Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on good reduction of simple Abelian varieties


Author: C. Adimoolam
Journal: Proc. Amer. Math. Soc. 64 (1977), 196-198
MSC: Primary 14K15; Secondary 14G25
MathSciNet review: 0447259
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note it is shown that the reduction of a simple abelian variety of dimension $ \geqslant 2$, defined over an algebraic number field, at any finite good prime need not be simple. We give an example of a two-dimensional simple abelian variety defined over an algebraic number field whose reduction at any finite prime is isogenous either to a product of ordinary elliptic curves or to a product of supersingular elliptic curves.


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

  • [1] C. Adimoolam, Moduli of polarized abelian varieties and complex multiplications, Ph.D. thesis, SUNY at Stony Brook, May 1975.
  • [2] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993 (49 #2762)
  • [3] Yasuo Morita, On potential good reduction of abelian varieties, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22 (1975), no. 3, 437–447. MR 0404269 (53 #8072)
  • [4] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985 (44 #219)
  • [5] Tadao Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4) 2 (1969), 63–135. MR 0241435 (39 #2775)
  • [6] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 0236190 (38 #4488)
  • [7] Goro Shimura, Reduction of algebraic varieties with respect to a discrete valuation of the basic field, Amer. J. Math. 77 (1955), 134–176. MR 0066687 (16,616d)
  • [8] Goro Shimura, On the zeta-functions of the algebraic curves uniformized by certain automorphic functions, J. Math. Soc. Japan 13 (1961), 275–331. MR 0142515 (26 #84)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 14K15, 14G25

Retrieve articles in all journals with MSC: 14K15, 14G25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0447259-X
PII: S 0002-9939(1977)0447259-X
Keywords: Simple abelian variety, good reduction, p-rank of an abelian variety, Barsotti-Tate group, Dieudonné module, indefinite quaternion algebra, ordinary and supersingular elliptic curves
Article copyright: © Copyright 1977 American Mathematical Society