Unconditional class group tabulation of imaginary quadratic fields to |Δ|<2⁴⁰

Mosunov, A.; M. J. Jacobson, Jr.

doi:10.1090/mcom3050

Unconditional class group tabulation of imaginary quadratic fields to $|\Delta | < 2^{40}$
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by A. S. Mosunov and Jr. M. J. Jacobson;

Math. Comp. 85 (2016), 1983-2009

DOI: https://doi.org/10.1090/mcom3050

Published electronically: November 3, 2015

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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 Jacobson, Ramachandran, and Williams 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 of conjectures is presented, along with new examples of class groups with exotic structures.

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