Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points

Source abstract

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 A_{g,1} . Specifically, we ask that the sizes of these orbits grow like a power of the discriminant of the underlying endomorphism algebra. We prove this unconditionally in the case g ≤ 6 g\leq 6 , and for all g 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.