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

Tsimerman, Jacob

doi:10.1090/S0894-0347-2012-00739-5

Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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

Armand Brumer and Joseph H. Silverman, The number of elliptic curves over $\mathbf Q$ with conductor $N$, Manuscripta Math. 91 (1996), no. 1, 95–102. MR 1404420, DOI 10.1007/BF02567942
C. Chai and F. Oort, Abelian Varieties Isogenous to a Jacobian, http://www. math. upenn. edu/$\sim$ chai/papers_ pdf/isogJac65. pdf, preprint, 2010
Bas Edixhoven, On the André-Oort conjecture for Hilbert modular surfaces, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 133–155. MR 1827018
S. J. Edixhoven, B. J. J. Moonen, and F. Oort, Open problems in algebraic geometry, Bull. Sci. Math. 125 (2001), no. 1, 1–22. MR 1812812, DOI 10.1016/S0007-4497(00)01075-7
Jordan S. Ellenberg and Akshay Venkatesh, Reflection principles and bounds for class group torsion, Int. Math. Res. Not. IMRN 1 (2007), Art. ID rnm002, 18. MR 2331900, DOI 10.1093/imrn/rnm002
Frank Gerth III, Ranks of 3-class groups of non-Galois cubic fields, Acta Arith. 30 (1976), no. 4, 307–322. MR 422198, DOI 10.4064/aa-30-4-307-322
B. Klingler and A. Yafaev, The André-Oort conjecture, http://www. math. jussieu. fr/$\sim$ klingler/papiers/KY12. pdf, preprint, 2008.
Tadasi Nakayama, Cohomology of class field theory and tensor product modules. I, Ann. of Math. (2) 65 (1957), 255–267. MR 90620, DOI 10.2307/1969962
Takashi Ono, On the Tamagawa number of algebraic tori, Ann. of Math. (2) 78 (1963), 47–73. MR 156851, DOI 10.2307/1970502
Takashi Ono, Arithmetic of algebraic tori, Ann. of Math. (2) 74 (1961), 101–139. MR 124326, DOI 10.2307/1970307
J. Pila, O-minimality and the André-Oort conjecture for $\mathbb {C}^n$, http://www.maths.ox.ac.uk/system/files/PilaAOMML4. pdf, to appear in Annals of Math.
Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998. MR 1492449, DOI 10.1515/9781400883943
Jih Min Shyr, On some class number relations of algebraic tori, Michigan Math. J. 24 (1977), no. 3, 365–377. MR 491596
E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties : towards the André-Oort conjecture. http://www.math.u-psud.fr/~ullmo/Prebublications/UllmoYafaev2(3)(2)(2).pdf
E. Ullmo and A. Yafaev, Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciaux. preprint.
Andrei Yafaev, A conjecture of Yves André’s, Duke Math. J. 132 (2006), no. 3, 393–407. MR 2219262, DOI 10.1215/S0012-7094-06-13231-3
Shou-Wu Zhang, Equidistribution of CM-points on quaternion Shimura varieties, Int. Math. Res. Not. 59 (2005), 3657–3689. MR 2200081, DOI 10.1155/IMRN.2005.3657