Abstract

AbstractSince Faltings proved Mordell’s conjecture in [16] in 1983, we have known that the sets of rational points on curves of genus at least $2$ are finite. Determining these sets in individual cases is still an unsolved problem. Chabauty’s method (1941) [10] is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. Under the condition that the Mordell–Weil rank is less than the genus, Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, has been applied successfully to determine all rational points in many cases.Minhyong Kim’s nonabelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$ ).This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ (line bundles over the Jacobian and models over the integers).

Highlights

  • Faltings proved in 1983 [16] that for every number field K and every curve C over K of genus at least 2, the set of K-rational points C(K) is finite

  • There is a fair amount of evidence that Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, does determine all rational points when r < g, with r the Mordell–Weil rank and g the genus of C

  • For a general introduction to Chabauty’s method and Coleman’s effective version of it, we highly recommend [24] and, for an implementation of it that is ‘geometric’ in the sense of this article, [17], in which equations for the curve embedded in the Jacobian are pulled back via local parametrisations of the closure of the Mordell–Weil group

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Summary

Introduction

Faltings proved in 1983 [16] that for every number field K and every curve C over K of genus at least 2, the set of K-rational points C(K) is finite. Chabauty’s method (1941) for determining C(Q) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. For a general introduction to Chabauty’s method and Coleman’s effective version of it, we highly recommend [24] and, for an implementation of it that is ‘geometric’ in the sense of this article, [17], in which equations for the curve embedded in the Jacobian are pulled back via local parametrisations of the closure of the Mordell–Weil group. This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ Many pictures can be found in [19], and some in [15]

Algebraic geometry
From algebraic geometry to formal geometry
Logarithm and exponential
Parametrisation by power series
The p-adic closure
Explicit description of the Poincare torsor
Norms along finite relative Cartier divisors
Explicit description of the Poincare torsor of a smooth curve
Explicit isomorphism for norms along equivalent divisors
Symmetry of the norm for divisors on smooth curves
Explicit residue disks and partial group laws
Extension of the Poincare biextension over Neron models
Explicit description of the extended Poincare bundle
Integral points of the extended Poincare torsor
Description of the map from the curve to the torsor
Some residue disks of the biextension
The rational points with a specific image mod 5
Some further remarks
Finiteness of rational points
The relation with p-adic heights
Full Text
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