 2 푖 3! 푖 푖=1

So, writing ∑︀∞ 푗 and using the identity ∑︀푑 gives 푓(푛) = 푗=0 푎푛푛 푖=1 log 푢푖 = 0

푑 푑 푎0 = 2 − (−1) ,

푎1 = 0 , 푑 푑−3 ∑︁ 2 푎2 = 2 (log 푢푖) , 푖=1

푎3 = 0 ,

푣3(푎푗) ≥ 3 for all 푗 ≥ 4 .

We have , so has no solution in when is not divisible by . But 푣(푎2) ≥ 2 푓(푛) Z3 푎0 9

푎0 is divisible by 9 if and only if 푑 is divisible by 3, which completes the proof.

Remark 6.3.3. Unfortunately, it seems unlikely that we can do any better in general by these methods.

Consider the number field 퐾 := Q[푧]/(푧3 + 3푧2 + 2푧 + 3). It is easy to check that splits completely in . A computation in Magma shows that × is generated by 3 퐾 풪퐾 푧2 + 1 and −1. Set 푢 = −(푧2 + 1)2 = −(9푧2 + 3푧 + 10) = −3(3푧2 + 푧 + 3) − 1.

110 Computing in Magma gives

1 2 3 log(푢1) = 0 · 3 + 2 · 3 + 푂(3 ) ,

1 2 3 log(푢2) = 2 · 3 + 2 · 3 + 푂(3 ) ,

1 2 3 log(푢3) = 1 · 3 + 1 · 3 + 푂(3 ) .

2 2 2 In this case, it is clear that 푣3(log(푢1) +log(푢2) +log(푢3) ) = 2 exactly. Write 푓(푛) = ∑︀∞ 푗 as in the proof of theorem 6.3.1. Then, we see that , 푗=0 푎푛푛 푣3(푎0) = 푣3(푎2) = 2 and all other coefficients have larger valuation. Moreover, 푓(푛) 2 . It 9 ≡ 1 − 푛 (mod 3) follows by Hensel’s Lemma that will have exactly two solutions in . 푓 Z3

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