Algorithms for $p$-adic Heights on Hyperelliptic Curves of Arbitrary Reduction
Francesca Bianchi, Enis Kaya, J. Steffen Müller
TL;DR
The paper develops a comprehensive algorithm to compute Coleman--Gross and Nekovář $p$-adic heights on hyperelliptic curves over number fields with arbitrary reduction above $p$, expressing the global height as a sum of ramified and unramified local terms. It integrates Vologodsky integration (via Katz–Kaya) for ramified places and offers an alternative genus-2 path through $p$-adic sigma functions, alongside Blakestad's canonical splitting in semistable ordinary reduction. It further connects to Mazur–Tate $p$-adic heights through $p$-adic Néron functions, with a detailed treatment of the canonical subspace in genus 2 and explicit formulas for the local heights via abelian logarithms, enabling quadratic Chabauty computations even at primes of bad reduction. The paper provides explicit algorithms, rigorous constructions, and numerical examples including the first quadratic Chabauty instance for a curve with bad reduction, and releases public code to reproduce the computations. This advances practical $p$-adic height calculations and their applications to Chabauty-type methods for rational and integral points on curves.
Abstract
In this paper, we develop an algorithm for computing Coleman--Gross (and hence Nekovář) $p$-adic heights on hyperelliptic curves over number fields with arbitrary reduction type above $p$. This height is defined as a sum of local heights at each finite place and we use algorithms for Vologodsky integrals, developed by Katz and the second-named author, to compute the local heights above $p$. We also discuss an alternative method to compute these for odd degree genus 2 curves via $p$-adic sigma functions, via work of the first-named author. For both approaches one needs to choose a splitting of the Hodge filtration. A canonical choice for this is due to Blakestad in the case of an odd degree curve of genus $2$ that has semistable ordinary reduction at $p$. We provide an algorithm to compute Blakestad's splitting, which is conjecturally the unit root splitting for the action of Frobenius. We give several numerical examples, including the first worked quadratic Chabauty example in the literature for a curve with bad reduction.