Seminar

Let $q,Q$ be two integral quadratic forms in $m < n$ variables. One can ask when $q$ can be represented by $Q$ - that is, whether there exists an $n \times m$-integer matrix $T$ such that $Q \circ T = q$. 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.