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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn033, 27 pages, doi:10.1093/imrn/rnn033 published on May 6, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Splitting of Abelian Varieties

V. Kumar Murty1 and Vijay M. Patankar2

1 Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada M5S 2E4
2 Microsoft Research Lab India, "Scientia", 196/36 2nd Main Road, Sadashivanagar, Bangalore - 560080, India

Correspondence: Correspondence to be sent to: vij{at}microsoft.com

We study a new local–global problem in the context of Abelian varieties: Given an absolutely simple Abelian variety over a number field K, find a necessary and sufficient condition for the existence of a place v of K (or infinitely many places, or a set of places of positive density) such that A remains absolutely simple modulo v. It is well known that for absolutely simple Abelian surfaces with multiplication by an indefinite quaternion algebra, A modulo v, denoted by Av, is always isogenous to the square of some elliptic curve. For absolutely simple Abelian varieties of complex multiplication type, we show that Av stays simple for a set of places of density 1. On the other hand, for absolutely simple Abelian varieties associated by Shimura to cusp forms of weight 2, if Av splits at a set of places of positive density, then the absolute endomorphism algebra is noncommutative. Based on these results, we then formulate a conjecture: An absolutely simple Abelian variety defined over a number field splits at a set of places of positive density if and only if its absolute endomorphism algebra is noncommutative.



References

  1. Albert A. A. Involutorial simple algebras and real Riemann matrices. Annals of Mathematics (1935) 36:886–994.[CrossRef][ISI]
  2. Adimoolam C. A note on good reduction of simple Abelian varieties. Proceedings of the American Mathematical Society (1977) 64:196–8.[CrossRef][ISI]
  3. Chavdarov N. The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. Duke Mathematical Journal (1997) 87:151–80.[CrossRef][ISI]
  4. Chai C.-L., Oort F. A note on the existence of absolutely simple Jacobians. Journal of Pure and Applied Algebra (2001) 155:115–20.[CrossRef][ISI]
  5. Harder G. Some results on the Eisenstein cohomology of arithmetic subgroups of GLn. In: Cohomology of Arithmetic Groups and Automorphic Forms—Labesse J.-P., Schwermer J., eds. (1990) Berlin: Springer. Lecture Notes in Mathematics, 1447.
  6. Mazur B. On the passage from local to global in number theory. American Mathematical Society. Bulletin (1993) 29:14–50.[CrossRef]
  7. Milne J. S. Abelian varieties with complex multiplication (for pedestrians). (1998) 6. Available at www.milne.org.
  8. Momose F. On the l-adic representations attached to modular forms. Journal of the Faculty of Science, University of Tokyo Sect. 1 A Mathematics (1981) 28:89–109.
  9. Mumford D. Abelian Varieties (1988) Oxford and Bombay: Oxford University Press and Tata Institute of Fundamental Research. (reprint of 1974 edition).
  10. Murty V. K. Splitting of Abelian varieties: A new local-global problem. In: Algebra and Number Theory—Tandon R., ed. (2005) New Delhi: Hindustan Book Agency. 258–68.
  11. Ogus A. Hodge cycles and crystalline cohomology. In: Hodge Cycles, Motives, and Shimura Varieties (1982) Berlin: Springer. 357–414. Lecture Notes in Mathematics, 900.
  12. Pohlmann H. Algebraic cycles on Abelian varieties of complex multiplication type. Annals of Mathematics (1968) 88:161–80.[CrossRef][ISI]
  13. Rajan C. S. On strong multiplicity one for l-adic representations. International Mathematics Research Notices (1998) 1998(3):161–72.[Free Full Text]
  14. Rajan C. S. Refinement of strong multiplicity one for automorphic representations of GL(n). Proceedings of the American Mathematical Society (2000) 128:691–700.[CrossRef][ISI]
  15. Ramakrishnan D. Fourier Analysis on Number Fields (2002) Berlin: Springer. Graduate Texts in Mathematics, 186.
  16. Ribet K. Twists of modular forms and endomorphisms of Abelian varieties. Mathematische Annalen (1980) 253:43–62.[CrossRef][ISI]
  17. Ribet K. Endomorphism algebras of Abelian varieties attached to newforms of weight 2. In: Séminaire de Théorie des Nombres, Paris 1979–80 (1981) Boston, MA: Birkhauser Boston. Progress in Mathematics, 12. (Séminaire Delange-Pisot-Poitou).
  18. Serre J.-P. Résumé des cours de 1985–1986, Annuaire du Collège de France (1986) 95–99. (See also Collected Papers, vol. 4, 33–37. Berlin: Springer, 2000.)
  19. Serre J.-P. Notes on article 133. In: Collected Papers (2000) 4(637). Berlin: Springer.
  20. Serre J.-P. Article 133, Lettres á Ken Ribet du 1/1/1981 et du 29/1/1981. In: Collected Papers (2000) 4. Berlin: Springer. 1–20.
  21. Serre J.-P., Tate J. Good reduction of Abelian varieties. Annals of Mathematics (1968) 88:492–517.[CrossRef][ISI]
  22. Shimura G. Introduction to the Arithmetic Theory of Automorphic Functions (1994) Princeton, NJ: Princeton University Press.
  23. Shimura G., Taniyama Y. Complex Multiplication of Abelian varieties and Its Applications to Number Theory (1961) Tokyo, Japan: Mathematical Society of Japan. Publications of the Mathematical Society of Japan, 6.
  24. Tate J. Algebraic cycles and poles of zeta functions. In: Arithmetic Algebraic Geometry—Schilling O. F. G., ed. (1965) New York: Harper and Row. 93–110.
  25. Tate J. Endomorphisms of Abelian varieties over finite fields. Inventiones Mathematicae (1966) 2:134–44. See also Appendix to [9].[CrossRef]
  26. Waterhouse W. C. Abelian varieties over finite fields. Annales Scientifiques de l'Ecole Normale Superieure (1969) 42:521–60.
  27. Waterhouse W. C., Milne J. S. Abelian varieties over finite fields. Proceedings of Symposia in Pure Mathematics (1971) 20:53–64.

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This Article
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Right arrow Alert me when this article is cited
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