Andrei Seymour-Howell

University of Bristol University of Bristol

Unconditional computation of the class groups of real quadratic fields

Linfoot Number Theory Seminar

1st November 2023, 11:00 am – 12:00 pm
Fry Building, 2.04

Computing the class number of quadratic fields has been studied by several people over the years, most notably Gauß, who used these computations to help inform his famous conjecture that there are infinitely many real quadratic fields of class number one. The current state of the art algorithms to compute class groups of real quadratic fields unfortunately all rely on GRH. In this talk, I will describe an algorithm to batch verify the output of the conditional algorithms, without assuming any unproven conjectures. Rather surprisingly, the main tool used will be the Selberg trace formula and explicit numerical computations of Maaß cusp forms. We have implemented this algorithm to compute class groups with discriminants up to X=10^11 and used the output to test various implications of the Cohen-Lenstra heuristics. This is joint work with Ce Bian, Andrew Booker, Austin Docherty and Michael Jacobson.

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