Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Abstract

We establish explicit Ichino's formulae for the central values of the triple product L-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 4.5 for the real place which is the main novelty of this article. As an application we prove the optimal upper bound of a sum of restricted 2-norms of the L 2 -normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods in [Blo13]. Contents 1. Introduction and main results 1 2. Proof of Theorem 1.2 3 3. Special value formulae 12 4. Local calculations 16 5. Proof of Proposition 4.5 28 References 40Here the L-functions are complete L-functions.

Quadratic Hecke sums and mass equidistribution

“…These local integrals have all already been computed in the literature, so our task is just to assemble the relevant computations. We rely primarily upon the works of Chen-Cheng [6] and Cheng [7].…”

“…It is shown in [7,Prop 4.12 (3)] that I ∞ is a nonzero constant. (Alternatively, we may reduce to Watson's calculations and the comparison of local integrals proved in [19] by noting that I ∞ is the same local integral that appears in the setting of Proposition 2.1.…”