Reduction of Elliptic Curves

The properties of reduction in this chapter, except for §3, are due to Deuring, who used them to give his algebraic proofs for complex multiplication. We shall not give any proofs. These can be given ad hoc, as Deuring did, for the elliptic curves, or one can develop a general reduction theory, as in Shimura [39]. No matter what, it is a pain to lay these foundations, but the results can be stated simply. Although classically one reduces over a discrete valuation ring, it is useful to deal with an arbitrary local ring.