Elliptic curves and their Frobenius fields

Manisha Kulkarni, Vijay M. Patankar, C. S. Rajan

Published 2014-03-22, updated 2015-04-01Version 2

Let \( E \) be an elliptic curve defined over a number field \( K \). For a place \( v \) of \( K \) of good reduction for \( E \), let \( F(E, v) \) denote the Frobenius field of \( E \) at \( v \), given by the splitting field of the characteristic polynomial of the Frobenius automorphism at \( v \) acting on the Tate module of \( E \). We show that the set of finite places \( v \) of \( K \) such that \( F(E,v) \) equals a fixed imaginary quadratic field \( F \) has positive upper density if and only if \( E \) has complex multiplication by \( F \). Given two elliptic curves \( E_1 \) and \( E_2 \) defined over a number field \( K \), with at least one of them without complex multiplication, we prove that the set of places \( v \) of \( K \) of good reduction such that \( F(E_1, v) = F (E_2, v) \) has positive upper density if and only if \( E_1 \) and \( E_2 \) are isogenous over some extension of \( K \).