Representations of Binary by Quaternary Quadratic Form

Let q,Q be two integral quadratic forms in m less than n

variables. One can ask when q can be represented by Q - that is,

whether there exists an n×m-integer matrix T such that Q∘T=q.  Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo N for every positive integer N. In the absence of local obstructions, does a (global) representation of q by Q

exist?

This question is particularly delicate when the codimension n−m

is small, with codimension 2 being the most challenging. In this talk, we

discuss joint work with Wooyeon Kim and Pengyu Yang where we establish

such a local-global principle for representations of binary by

quaternary quadratic forms (when m=2 and n=4) under two Linnik-type

splitting conditions. Our proof uses a recent measure rigidity result of

Einsiedler and Lindenstrauss for higher-rank diagonalizable actions and

the determinant method.