Computing L-series of geometrically hyperelliptic curves of genus three

TL;DR

This paper presents an algorithm for computing local zeta functions of genus three hyperelliptic curves over Q, extending previous methods to a broader class of curves with efficient implementation.

Contribution

It introduces a novel algorithm adapting the accumulating remainder tree technique to quadratic fields for genus three hyperelliptic curves over Q.

Findings

Algorithm successfully computes local zeta functions at all odd primes of good reduction.

Implementation demonstrates improved performance over previous methods for similar curves.

Extends computational techniques to non-standard hyperelliptic models over Q.

Abstract

Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N. The algorithm relies on an adaptation of the "accumulating remainder tree" to matrices with entries in a quadratic field. We report on an implementation, and compare its performance to previous algorithms for the ordinary hyperelliptic case.