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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A note on good reduction of simple Abelian varieties
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by C. Adimoolam PDF
Proc. Amer. Math. Soc. 64 (1977), 196-198 Request permission


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.
    C. Adimoolam, Moduli of polarized abelian varieties and complex multiplications, Ph.D. thesis, SUNY at Stony Brook, May 1975.
  • 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) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
  • 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
  • David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
  • Tadao Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4) 2 (1969), 63–135. MR 241435
  • Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
  • Goro Shimura, Reduction of algebraic varieties with respect to a discrete valuation of the basic field, Amer. J. Math. 77 (1955), 134–176. MR 66687, DOI 10.2307/2372425
  • Goro Shimura, On the zeta-functions of the algebraic curves uniformized by certain automorphic functions, J. Math. Soc. Japan 13 (1961), 275–331. MR 142515, DOI 10.2969/jmsj/01330275
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 196-198
  • MSC: Primary 14K15; Secondary 14G25
  • DOI:
  • MathSciNet review: 0447259