Kernels of L-functions of cusp forms

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.

Simultaneous nonvanishing of products of L-functions associated to elliptic cusp forms

“…Double Eisenstein Series and its Fourier expansion. We recall the basics on double Eisenstein series (see [4,3]), and following the lines in [7], we compute its Fourier expansion.…”

On integral kernels for Dirichlet series associated to Jacobi forms

“…Remark 5.2. The analytic continuation of the series Λ μ (f, s) established in Proposition 1.3 together with Lemma 5.1 and Theorem 1.1, allow us to replace the condition 3 2 < (s) < k 2 − 2 in our theorem by the less restrictive set of inequalities 3 2 < (s) < k − 3.…”

“…and (s) < k − 3 are used in the last step. From this relation, one deduces that the integral (5.2) is, as a function of s, absolute and uniformly convergent on any compact subset of the vertical strip3 2 < (s) < k − 3. At this point, we recall a standard result in complex analysis (see[17, p. 392], for example) in order to conclude from the previous statement and the holomorphicity of ζ 4mZ (τ + n 0 , s − 1 2 ) on (s) >3 2 that (5.2) defines a holomorphic function of s.Proof of Proposition 1.3.…”

“…They were first considered by Cohen [2], and have been studied and used by several authors since then. (For example, Diamantis and O'Sullivan [3], Fukuhara [5], Fukuhara and Yang [6], Kohnen and Zagier [16].) For a recent, comprehensive look at the properties of (1.2), see [3].…”

“…Alternatively, one could use the functional equations (1.5) in order to get an integral representation for every Λ μ (f, s − 1 2 ). The restriction 3 2 < (s) < k 2 − 2 in the theorem can be replaced by the extended vertical strip 3 2 < (s) < k − 3 via the analytic continuation of both sides of the equation (see final remark of the paper).…”