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Lethbridge Number Theory and Combinatorics Seminar
Date: June 28, 2016
Time: 2:00-2:50pm
Lecturer(s): Vijay Patankar (Jawaharlal Nehru University)
Location: University of Lethbridge, C630
Topic: Pairs of elliptic curves and their Frobenius fields
Description (in plain text format): 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.
If time permits 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.
Other Information:
Contact:
Barb Hodgson | hodgsonb@uleth.ca | (403) 329-2470 | uleth.ca/artsci/math-computer-science
