Hensel-lifting torsion points on Jacobians and Galois representations

TL;DR

This paper presents a method to compute mod  Galois representations from Frobenius data and Jacobians by p-adically lifting torsion points using Makdisi's algorithms.

Contribution

It introduces a novel p-adic lifting technique for torsion points on Jacobians to determine Galois representations from limited Frobenius information.

Findings

Effective computation of  Galois representations from Frobenius data.

A new p-adic lifting method for torsion points on Jacobians.

Application of Makdisi's algorithms to Galois representation problems.

Abstract

Let $\rho$ be a mod $\ell$ Galois representation. We show how to compute $\rho$, given the characteristic polynomial of the image of the Frobenius at one prime $p$ and a curve $C$ whose Jacobian contains $\rho$ in its $\ell$-torsion. The main ingredient is a method to $p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.