p-adic elliptic polylogarithms and cubic Chabauty

TL;DR

This paper develops explicit formulas involving p-adic elliptic polylogarithms to describe the Chabauty--Kim set for elliptic curves, enabling verification of Kim's conjecture in new cases.

Contribution

It provides a novel explicit formula for the Chabauty--Kim set in depth 3 for elliptic curves of rank at most 2, using p-adic elliptic polylogarithms.

Findings

Explicit formula for Chabauty--Kim set in depth 3

Verification of Kim's conjecture in new instances

Finite set described by zeros of p-adic elliptic polylogarithm polynomials

Abstract

The Chabauty--Coleman--Kim method, under favourable circumstances, describes the set of integral points of a hyperelliptic curve inside the $p$-adic zeroes of certain transcendental functions. For an elliptic curve of Mordell--Weil rank one, the Chabauty--Coleman--Kim set in depth 2 is given by the zeroes of a (finite union of) quadratic polynomial(s) in the $p$-adic logarithm of the elliptic curve and the local $p$-adic height at $p$. Here, we give an explicit formula for a finite set containing the Chabauty--Coleman--Kim set in depth 3 for an elliptic curve of rank at most 2 under an assumption on non-vanishing of a special value of a $p$-adic $L$-function. The finite set is given by the zeroes of a polynomial in $p$-adic elliptic polylogarithms. We use these formulas to verify new instances of Kim's conjecture.