Test vectors for Waldspurger's period integral and application to the mass equidistribution on nonsplit torus

Abstract

In this paper we provide local test vector for Waldspurger's period integral, when the level of the representation $\pi_v$ is sufficiently large compared to the level of the character $\Omega_v$ over quadratic extension, while allowing joint ramifications. The test vectors we shall use are variants of classical newforms, and the size of the resulting local integral is asymptotically the inverse of convexity bound for $L(\Pi\otimes\Omega,1/2)$. Such test vectors are used to recover Gross-Prasad type test vectors. We also get vanishing result for local integral when using other test vectors. This phenomenon is used to prove the mass equidistribution of cuspidal newforms on nonsplit torus in depth aspect.