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

Abstract

We analyze the complexity of the computation of the class group structure, regulator, and a system of fundamental units of an order in a certain class of number fields. Our approach differs from Buchmann’s, who proved a complexity bound under the generalized Riemann hypothesis of L ( 1 / 2 , O ( 1 ) ) L(1/2,O(1)) when the discriminant tends to infinity with fixed degree. We achieve a heuristic subexponential complexity in O ( L ( 1 / 3 , O ( 1 ) ) ) O(L(1/3,O(1))) under the generalized Riemann hypothesis when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge, Gaudry and Thomé in the context of algebraic curves over finite fields. We also address rigorously the problem of the precision of the computation of the regulator.