Simultaneous equidistribution of toric periods and fractional moments of $L$-functions

Abstract

The embedding of a torus into an inner form of PGL2 defines an adelic toric period. A general version of Duke’s theorem states that this period equidistributes as the discriminant of the splitting field tends to infinity. In this paper we consider a torus embedded diagonally into two distinct inner forms of PGL2. Assuming the Generalized Riemann Hypothesis (and some additional technical assumptions), we show simultaneous equidistribution as the discriminant tends to infinity, with an effective logarithmic rate. Our proof is based on a probabilistic approach to estimating fractional moments of $L$-functions twisted by extended class group characters.