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 6.8 which generalizes a result of Michel-Venkatesh. As an application we prove the optimal upper bound of a sum of restricted $L^2$-norms of the $L^2$-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods of Blomer.

arXiv commentFinal version. Accepted by Forum Math

References in corpus (3)

• The subconvexity problem for $\GL_{2}$
• On Deligne's conjecture for certain automorphic $L$-functions for $\mathrm{GL}(3)\times \mathrm{GL}(2)$
• Gamma factors for Asai representations of ${\mathrm{GL}}_2$