虚二次域到$|Δ| < 2^{40}$的无条件类群制表
Unconditional Class Group Tabulation of Imaginary Quadratic Fields to $|Δ| < 2^{40}$
摘要：
提出了一种改进的有界判别式虚二次域类组列表算法。我们的方法使用经典的类数公式，涉及θ-系列计算的群阶无条件地为所有$Δ\not \equiv 1 \pmod{8}.$的群结构是解决使用的群阶的因式分解。$1 \bmod 8$案例使用\cite{jacobson}的方法进行处理，包括基于Eichler-Selberg迹公式的批量验证方法，以消除对扩展黎曼假设的依赖。我们的新方法使我们能够扩展以前的界限$|Δ| < 2 \cdot 10^{11}$的$2^{40}$。统计数据支持的各种结构，以及新的例子与异国情调的结构类群。
英文摘要：
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 $Δ\not \equiv 1 \pmod{8}.$ 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 $|Δ| < 2 \cdot 10^{11}$ 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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