Local-global principles for representations of quadratic forms

Jordan Ellenberg, Akshay Venkatesh

Published 2006-04-11, updated 2006-04-13Version 2

We prove the local-global principle holds for the problem of representations of quadratic forms by quadratic forms, in codimension $\geq 7$. The proof uses the ergodic theory of $p$-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.