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Abelian varieties over $\mathbb{Q}$ with bad reduction in one prime only

Published online by Cambridge University Press:  21 June 2005

René Schoof
Affiliation:
Dipartimento di Matematica, 2a Università di Roma ‘Tor Vergata’, I-00133, Roma, Italyschoof@science.uva.nl
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Abstract

We show that for the primes l = 2, 3, 5, 7 or 13, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ that have good reduction at every prime different from l and are semi-stable at l. We show that any semi-stable abelian variety over $\mathbb{Q}$ with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X0(11). In addition, we show that for l = 2, 3 and 5, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ with good reduction outside l that acquire semi-stable reduction at l over a tamely ramified extension.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2005

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