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Title: Unconditional Class Group Tabulation to 2⁴⁰
Author: Mosunov, Anton
Advisor: Jacobson, Michael
Keywords: Mathematics
Issue Date: 21-Jul-2014
Abstract: In this thesis, we aim to tabulate the class groups of binary quadratic forms for all fundamental discriminants D < 0 satisfying |D| < 2⁴⁰. Our computations are performed in several stages. We first multiply large polynomials in order to produce class numbers h(D) for D != 1 (mod 8). This stage is followed by the resolution of class groups Cl(D) with the Buchmann-Jacobson-Teske algorithm, which can be significantly accelerated when the class numbers are known. In order to determine Cl(D) for D = 1 (mod 8), we use this algorithm in conduction with Bach's conditional averaging method and the Eichler-Selberg trace formula, required for unconditional verification. Our novel class group tabulation method allowed us to gather unconditional numerical evidence in support of certain hypotheses, such as the Littlewood's bounds on L(1,x) and the Cohen-Lenstra heuristics.
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