Related papers: p-adic elliptic polylogarithms and cubic Chabauty

We describe an algorithm to compute the local Coleman-Gross p-adic height at p on a hyperelliptic curve. Previously, this was only possible using an algorithm due to Balakrishnan and Besser, which was limited to odd degree. While we follow…

Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the…

The main point of the paper is to take the explicit motivic Chabauty-Kim method developed in papers of Dan-Cohen--Wewers and Dan-Cohen and the author and make it work for non-rational curves. In particular, we calculate the abstract form of…

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…

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…

We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set $X(\mathbb{Q}_2 )_2 $, when $X$ is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions…

We give a generalization of the method of "Elliptic Curve Chabauty" to higher genus curves and their Jacobians. This method can sometimes be used in conjunction with covering techniques and a modified version of the Mordell-Weil sieve to…

In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good…

In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of…

Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic…

The Chabauty-Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell-Weil rank $r < g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was…

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement…

We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of…

Kantor's Thesis was the first step in unifying the Chabauty-Kim and Lawrence-Venkatesh methods via relative completion. In this work, we refine Kantor's approach by addressing its limitations, achieving the first unification where a…

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus $g \ge 2$ over…

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…

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) 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…

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed…

We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis,…