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

TL;DR

This paper develops new local test vectors for Waldspurger's period integral using minimal vectors, enabling a unified approach for different algebra cases and leading to improved subconvexity bounds in number theory.

Contribution

It introduces a novel construction of local test vectors for Waldspurger's integral using minimal vectors, applicable to both matrix and division algebra cases, and links these to subconvexity bounds.

Findings

Constructed explicit local test vectors for Waldspurger's period integral.

Established a relation between local integral size and finite conductor.

Proved a hybrid subconvexity bound comparable to Weyl's bound.

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.