Analytic and Geometric Number Theory Seminar

Let K/Q be an extension of number fields. The Hasse norm theorem states that when K is cyclic any non-zero element of Q can be represented as a norm from K globally if and only if it can be represented everywhere locally. In this talk I will discuss the harder problem of representing non-constant polynomials by norms, focusing on the case of irreducible quadratics. Here it transpires that ideas stemming from Linnik's dispersion method and the theory of bilinear sums can lead to a proof of the Hasse principle in this case. This is joint work with Roger Heath-Brown.