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

Yueke Hu Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Paul D.Nelson Department of Mathematics, ETH Zurich, Zurich, Switzerland

Number Theory mathscidoc:2206.24013

In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on 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(π×π_{χ^{−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.
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@inproceedings{yuekenew,
  title={New test vector for Waldspurger's period integral, relative trace formula, and hybrid subconvexity bounds},
  author={Yueke Hu, and Paul D.Nelson},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220629151201386657489},
}
Yueke Hu, and Paul D.Nelson. New test vector for Waldspurger's period integral, relative trace formula, and hybrid subconvexity bounds. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220629151201386657489.
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