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arXiv · 2604.20662

p-adic elliptic polylogarithms and cubic Chabauty

Jennifer S. Balakrishnan·Francesca Bianchi·Netan Dogra
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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.

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Jennifer S. Balakrishnan, Francesca Bianchi, Netan Dogra. 2026-04-22. p-adic elliptic polylogarithms and cubic Chabauty. https://arxiv.org/abs/2604.20662

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