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arXiv · 1502.07953

Unconditional Class Group Tabulation of Imaginary Quadratic Fields to $|\Delta| < 2^{40}$

A. S. Mosunov·M. J. Jacobson Jr
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Abstract

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all $\Delta \not \equiv 1 \pmod{8}.$ The group structure is resolved using the factorization of the group order. The $1 \bmod 8$ case was handled using the methods of \cite{jacobson}, including the batch verification method based on the Eichler-Selberg trace formula to remove dependence on the Extended Riemann Hypothesis. Our new method enabled us to extend the previous bound of $|\Delta| < 2 \cdot 10^{11}$ to $2^{40}$. Statistical data in support of a variety conjectures is presented, along with new examples of class groups with exotic structures.

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BibTeXRIS

A. S. Mosunov, M. J. Jacobson Jr. 2015-02-27. Unconditional Class Group Tabulation of Imaginary Quadratic Fields to $|\Delta| < 2^{40}$. https://arxiv.org/abs/1502.07953

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