Abelian varieties over cyclotomic fields with good reduction everywhere |
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Authors: | René Schoof |
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Affiliation: | (1) Departimento di Mathematica, 2a Università di Roma ``Tor Vergata', I-00133 Roma, Italy (e-mail: schoof@science.uva.nl), IT |
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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 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. Received: 19 April 2001 / Revised version: 21 October 2001 / Published online: 10 February 2003 |
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