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

Author: Jacob Tsimerman
Journal: J. Amer. Math. Soc. 25 (2012), 1091-1117
MSC (2010): Primary 11G15
Published electronically: April 12, 2012
MathSciNet review: 2947946
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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}$. 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\leq 6$, 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.

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Additional Information

Jacob Tsimerman
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544-1000
Address at time of publication: Department of Mathematics, Faculty of Arts & Sciences, Harvard University, One Oxford Street, Cambridge MA 02138

Received by editor(s): May 29, 2011
Received by editor(s) in revised form: February 27, 2012, and March 23, 2012
Published electronically: April 12, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.