Mathematische Annalen

, Volume 325, Issue 3, pp 413–448

Abelian varieties over cyclotomic fields with good reduction everywhere

  • René Schoof

DOI: 10.1007/s00208-002-0368-7

Cite this article as:
Schoof, R. Math. Ann. (2003) 325: 413. doi:10.1007/s00208-002-0368-7

Abstract

 For every conductor f{1,3,4,5,7,8,9,11,12,15} there exist non-zero abelian varieties over the cyclotomic field Qf) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X1(f) are examples of such abelian varieties. In the other direction we show that for all f in the above set there do not exist any non-zero abelian varieties over Qf) with good reduction everywhere except possibly when f=11 or 15. Assuming the Generalized Riemann Hypothesis (GRH) we prove the same result when f=11 and 15.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • René Schoof
    • 1
  1. 1.Departimento di Mathematica, 2a Università di Roma ``Tor Vergata'', I-00133 Roma, Italy (e-mail: schoof@science.uva.nl)IT