Arithmetic of Abelian Varieties - Reduction Mod p

Reduction Mod p

Reduction of an abelian variety A modulo a prime ideal of (the integers of) K — say, a prime number p — to get an abelian variety A_p over a finite field, is possible for almost all p. The 'bad' primes, for which the reduction degenerates by acquiring singular points, are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory.

Here a refined theory of (in effect) a right adjoint to reduction mod p — the Néron model — cannot always be avoided. In the case of an elliptic curve there is an algorithm of John Tate describing it.