An L(1/3) algorithm for ideal class group and regulator computation in certain number fields

TL;DR

This paper presents an L(1/3) subexponential algorithm for computing the class group, regulator, and fundamental units in certain number fields, improving upon previous methods with a different complexity bound.

Contribution

It introduces a novel L(1/3) complexity algorithm for number field invariants, extending the scope beyond fixed-degree cases.

Findings

Achieves subexponential complexity in L(1/3,O(1))

Extends applicability to number fields with both discriminant and degree tending to infinity

Uses techniques adapted from algebraic curves over finite fields

Abstract

We analyse the complexity of the computation of the class group structure, regulator, and a system of fundamental units of a certain class of number fields. Our approach differs from Buchmann's, who proved a complexity bound of L(1/2,O(1)) when the discriminant tends to infinity with fixed degree. We achieve a subexponential complexity in O(L(1/3,O(1))) when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge and Gaudry in the context of algebraic curves over finite fields.