Compositio Mathematica



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


René Schoof a1
a1 Dipartimento di Matematica, 2a Università di Roma ‘Tor Vergata’, I-00133, Roma, Italy schoof@science.uva.nl

Article author query
schoof r   [Google Scholar] 
 

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.

(Received June 15 2003)
(Accepted April 14 2004)
(Published Online June 21 2005)


Key Words: abelian varieties; number fields; group schemes; semi-stable reduction.

Maths Classification

11G10; 14L15.