Joint equidistribution of newforms

Abstract

Let $Y_1$ be a compact arithmetic hyperbolic surface associated to a maximal quaternion order, let $Y_q$ be a cover associated to an Eichler suborder of prime level $q$, and let ${\hat{I}}_q^1$ be embedding of $Y_q$ as the Hecke correspondence into $Y_1\times Y_1$. Let $\hat{I}{¼}_1$ and $\hat{I}{¼}_q$ be the invariant probability measures on $Y_1$ and $Y_q$, respectively. If $F$ is a newform on $Y_q$, we conjecture that the pushforward measure $({\hat{I}}_q^1{)}_{\ast }(|F{|}^2\hat{I}{¼}_q)$ converges weakly to the uniform measure $\hat{I}{¼}_1\times \hat{I}{¼}_1$, as $q$ tends to infinity. We prove this conjecture with an effective rate of equidistribution, assuming GRH.

51 pages