Cusp Forms Research Articles

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

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

Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

On the density hypothesis for $L$-functions associated with holomorphic cusp forms

On the density hypothesis for $L$-functions associated with holomorphic cusp forms

𝐿-values of harmonic Maass forms

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the “central L L -value” of the modular j j -invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this “ L L -value” as the value of an actual L L -series, and extend it to all integral arguments and to a large class of harmonic Maass forms which includes all weakly holomorphic cusp forms. The context and relation to previously defined L L -series for weakly holomorphic and harmonic Maass forms are discussed. These formulas suggest possible arithmetic or geometric meaning of L L -values in these situations. The key ingredient of the proof is to apply a recent theory of Lee, Raji, and the authors to describe harmonic Maass L L -functions using test functions.

Quantum statistical mechanics and the boundary of modular curves

The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL2 and an analog for SL2. The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

ESTIMATE FOR HIGHER MOMENTS OF CUSP FORM COEFFICIENTS OVER SUM OF TWO SQUARES COEFFICIENTS OVER SUM OF TWO SQUARES

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k1 and k2 for the full modular group Γ = SL(2, Z), respectively. Denote by λf (n) and λg(n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we consider the summatory function X n=a2+b2≤x λf (n)iλg(n)j , for x ≥ 2, where a, b ∈ Z and i, j ≥ 1 are positive integers.

Shifted convolution sums for GL(3)×GL(2)$GL(3)\times GL(2)$ averaged over weighted sets

AbstractLet be the Fourier coefficients of an Hecke–Maass cusp form and be those of an Hecke holomorphic or Hecke–Maass cusp form . Let and be a sequence. We show that if for some , for any , and a similar bound holds when . This improves Sun's bound and generalizes it to an average with arbitrary weights . Moreover, we demonstrate how one can recover the factorizable moduli structure given by Jutila's variant of the circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using Jutila's circle method.

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for [Formula: see text]. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for [Formula: see text]. Furthermore, we present a conditional result regarding sign changes of these coefficients.

On products of Fourier coefficients of half-integral weight cusp forms

On products of Fourier coefficients of half-integral weight cusp forms

Shifted convolution sums of divisor functions with Fourier coefficients

Let f(z) be a holomorphic cusp form of weight κ for the full modular group SL2(Z). Denote its n-th normalized Fourier coefficient by λf(n). Let τk(n) denote that k-th divisor function with k≥4. In this paper, we consider the shifted convolution sum∑n≤Xτk(n)λf(n+h). We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter h.

Zeros of modular forms and Faber polynomials

AbstractWe study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near vertical lines, with the zeros of a weight form lying at height approximately . This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree .

Regular Friedmann Universes and Matter Transformations

Regular Friedmann Universes and Matter Transformations

Reflective obstructions of unitary modular varieties

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than 138. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula.

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.

Bounds for the Bergman Kernel and the Sup-Norm of Holomorphic Siegel Cusp Forms

Abstract We prove “polynomial in $k$” bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds, while the lower bounds match for all $n \ge 1$. For an $L^{2}$-normalized Siegel cusp form $F$ of degree $2$, our bound for its sup-norm is $O_{\epsilon } (k^{9/4+\epsilon })$. Further, we show that in any compact set $\Omega $ (which does not depend on $k$) contained in the Siegel fundamental domain of $\textrm {Sp}(2,\mathbb {Z})$ on the Siegel upper half space, the sup-norm of $F$ is $O_{\Omega }(k^{3/2 - \eta })$ for some $\eta>0$, going beyond the “generic” bound in this setting.

Deterministic comparison of cusp form coefficients over certain sequences

Deterministic comparison of cusp form coefficients over certain sequences

Evaluation of and period polynomial relations

Abstract In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration [9]. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for $\operatorname {\mathrm {SL}}_2({\mathbb {Z}})$ . In contrast, a simple combinatorial filtration, the block filtration [13, 28] is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of $\zeta (2,\ldots ,2,4,2,\ldots ,2)$ as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple t values [22] $t(2\ell ,2k)$ in terms of classical double zeta values.

Extending the support of 1- and 2-level densities for cusp form $L$-functions under square-root cancellation hypotheses

Extending the support of 1- and 2-level densities for cusp form $L$-functions under square-root cancellation hypotheses

Symmetries and spectral statistics in chaotic conformal field theories. Part II. Maass cusp forms and arithmetic chaos

We continue the study of random matrix universality in two-dimensional conformal field theories. This is facilitated by expanding the spectral form factor in a basis of modular invariant eigenfunctions of the Laplacian on the fundamental domain. The focus of this paper is on the discrete part of the spectrum, which consists of the Maass cusp forms. Both their eigenvalues and Fourier coefficients are sporadic discrete numbers with interesting statistical properties and relations to analytic number theory; this is referred to as ‘arithmetic chaos’. We show that the near-extremal spectral form factor at late times is only sensitive to a statistical average over these erratic features. Nevertheless, complete information about their statistical distributions is encoded in the spectral form factor if all its spin sectors exhibit universal random matrix eigenvalue repulsion (a ‘linear ramp’). We ‘bootstrap’ the spectral correlations between the cusp form basis functions that correspond to a universal linear ramp and show that they are unique up to theory-dependent subleading corrections. The statistical treatment of cusp forms provides a natural avenue to fix the subleading corrections in a minimal way, which we observe leads to the same correlations as those described by the [torus]×[interval] wormhole amplitude in AdS3 gravity.