Lethbridge Number Theory and Combinatorics Seminar: Vijay Patankar

Given an elliptic curve E over a number field K. The Frobenius field attached to E at a prime p is the splitting field of the characteristic polynomial of the Frobenius endomorphism acting on the ell-adic Tate module of E (ell a prime different from p) over the rationals. Thus, the splitting field is either of degree 1 or degree 2 over the rationals.

Let E_1 and E_2 be elliptic curves 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 the corresponding Frobenius fields are equal has positive upper density if and only if E_1 and E_2 are isogenous over some extension of K.

For an elliptic curve E defined over a number field K, we show that the set of finite places of K such that the Frobenius field at v equals a fixed imaginary quadratic field F has positive upper density if and only if E has complex multiplication by F.

Time permiting we will provide a sketch of a result about two dimensional ell-adic Galois representations that we will need using an algebraic density theorem due to Rajan.