arXiv subjects

L. Alexander Betts

Publications and source records attributed to L. Alexander Betts.

17 recordsLinked to original sources

arXiv2026

Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections

Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for $X$ which everywhere locally comes from a point of $X$ in fact globally comes from a point of $X$. We show that $X/\mathbb{Q}$ satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime $p$, and give the appropriate generalisation to $S$-integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over $\mathbb{Z}[1/2]$.

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arXiv2025

A motivic Weil height machine for curves

The rational points of a smooth curve $X$ over a number field $k$ map to the set of augmentations of the associated motivic algebra. An expectation, related to Kim's conjecture, is that for $X$ hyperbolic, the set of augmentations which come locally at each place of $k$ from a point is equal to the set of rational points. Our view is that this should provide a relative of the Grothendieck section conjecture which may be both more accessible, and more directly applicable, than the latter. As a first step in this direction, we extend aspects of the ``Weil height machine'' to the set of such augmentations, and use this to prove a Manin--Dem'janenko-style finiteness result for motivic augmentations for particular curves. Along the way, we determine the structure of the cohomological motive of a $\mathbb{G}_m$-bundle over an algebraic variety as a highly structured algebra in the derived $\infty$-category of mixed motives with rational coefficients.

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arXiv2025

On a non-abelian analogue of a conjecture of Michael Stoll

We formulate a non-abelian generalisation of a conjecture of Stoll, which conjecturally describes the structure of the loci cut out by Kim's method of non-abelian Chabauty. We prove the rank 0 quadratic case of this conjecture, which in particular determines the structure of the quadratic Chabauty locus for once-punctured elliptic curves of rank 0. The proof involves using a variant of the geometric quadratic Chabauty method of Edixhoven and Lido to reduce to an unlikely intersections problem, and ultimately to known results about the relative Manin--Mumford Conjecture.

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arXiv2025

Local heights on hyperelliptic curves and quadratic Chabauty

Quadratic Chabauty is a $p$-adic method for determining rational points on curves. Local heights are arithmetic invariants used in the quadratic Chabauty method. We present an algorithm to compute these local heights for hyperelliptic curves at odd primes $\ell \neq p$. This algorithm significantly broadens the applicability of quadratic Chabauty to curves which were previously inaccessible due to the presence of non-trivial local heights. We provide numerous examples, including the first quadratic Chabauty computation for a curve having two primes with non-trivial local heights.

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arXiv2025

Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields

Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL$_2(\mathbf{Z}_3)$ can occur as the image of the $3$-adic Galois representation attached to a non-CM elliptic curve over $\mathbf{Q}$, with a single exception: the normaliser of the non-split Cartan subgroup of level 27. In this paper, we complete the classification of 3-adic Galois images by showing that the normaliser of the non-split Cartan subgroup of level 27 cannot occur as a 3-adic Galois image of a non-CM elliptic curve. Our proof proceeds via computing the $\mathbf{Q}(ζ_3)$-rational points on a certain smooth plane quartic curve $X'_H$ (arising as a quotient of the modular curve $X_{ns}^+(27)$) defined over $\mathbf{Q}(ζ_3)$ whose Jacobian has Mordell--Weil rank 6. To this end, we describe how to carry out the quadratic Chabauty method for a modular curve $X$ defined over a number field $F$, which, when applicable, determines a finite subset of $X(F\otimes\mathbf{Q}_p)$ in certain situations of larger Mordell--Weil rank than previously considered. Together with an analysis of local heights above 3, we apply this quadratic Chabauty method to determine $X'_H(\mathbf{Q}(ζ_3))$. This allows us to compute the set $X_{ns}^+(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois.

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arXiv2023

Bounds on the Chabauty--Kim Locus of Hyperbolic Curves

Conditionally on the Tate--Shafarevich and Bloch--Kato Conjectures, we give an explicit upper bound on the size of the $p$-adic Chabauty--Kim locus, and hence on the number of rational points, of a smooth projective curve $X/\mathbb{Q}$ of genus $g\geq2$ in terms of $p$, $g$, the Mordell--Weil rank $r$ of its Jacobian, and the reduction types of $X$ at bad primes. This is achieved using the effective Chabauty--Kim method, generalising bounds found by Coleman and Balakrishnan--Dogra using the abelian and quadratic Chabauty methods.

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arXiv2023

Refined Selmer equations for the thrice-punctured line in depth two

In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on $\mathbb P^1_{\mathbb Z}\setminus\{0,1,\infty\}$. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of $S$ increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where $S$ has size $2$ which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim.

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arXiv2022

Local constancy of pro-unipotent Kummer maps

It is a theorem of Kim-Tamagawa that the $\mathbb Q_\ell$-pro-unipotent Kummer map associated to a smooth projective curve $Y$ over a finite extension of $\mathbb Q_p$ is locally constant when $\ell\neq p$. The present paper establishes two generalisations of this result. Firstly, we extend the Kim-Tamagawa Theorem to the case that $Y$ is a smooth variety of any dimension. Secondly, we formulate and prove the analogue of the Kim-Tamagawa Theorem in the case $\ell = p$, again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale-de Rham comparison theorem for pro-unipotent fundamental groupoids using methods of Scholze and Diao-Lan-Liu-Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids.

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arXiv2022

Galois sections and $p$-adic period mappings

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\geq2$, we prove that the image of the "Selmer" part of Grothendieck's section set inside the $K_v$-rational points $Y(K_v)$ is finite for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck's section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations.

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arXiv2022

The motivic anabelian geometry of local heights on abelian varieties

We study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent étale realisations when the base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de Rham realisation when the base field is archimedean. In the course of proving the $p$-adic instance of these theorems, we develop a new technique for studying local non-abelian Bloch--Kato Selmer sets, working with certain explicit cosimplicial group models for these sets and using methods from homotopical algebra. Among other uses, these models enable us to construct a non-abelian generalisation of the Bloch--Kato exponential sequence under minimal conditions.

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arXiv2021

Weight filtrations on Selmer schemes and the effective Chabauty--Kim method

We develop an effective version of the Chabauty--Kim method which gives explicit upper bounds on the number of $S$-integral points on a hyperbolic curve in terms of dimensions of certain Bloch--Kato Selmer groups. Using this, we give a new "motivic" proof that the number of solutions to the $S$-unit equation is bounded uniformly in terms of $\#S$.

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arXiv2021

A user's guide to the local arithmetic of hyperelliptic curves

A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.

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arXiv2021

Semisimplicity and weight-monodromy for fundamental groups

Let X be a smooth, geometrically connected variety over a p-adic local field. We show that the pro-unipotent fundamental group of X (in both the etale and crystalline settings) satisfies the weight-monodromy conjecture, following Vologodsky. We deduce (in the etale setting) that Frobenii act semisimply on the Lie algebra of the pro-unipotent fundamental group of X, and (in the crystalline setting) that the same is true for a K-linear power of the crystalline Frobenius. We give applications to the representability and geometry of the Selmer varieties appearing in the Chabauty-Kim program, even in cases of bad reduction.

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arXiv2020

Variation of Tamagawa numbers of Jacobians of hyperelliptic curves with semistable reduction

We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as one varies the base field or the curve, in the case of semistable reduction. We find that there are strong constraints on the behaviour that appears, some of which are unexpected and specific to hyperelliptic curves. Our methods are explicit and allow one to write down formulae for Tamagawa numbers of infinite families of hyperelliptic curves, of the kind used in proofs of the parity conjecture for Jacobians of curves of small genus.

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arXiv2020

The local theory of unipotent Kummer maps and refined Selmer schemes

We study the Galois action on paths in the $\mathbb{Q}_\ell$-pro-unipotent étale fundamental groupoid of a hyperbolic curve $X$ over a $p$-adic field with $\ell\neq p$. We prove an Oda--Tamagawa-type criterion for the existence of a Galois-invariant path in terms of the reduction of $X$, as well as an anabelian reconstruction result determining the stable reduction type of $X$ in terms of its fundamental groupoid. We give an explicit combinatorial description of the non-abelian Kummer map of $X$ in arbitrary depth, and deduce consequences for the non-abelian Chabauty method for affine hyperbolic curves and for explicit quadratic Chabauty.

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arXiv2015

Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties

We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this translates into a purely algebraic problem concerning the number of $H$-invariant points on a quotient of $C_n$-lattices $Λ/eΛ'$ for varying subgroups $H$ of $C_n$ and integers $e$. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions (corresponding to varying $e$) and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the $p$-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties, that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the form $y^2=f(x)$, under some simplifying hypotheses.

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arXiv2015

The first-order theory of geometric points of schemes: Chevalley's theorem and quantifier elimination

Chevalley's theorem on the images of morphisms of schemes and the principle of quantifier elimination for the theory of algebraically closed fields are widely understood to be two perspectives on the same theorem. In this paper, we demonstrate that both results can easily be proven simultaneously, using a mixture of geometric and logical techniques. In doing so, we give logical meaning to geometric points of schemes and to finitely presented morphisms thereof, in a manner reminiscent of Spencer Breiner's logical schemes.

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