Open AccessAbelian varieties over cyclotomic fields with good reduction everywhere
Author(s) -
Ren x E Schoof
Publication year - 2003
Publication title -
mathematische annalen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.235
H-Index - 75
eISSN - 1432-1807
pISSN - 0025-5831
DOI - 10.1007/s00208-002-0368-7
Subject(s) - mathematics , isogeny , abelian group , abelian variety , good reduction , jacobian matrix and determinant , reduction (mathematics) , riemann hypothesis , cyclotomic field , arithmetic of abelian varieties , pure mathematics , almost everywhere , zero (linguistics) , discrete mathematics , elementary abelian group , rank of an abelian group , geometry , medicine , linguistics , philosophy , surgery
For every conductor f{1,3,4,5,7,8,9,11,12,15} there exist non-zero abelian varieties over the cyclotomic field Q(ζ f ) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X 1 (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 Q(ζ f ) 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.
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