Unconditional computation of the class groups of real quadratic fields

Abstract

We describe an algorithm, based on the Selberg trace formula and explicit numerical computations of Maaß cusp forms, for computing the class groups and regulators of all real quadratic fields of discriminant Δ ≤ X \Delta \le X in time O ( X 5 / 4 + o ( 1 ) ) O(X^{5/4+o(1)}) , without assuming any unproven conjectures. We applied the algorithm to compute up to X = 10 11 X=10^{11} and used the output to test various implications of the Cohen–Lenstra heuristics.