Genus computation of global function fields

TL;DR

This paper introduces a fast algorithm for computing the genus of global function fields using lattice methods and the Montes algorithm, avoiding basis computations.

Contribution

It presents a novel genus computation algorithm that leverages lattice theory and the Montes algorithm, improving efficiency over previous methods.

Findings

The algorithm computes genus efficiently without basis calculations.

It expresses genus in terms of field extension degrees and order indices.

The method is applicable to function fields over finite fields.

Abstract

In this paper we present an algorithm that computes the genus of a global function field. Let F/k be function field over a field k, and let k0 be the full constant field of F/k. By using lattices over subrings of F, we can express the genus g of F in terms of [k0 : k] and the indices of certain orders of the finite and infinite maximal orders of F . If k is a finite field, the Montes algorithm computes the latter indices as a by-product. This leads us to a fast computation of the genus of global function fields. Our algorithm does not require the computation of any basis, neither of finite nor infinite maximal order.