Speaker: Ohad Avnery
(Open University):
Title:
Elementary solution to primes of the form x^2+ny^2

Abstract:
Using class field theory, Cox gave a characterization of the primes p represented by the quadratic form x^2+ny^2. The characterization has to do with an irreducible integral polynomial f_n and its roots modulo p.

We give a new proof of this result, inspired by explicit class field theory and formulas for x and y. For us, the polynomial f_n is just the minimal polynomial of the j-invariant of a suitably chosen elliptic curve.

The proof also gives us an explicit formula for x in the representation of p as x^2 + ny^2, as an exponential sum over F_p (or equivalently, as the normalized number of points on an elliptic curve E over F_p). This simplifies and generalizes formulas of Gauss, Eisenstein, and others.
 
This is a joint work with Mark Shusterman.