Some theorems on equality of L-functions of elliptic curves

Ivica Gusić, Zagreb, Croatia

Keywords: Galois field, discrete valuation, elliptic curve, abelian variety, l-adic representation, Frobenius element, inertia subgroup, L-function, Euler factor, isogeny, simple abelian variety, Weierstrass equation, index of inertia, Dirichlet density, good reduction of elliptic curve, elliptic curve with complex multiplication AMS Subject classification: 11G40, 14G10, 14H52

Abstract: Problem of equality L(A, K, s) = L(B, M, s) where A, B are elliptic curves over Galois number fields K, M respectively, is studied. The problem is solved in the case if A, B are defined over Q and K, M are not quadratic over K ∩ M (Theorem 1), as well as in the case if K ∩ M = Q and (6, n) = 1, where n = (K : Q) (Theorem 2).