Brauer-Siegel for Arithmetic Tori and lower bounds for Galois orbits of special points

TL;DR

This paper establishes a Brauer-Siegel type formula for arithmetic tori, linking discriminant, regulator, and class number, and applies it to analyze Galois orbits of CM points with new bounds and asymptotics.

Contribution

It derives a Brauer-Siegel formula for tori based on Shyr's class number formula and applies it to obtain bounds on Galois orbits of CM points, including unconditional results for small dimensions.

Findings

Unconditional growth bounds for Galois orbits when g ≤ 5.

Conditional bounds under the Generalized Riemann Hypothesis for all g.

A transfer principle for torsion in class groups of number fields.

Abstract

In \cite{S}, Shyr derived an analogue of Dirichlet's class number formula for arithmetic Tori. We use this formula to derive a Brauer-Siegel formula for Tori, relating the Discriminant of a torus to the product of its regulator and class number. We apply this formula to derive asymptotics and lower bounds for Galois orbits of CM points in the Siegel modular variety $A_{g,1}$. Specifically, we ask that the sizes of these orbits grows like a power of Discriminant of the underlying endomorphism algebra. We prove this unconditionally in the case $g\leq 5$, and for all $g$ under the Generalized Riemann Hypothesis for CM fields. Along the way we derive a general transfer principle for torsion in ideal class groups of number fields.