About the choice of a basis in Kedlaya's algorithm

TL;DR

This paper discusses the choice of basis in Kedlaya's algorithm for counting points on hyperelliptic curves, focusing on Edixhoven's basis that simplifies p-adic computations and reduces precision requirements.

Contribution

It provides a detailed explanation and proof of the suitability of Edixhoven's basis in Kedlaya's algorithm, enhancing computational efficiency.

Findings

Edixhoven's basis ensures p-adic integrality of the Frobenius matrix coefficients.

Using this basis reduces the required p-adic precision in computations.

The paper confirms the mathematical validity of Edixhoven's basis for Kedlaya's algorithm.

Abstract

Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic lift of the curve. The Frobenius morphism of the curve induces an automorphism of this cohomological space. The key step of the algorithm is to determine this automorphism with a sufficiently high p-adic precision: it is given in the form of a matrix with respect to a certain basis. Edixhoven has found a basis that has the property that the coefficients of the matrix are p-adically integral. This allows a smaller required precision, because a (semi-linear) power of this matrix must be computed up to some given precision. This text describes Edixhoven's basis and provides a proof of the fact that the basis is suitable.