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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.

References [Enhancements On Off] (What's this?)

  • 1. A. Brumer and J. H. Silverman, The number of elliptic curves over $ \mathbb{Q}$ with conductor $ N$, Manuscripta Math. 91 (1996), no. 1, pp. 95-102. MR 1404420 (97e:11062)
  • 2. C. Chai and F. Oort, Abelian Varieties Isogenous to a Jacobian, http://www. math. upenn. edu/$ \sim $ chai/papers_ pdf/isogJac65. pdf, preprint, 2010
  • 3. B. Edixhoven, On the André-Oort conjecture for Hilbert modular surfaces, Moduli of abelian varieties (Texel Island, 1999), Progr. Math. 195, Birkhäuser, Basel, 2001, pp. 133-155. MR 1827018 (2002c:14042)
  • 4. B. Edixhoven, B. Moonen and F. Oort, Open problems in algebraic geometry, Bull. Sci. Math., vol. 125. (2001), pp. 1-22. MR 1812812 (2002a:14001)
  • 5. J. Ellenberg and A. Venkatesh, Reflection principles and bounds for class group torsion, Int. Math. Res. Not. 2007, no. 1. MR 2331900 (2008f:11124)
  • 6. F. Gerth, Ranks of 3-Class Groups of non-Galois Cubic Fields, Acta Arithmetica, XXX, 1976, pp. 307-322. MR 0422198 (54:10190)
  • 7. B. Klingler and A. Yafaev, The André-Oort conjecture, http://www. math. jussieu. fr/$ \sim $ klingler/papiers/KY12. pdf, preprint, 2008.
  • 8. T. Nakayama, Cohomology of class field theory and tensor product modules, Ann. of Math. (2), 65 (1957), pp. 255-267. MR 0090620 (19:841b)
  • 9. T. Ono, On the Tamagawa Number of Algebraic Tori, Annals of Mathematics, Vol. 78, No. 1, July, 1963. MR 0156851 (28:94)
  • 10. T. Ono, Arithmetic of Algebraic Tori, Annals of Mathematics, Vol. 74, No. 1, July, 1961. MR 0124326 (23:A1640)
  • 11. J. Pila, O-minimality and the André-Oort conjecture for $ \mathbb{C}^n$, pdf, to appear in Annals of Math.
  • 12. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, 1993. ISBN0125581807 MR 1278263 (95b:11039)
  • 13. G. Shimura, On Abelian Varieties with Complex Multiplication, Princeton Mathematical Series 46, Princeton University Press, 1998. MR 1492449 (99e:11076)
  • 14. J. Shyr, On Some Class Number Relations of Algebraic Tori, Michigan Math Journal, Vol. 24, Issue 3, 1977, pp. 365-377. MR 0491596 (58:10819)
  • 15. E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties : towards the André-Oort conjecture.
  • 16. E. Ullmo and A. Yafaev, Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciaux. preprint.
  • 17. A. Yafaev, A conjecture of Yves André's, Duke Mathematical Journal, Vol. 132, No. 3, 2006, pp. 393-408. MR 2219262 (2007b:11089)
  • 18. S. Zhang, Equidistribution of CM points on Quaternionic Shimura Varieties, International Mathematics Research Notices, No. 59, 2005, pp. 3657-3689. MR 2200081 (2007g:11067)

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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.

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