Kernels of $L$-functions of cusp forms

TL;DR

This paper introduces a new expression for the inner product of kernel functions related to cusp forms, extending known formulas and providing alternative proofs for classical theorems in the theory of automorphic forms.

Contribution

It presents a novel formulation of kernel inner products, generalizes Cohen's series representation, and extends formulas of Kohnen and Zagier, as well as Manin's Periods Theorem.

Findings

New expression for kernel inner product derived

Extension of Kohnen and Zagier's formula achieved

Alternative proof of Manin's Periods Theorem provided

Abstract

We give a new expression for the inner product of two kernel functions associated to a cusp form. Among other applications, it yields an extension of a formula of Kohnen and Zagier, and another proof of Manin's Periods Theorem. Cohen's representation of these kernels as series is also generalized.