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 $\iota_q$ be embedding of $Y_q$ as the Hecke correspondence into $Y_1 \times Y_1$. Let $\mu_1$ and $\mu_q$ 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 $(\iota_q)_\ast(\lvert F \rvert^2 \mu_q)$ converges weakly to the uniform measure $\mu_1 \times \mu_1$, as $q$ tends to infinity. We prove this conjecture with an effective rate of equidistribution, assuming GRH.