arXiv subjects

Martin Lüdtke

Publications and source records attributed to Martin Lüdtke.

10 recordsLinked to original sources

arXiv2026

Affine Chabauty I

We prove finiteness and give an explicit upper bound on the number of $S$-integral points on affine curves satisfying a certain rank-genus inequality. We achieve this by developing an analogue of the Chabauty method, embedding the curve into its generalised Jacobian and bounding the Abel-Jacobi image of the $S$-integral points using arithmetic intersection theory. Our results also provide the foundations for an algorithm to determine the set of $S$-integral points on affine curves presented in a follow-up article.

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arXiv2026

Nonabelian Chabauty for the Thrice-punctured Line over Cyclotomic Fields

In this paper we study the motivic Chabauty--Kim method, which aims to determine the set of $S$-integral points of $\mathbb{P}^1\smallsetminus \{0,1,\infty\}$, over cyclotomic fields. We focus on the case $K=\mathbb{Q}(ζ_8)$ and $S=\left\{(1-ζ_8)\right\}$, where we obtain explicit polylogarithmic Kim functions up to depth $4$ and verify Kim's Conjecture for several primes. We also observe and explain that the Chabauty--Kim locus for the polylogarithmic quotient contains, in addition to the $S$-integral points, certain exceptional points arising from roots of unity in $\mathbb{Q}_p$.

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arXiv2026

Polylogarithmic Chabauty--Kim loci over number fields

We study polylogarithmic Chabauty--Kim loci for $S$-integral points on $\mathbb{P}^1\smallsetminus\{0,1,\infty\}$ over number fields. We compare motivic and étale Selmer schemes, describe Galois actions on Selmer schemes and period rings, and give an explicit formula for the localisation map on the polylogarithmic Selmer scheme. For imaginary and real quadratic fields, we derive equations and determine several polylogarithmic and full Chabauty--Kim loci and show that Kim's Conjecture holds in several new cases. In other cases, we show that the polylogarithmic Chabauty--Kim method is insufficient to cut out precisely the $S$-integral points, even when combined with $S_3$-symmetrisation, due to additional points arising from $p$-adic roots of unity. As a further application, we give a Chabauty--Kim theoretic proof of the $S$-Selmer Section Conjecture for imaginary quadratic fields when $S$ is empty or $S$ consists of a single prime not fixed by complex conjugation.

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arXiv2026

The motivic Selmer scheme of the thrice-punctured line

Let $X = \mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ be the thrice-punctured over a ring of $S$-integers $\mathcal{O}_{K,S}$ in a number field~$K$. For any quotient $π_1^{\mathrm{mot}}(X,0) \twoheadrightarrow Π$ of Deligne--Goncharov's motivic fundamental group there is an associated Selmer scheme which parametrises $Π$-torsors with a mixed Tate motive structure. We give several descriptions of the motivic Selmer scheme which make it amenable to computations, using $\mathbb{G}_m$-equivariant cocycles of algebraic groups, Lie algebras, and complete Hopf algebras. We prove that the Selmer scheme is isomorphic to an affine space $\mathbb{A}^N_{\mathbb{Q}}$, and we construct coordinates realising this isomorphism. This is a key ingredient for making the motivic Chabauty--Kim method explicit in a general setting, without restrictions on the base field or the choice of fundamental group quotient.

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arXiv2026

Affine Chabauty II

We present an algorithm for determining the set of $S$-integral points on an affine curve based on the Affine Chabauty method developed in the first part of this series. We achieve this by constructing explicit logarithmic differentials whose integrals take on prescribed values on $S$-integral points. Along the way, we prove a $p$-adic residue theorem for Coleman integrals of log differentials.

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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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arXiv2024

Refined Chabauty--Kim computations for the thrice-punctured line over $\mathbb{Z}[1/6]$

The Chabauty--Kim method and its refined variant by Betts and Dogra aim to cut out the $S$-integral points $X(\mathbb{Z}_S)$ on a curve inside the $p$-adic points $X(\mathbb{Z}_p)$ by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line when $S$ contains two primes. We describe an algorithm for computing refined Chabauty--Kim loci and verify Kim's conjecture over $\mathbb{Z}[1/6]$ for all choices of auxiliary prime $p < 10{,}000$.

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arXiv2023

Linear and quadratic Chabauty for affine hyperbolic curves

We give sufficient conditions for finiteness of linear and quadratic refined Chabauty-Kim loci of affine hyperbolic curves. We achieve this by constructing depth $\leq 2$ quotients of the fundamental group, following a construction of Balakrishnan-Dogra in the projective case. We also apply Betts' machinery of weight filtrations to give unconditional explicit upper bounds on the number of S-integral points when our conditions are satisfied.

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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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arXiv2018

A Birational Anabelian Reconstruction Theorem for Curves over Algebraically Closed Fields in Arbitrary Characteristic

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover $K|k$ from its absolute Galois group alone, we prove that it can be recovered from the pair $(\mathrm{Aut}(\overline{K}|k),\mathrm{Aut}(\overline{K}|K))$, consisting of the absolute Galois group of $K$ and the larger group of field automorphisms fixing only the base field.

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