Quadratic Chabauty:p-adic heights and integral points on hyperelliptic curves

Abstract

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with Q.

Quadratic Chabauty and Rational Points II: Generalised Height Functions on Selmer Varieties

“…Furthermore, by relating the mixed extensions A Z (x) to the ones arising in Nekovář's theory, we gave a formula for h v (A Z (x)) as a local height pairing h v (A Z (x − b, D Z (x − b)) between divisors. This was inspired by earlier uses of p-adic heights to obtain quadratic Chabauty formulae for integral points on elliptic and hyperelliptic curves in papers of Kim [28] and of the first author with Kedlaya and Kim [6] and Besser and Müller [4].…”

Explicit Chabauty–Kim for the split Cartan modular curve of level 13

“…Quadratic Chabauty pairs for rational points. The definition of quadratic Chabauty pairs is inspired by an approach for computing integral points on rank 1 elliptic curves [BB15], and more generally, on odd degree hyperelliptic curves [BBM16], which satisfy the assumptions of §1.4, as follows. Let h : J(Q) → Q p denote the p-adic height function [CG89].…”

An effective Chabauty–Kim theorem

“…where the m v are local constants as in [BBM16], and W denotes the scheme of Weierstrass points not equal to ∞.…”