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

TL;DR

This paper introduces an improved unconditional algorithm for computing class groups of imaginary quadratic fields with discriminants up to 2^40, extending previous bounds significantly and providing new data on class group structures.

Contribution

The authors develop a new method combining classical formulas and trace formulas to compute class groups unconditionally up to discriminant 2^40, surpassing previous limits.

Findings

Extended the discriminant bound from 2×10^{11} to 2^{40}.

Provided statistical data supporting conjectures about class groups.

Discovered new examples of class groups with exotic structures.

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.