ENDOMORPHISM RINGS OF REDUCTIONS OF ELLIPTIC CURVES AND ABELIAN VARIETIES (200,00 руб.)

Let E be an elliptic curve without CM that is defined over a number field A". <...> For all but finitely many non-Archimedean places v of A"" 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 u's with ordinary E(v) has density 1 (Serre). <...> For such v the endomorphism ring End(E(v)) 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 A" such that the discriminant A(v) of End(I?(t;)) is divisible by N and the ratio —^ is 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 ^-adic representations and elliptic curves" and questions of Mihran Pa-pikian and Alina Cojocaru.! <...>