Статья «ENDOMORPHISM RINGS OF REDUCTIONS OF ELLIPTIC CURVES AND ABELIAN VARIETIES, "Алгебра и анализ"»

Let E be an elliptic curve without CM that is defined over a number field K. For all but finitely many non-Archimedean places v of K there is a reduction E(v) of E at v that is an elliptic curve over the residue field k(v) at v. The set of v’s with ordinary E(v) has density 1 (Serre). For such v the endomorphism ring End(E(w)) of E(v) is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers N and M there are infinitely many non-Archimedean places v of K such that the discriminant A(v) of End(E(«)) is divisible by N and the ratio AM. ?relatively prime to NM. We also discuss similar questions for reductions of Abelian varieties. The subject of this paper was inspired by an exercise in Serre’s “Abelian f-adic representations and elliptic curves” and questions of Mihran Pa- pikian and Alina Cojocaru.