Title: Pairs of quadratic forms over p-adic fields

 

Abstract: The Hasse-Minkowski theorem implies that if a quadratic form over a number field k has a nontrivial zero over every achimedean and non-achimedean completion of k, then the form will have a nontrivial zero over k. It's natural to ask whether this is true for common nontrivial zeros of pairs of quadratic forms. In an effort to answer this question, new results about pairs of quadratic forms over p-adic fields have been proven. One such result, by Heath-Brown, deals with finding forms in the pencil generated by a pair of quadratic forms over a p-adic field in 8 variables that split off 3 hyperbolic planes. In this talk, we will examine this result by Heath-Brown, and we will discuss the ongoing effort of generalizing Heath-Brown's hyperbolic plane result to pairs of quadratic forms over a p-adic field in an arbitrary number of variables.