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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points
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by Jacob Tsimerman
J. Amer. Math. Soc. 25 (2012), 1091-1117
DOI: https://doi.org/10.1090/S0894-0347-2012-00739-5
Published electronically: April 12, 2012
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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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Bibliographic 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
  • MR Author ID: 896479
  • Email: jacobt@math.harvard.edu
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 1091-1117
  • MSC (2010): Primary 11G15
  • DOI: https://doi.org/10.1090/S0894-0347-2012-00739-5
  • MathSciNet review: 2947946