Cusp Forms Research Articles

Correction to: Period-like polynomials for L-series associated with half-integral weight cusp forms

AbstractWe correct an error affecting Propositions 5.2 and Theorem 5.4 of [1]

Construction and non-vanishing of a family of vector-valued Siegel Poincaré series

Using Poincaré series of K-finite matrix coefficients of integrable antiholomorphic discrete series representations of Sp2n(R), we construct a spanning set for the space Sρ(Γ) of Siegel cusp forms of weight ρ for Γ, where ρ is an irreducible polynomial representation of GLn(C) of highest weight ω∈Zn with ω1≥…≥ωn>2n, and Γ is a discrete subgroup of Sp2n(R) commensurable with Sp2n(Z). Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.

Convolution identities for divisor sums and modular forms

We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums [Formula: see text]where [Formula: see text] is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to [Formula: see text], do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.

Non-holomorphic modular forms from zeta generators

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Fourier coefficients of Jacobi Poincaré series and applications

We define Jacobi Poincaré series over Cayley numbers and explicitly compute its Fourier coefficients. As an application, we obtain an estimate for the Fourier coefficients of a Jacobi cusp form. We also evaluate certain Petersson scalar products involving Jacobi cusp forms and Poincaré series. This evaluation yields certain special values of shifted convolution of Dirichlet series of Rankin-Selberg type associated to Jacobi cusp forms in consideration.

Special values of shifted Dirichlet series and quasi-modular forms

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form.

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

On the Gross–Prasad conjecture with its refinement for (SO(5), SO(2)) and the generalized Böcherer conjecture

We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\mathrm {SO}(5), \mathrm {SO}(2))$. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

Large Fourier coefficients of half-integer weight modular forms

abstract: This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigenforms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants $D$ such that the Fourier coefficients evaluated at $|D|$ are non-zero. By adapting the resonance method, we also demonstrate that such Fourier coefficients must take quite large values.

Construction of Jacobi forms using adjoint of the Jacobi–Serre derivative

In the article, we study the Oberdieck derivative defined on the space of weak Jacobi forms. We prove that the Oberdieck derivative maps a Jacobi form to a Jacobi form. Moreover, we study the adjoint of the Oberdieck derivative of a Jacobi cusp form with respect to the Petersson scalar product defined on the space of Jacobi forms. As a consequence, we also obtain the adjoint of the Jacobi–Serre derivative (defined in an unpublished work of Oberdieck). As an application, we obtain certain relations among the Fourier coefficients of Jacobi forms.

Quadratic base change and resonance sums for holomorphic cusp forms on Γ0(N)

Let D,k be integers with D square free and k even. Let N be a positive integer so that (N,D)=1 when D has residue one modulo four and (N,4D)=1 when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum SX(α,β;π) attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by D is computed. First a Voronoi summation formula is derived that expresses SX(α,β;π) in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of SX(α,β;π) as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N, which is a special case of the multiplicity one theorem.

On universality in short intervals for zeta-functions of certain cusp forms

Abstract In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions.

On the Fourier Coefficients for General Product 𝐿-Functions

Let 𝑓 be a normalized primitive cusp form of even integral weight for the full modular group Γ = 𝑆𝐿(2, ℤ). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients 𝜆𝑓 ⊗𝑓 ⊗⋯⊗𝑙𝑓 (𝑛) of 𝑙-fold product 𝐿-functions, where 𝑓 ⊗ 𝑓 ⊗ ⋯ ⊗𝑙 𝑓 denotes the 𝑙-fold product of 𝑓. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product 𝐿-functions.

Period-like polynomials for L-series associated with half-integral weight cusp forms

Given the L-series of a half-integral weight cusp form, we construct polynomials behaving similarly to the classical period polynomial of an integral weight cusp form. We also define a lift of half-integral weight cusp forms to integral weight cusp forms that are compatible with the L-series of the respective forms.

On the sign changes of the coefficients attached to triple product L-functions

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights [Formula: see text] and [Formula: see text] for the full modular group SL[Formula: see text], respectively. In this paper, we investigate the sign changes of the sequence [Formula: see text] in short intervals, where [Formula: see text] denote the normalized coefficients of triple product L-functions [Formula: see text] attached to f and [Formula: see text], respectively. We provide quantitative results for the number of sign changes of the sequences mentioned above in short intervals.

Riemann hypothesis for period polynomials for cusp forms on Γ0(N)

We prove that for even integer k, almost all of zeros of the period polynomial associated to a cusp form of weight k on [Formula: see text] are on the circle [Formula: see text] under some conditions.

Bessel periods and anticyclotomic 𝑝-adic spinor 𝐿-functions

We construct the anticyclotomic p p -adic L L -function that interpolates a square root of central values of twisted spinor L L -functions of a quadratic base change of a Siegel cusp form of genus 2 2 with respect to a paramodular group of square-free level.

An asymptotic expansion for a Lambert series associated with Siegel cusp forms

An asymptotic expansion for a Lambert series associated with Siegel cusp forms

Structure of the space of polyharmonic Maass forms with an application to L-values

A polyharmonic Maass form introduced by Lagarias and Rhoades is a generalization of a harmonic Maass form in a sense. In this paper, we construct some bases for the spaces of polyharmonic Maass forms from the structure of the spaces of weakly holomorphic modular forms for prime levels or square-free levels. Furthermore, we relate the polyharmonic Maass forms of depth 1 or depth 3/2 to the L-series of holomorphic cusp forms for square-free levels.

On Möbius Functions from Automorphic Forms and a Generalized Sarnak’s Conjecture

Abstract In this paper, we consider generalized Möbius functions associated with two types of L-functions: Rankin–Selberg L-functions of symmetric powers of distinct holomorphic cusp forms and L-functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak’s conjecture holds for these two types of Möbius functions.