Theoretical Statistics and Mathematics Unit, ISI Delhi

September 16, 2015 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Vijay M. Patankar,
Jawaharlal Nehru University

Title:
Pairs of elliptic curves and their Frobenius fields

Abstract of Talk

For an elliptic curve E defined over a number field K and a finite prime v of K of good reduction for E, let F(E,v) denote the Frobenius field of E at v defined as the splitting field of the characteristic polynomial of the Frobenius automorphism at v acting on the Tate module of E. Let us recall that for an elliptic curve defined over the rational numbers and for a prime p, the characteristic polynomial of equals t^2 - a_p (E) t + p, where a_p(E) = p +1 - N_p(E) where N_p(E) is the number points point on E modulo p over the finite field with p elements.

We prove the following result about Frobenius fields of pairs of elliptic curves.

Suppose E1 and E2 are two elliptic curves defined over a number field K with at least one of them without complex multiplication. We prove that the set of primes v of K of good reduction such that the corresponding Frobenius fields are equal has positive density if and only if E1 and E2 are isogenous over some extension of K.

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

This is a joint work with Manisha Kulkarni (IIITB, Bangalore) and C. S. Rajan (TIFR).