By explicitly calculating and then analytically continuing the Fourier expansion of twisted double Eisenstein series $E_{s,k-s}^{*}(z,w; 1/2)$ Diamantis O'Sullivan, we prove a formula Petersson inner product Cohen's kernel one its twists, obtain rationality result. This extends result Kohnen Zagier.

‎We show that for all normalized Hecke eigenforms $f$‎ ‎with weight one and of CM type‎, ‎the number $(f,f)$ where $(cdot‎, ‎cdot )$ denotes‎ ‎the Petersson inner product‎, ‎is a linear form in logarithms and‎ ‎hence transcendental‎.

Let κ′, κ ≥ 3 be two odd integers. Let f (resp. g) be a normalized holomorphic modular form of weight 2κ (resp. κ′ + 1) and level one on the upper half plane h. Assume that they are Hecke eigenforms. Let L(s, Sym g×f) be the completed degree six L-function and we normalize so that s = 1 2 is the center of symmetry. Let 〈−,−〉 be the Petersson inner product, defined using the usual measure on h s...

We give a new construction of a p-adic L-function L(f,Ξ), for f a holomorphic newform and Ξ an anticyclotomic family of Hecke characters of Q( √ −d). The construction uses Ichino’s triple product formula to express the central values of L(f, ξ, s) in terms of Petersson inner products, and then uses results of Hida to interpolate them. The resulting construction is well-suited for studying what ...

In this paper we discuss the problem of numerically computing Petersson inner products of modular forms, given their q-expansion at∞. A formula of Nelson [Nel15] reduces this to obtaining q-expansions at all cusps, and we describe two algorithms based on linear interpolation for numerically obtaining such expansions. We apply our methods to numerically verify constants arising in an explicit ve...

We investigate differential operators and their compatibility with subgroups of SL2(R) n. In particular, we construct Rankin–Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin– Cohen bracket of a Hilbert–Eisenstein series and an arbitrary Hilbert modular for...

Every Jacobi cusp form of weight k and index m over SL2(Z) Z 2 is in correspondence with 2m Dirichlet series constructed with its Fourier coefficients. The standard way to get from one to the other is by a variation of the Mellin transform. In this paper, we introduce a set of integral kernels which yield the 2m Dirichlet series via the Petersson inner product. We show that those kernels are Ja...

Let Eλ be a Hilbert space, whose elements are functions spanned by the eigenfunctions of the Laplace-Beltrami operator associated with an eigenvalue λ > 0. The norm of elements in this space is given by the Petersson inner product. In this paper, the trace of Hecke operators Tn acting on the space Eλ is computed for congruence subgroups Γ0(N) of square free level, which may be considered as the...