Periods and Reciprocity II

Abstract

Let $\mathbf{F}$ be a number field and $\mathfrak{q},\mathfrak{l}$ two coprime integral ideals with $\mathfrak{q}$ squarefree and ${\pi }_1,{\pi }_2$ two fixed unitary automorphic representations of ${\mathrm{PGL}}_2({\mathbb{A}}_{\mathbf{F}})$ unramified at all finite places. In this paper, we use regularized integrals to obtain a formula that links the first moment of $L(\pi \otimes {\pi }_1\otimes {\pi }_2,\frac{1}{2})$ twisted by the Hecke eigenvalues ${\lambda }_{\pi }(\mathfrak{l})$, where $\pi$ runs through unitary automorphic representations of ${\mathrm{PGL}}_2({\mathbb{A}}_{\mathbf{F}})$ with conductor dividing $\mathfrak{q}$, with some spectral expansion of periods over representations of conductor dividing $\mathfrak{l}$. In the special case where ${\pi }_1={\pi }_2=\sigma$, this formula becomes a reciprocity relation between moments of $L$-functions. As applications, we obtain a subconvex estimate in the level aspect for the central value of the triple product $L(\pi \otimes {\pi }_1\otimes {\pi }_2,\frac{1}{2})$ and a simultaneous non-vanishing result for $L({\mathrm{Sym}}^2(\sigma )\otimes \pi ,\frac{1}{2})$ and $L(\pi ,\frac{1}{2})$.

30 pages

References in corpus (2)

• On the reciprocity law for the twisted second moment of Dirichlet $L$-functions
• The mean-square of Dirichlet L-functions

Cited by in corpus (1)

• Spectral Reciprocity for the product of Rankin-Selberg $L$-functions