Vol. 2, 2019

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Fast multiquadratic $S$-unit computation and application to the calculation of class groups

Jean-François Biasse and Christine Van Vredendaal

Vol. 2 (2019), No. 1, 103–118
Abstract

Let L = (d1,,dn) be a real multiquadratic field and S be a set of prime ideals of L. In this paper, we present a heuristic algorithm for the computation of the S-class group and the S-unit group that runs in time Poly(log(Δ),Size(S))eO˜(ln d) where d = indi and Δ is the discriminant of L. We use this method to compute the ideal class group of the maximal order OL of L in time Poly(log(Δ))eO˜(log d). When log(d) log(log(Δ))c for some constant c < 2, these methods run in polynomial time. We implemented our algorithm using Sage 7.5.1.

Keywords
ideal class group, $S$-unit group, multiquadratic fields
Mathematical Subject Classification 2010
Primary: 11R04, 11R29, 11R65, 11Y99
Secondary: 11R11, 11R16, 11S20, 11Y50
Milestones
Received: 2 March 2018
Revised: 21 September 2018
Accepted: 28 September 2018
Published: 13 February 2019
Authors
Jean-François Biasse
Department of Mathematics and Statistics
University of South Florida
Tampa, FL
United States
Christine Van Vredendaal
NXP Semiconductors
Eindhoven
Netherlands