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.
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Received: 19 April 2001 / Revised version: 21 October 2001 / Published online: 10 February 2003
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Schoof, R. Abelian varieties over cyclotomic fields with good reduction everywhere. Math. Ann. 325, 413–448 (2003). https://doi.org/10.1007/s00208-002-0368-7
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DOI: https://doi.org/10.1007/s00208-002-0368-7