New test vector for Waldspurger's period integral, relative trace formula, and hybrid subconvexity bounds

Abstract

In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on ${\text{GL}}_2$) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor $C(\pi \times {\pi }_{{\chi }^{-1}})$. As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type subconvexity bound which can be as strong as the Weyl bound in proper range.

arXiv commentUpdated version contains application to relative trace formula computations and hybrid subconvexity bounds

References in corpus (8)

• The Weyl bound for Dirichlet $L$-functions of cube-free conductor
• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• The orbit method and analysis of automorphic forms
• Waldspurger's period integral for newforms
• Test vectors for Waldspurger's period integral and application to the mass equidistribution on nonsplit torus
• Subconvex bounds for compact toric integrals
• Some analytic aspects of automorphic forms on GL(2) of minimal type
• Sup norm on ${\text{PGL}}_n$ in depth aspect

Cited by in corpus (3)

• Waldspurger's period integral for newforms
• Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity
• The Petersson/Kuznetsov trace formula with prescribed local ramifications