Abstract
This chapter presents a complete picture for all possible rings and algebras for elliptic curves over arbitrary fields. It is clear what happens with respect to reduction modulo p and lifting to characteristic zero: E has potentially good reduction modulo ñ if and only if j(E) is integral at p every endomorphism φ0 ∈ End(E0) can be lifted from characteristic p to characteristic zero; however, a non-commutative End(E0) cannot be lifted. For higher dimensions, all these questions are more difficult. Endomorphism algebras for abelian varieties over finite fields follow from the theory of Tate. The chapter presents some general facts and related theorems concerning endomorphism algebras.
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