Geometric Quadratic Chabauty

Abstract

Since 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).

Geometric quadratic Chabauty over number fields

“…Let us fix a prime p where C has good reduction while satisfying some additional mild ramification conditions (see Assumption 4.1). In its crudest form, our main result is an effective bound on C(K) of the following form, which generalizes the work of Edixhoven and Lido [11].…”

“…Now we briefly explain the proof of Theorem 1.1. The strategy, following [11] (summarized in Section 1.3), is to replace the Jacobian in Chabauty's original approach by something higher dimensional. More precisely, we construct a certain G ρ−1 m -torsor T over J, which will replace J.…”

“…We let O K,p := O K ⊗ Z Z p , and consider the following quadratic Chabauty diagram ( 6) The key of the approach is to analyze the p-adic closure Y of the O K -points of the torsor T. If K = Q, this can be done by parametrizing the p-adic closure of J(Z) = J(Q), as G m (Z) = ±1. This is a major simplification and essentially why [11] decided to work over Q. In fact, it was suggested to us by Edixhoven and Lido that a restriction of scalars approach might reduce the case of general number fields K back to the case of Q.…”

“…For the convenience of the reader we provide a list of notations used in the main body of the paper. We have chosen to use notations similar to [11] in order to facilitate the comparison with the original approach over Q.…”

“…Geometric quadratic Chabauty (over Q). Recently, Edixhoven and Lido have explored in [11] a different, and arguably more direct, approach to quadratic Chabauty. Their method is referred to as the geometric quadratic Chabauty method (with an emphasis on geometric), and works under the same condition r < g+ρ−1 as in Section 1.2.2.…”