Cusp Forms Research Articles

Equivalent criterion for the grand Riemann hypothesis associated with Maass cusp forms

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated with the primitive Maass cusp form over $\Gamma _0(N)$.

Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus

We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of quantum ergodicity type for eigenfunctions of the Laplacian on hyperbolic surfaces of finite area that Benjamini-Schramm converge to the hyperbolic plane. We show that this is generic for Mirzakhani’s model of random surfaces chosen uniformly with respect to the Weil-Petersson volume. Depending on the particular sequence of surfaces considered this gives a result of delocalisation of most cusp forms or of Eisenstein series.

The cubic moment of Hecke–Maass cusp forms and moments of L-functions

The cubic moment of Hecke–Maass cusp forms and moments of L-functions

On the Rankin-Selberg problem in families

In this paper, we investigate the Rankin-Selberg problem over short intervals in families of holomorphic modular forms and Hecke-Maass cusp forms. We assume a Lindelöf-on-average bound for holomorphic modular forms. On the other hand, we make no assumptions for Hecke-Maass cusp forms.

On the interlacing of the zeros of certain Poincaré cusp forms

Rankin proved that a specific family of Poincaré series (Gk(z,1))k≥24 which are cusp forms have all their finite zeros on the unit circle of the standard fundamental domain F for the action of SL(2,Z) on the complex upper half-plane H. In this article, we show that these finite zeros of Gk(z,1) and those of Gk+12(z,1) interlace on {eιθ:θ∈(π/2,2π/3)}.

On the average behavior of coefficients related to triple product $L$-functions

In this paper, we study the average behaviour of the coefficients of triple product $L$-functions and some related $L$-functions corresponding to normalized primitive holomorphic cusp form $f(z)$ of weight $k$ for the full modular group $SL(2,\mathbbZ).$ Here we call $f(z)$ a primitive cusp form if it is an eigenfunction of all Hecke operators simultaneously.

On sign changes of primitive Fourier coefficients of Siegel cusp forms

On sign changes of primitive Fourier coefficients of Siegel cusp forms

Integral mean-square estimation of error terms of Fourier coefficients attached to cusp forms

Integral mean-square estimation of error terms of Fourier coefficients attached to cusp forms

Bergman space zero sets, modular forms, von Neumann algebras and ordered groups

A^2_{\alpha} will denote the weighted L^2 Bergman space. Given a subset S of the open unit disc we define \Omega(S) to be the infimum of \{s \mid \exists f \in A^2_{s-2}, \; f\neq 0, having S as its zero set \} . By classical results on Hardy space there are sets S for which \Omega(S)=1 . Using von Neumann dimension techniques and cusp forms we give examples of S where 1<\Omega(S)<\infty . By using a left order on certain Fuchsian groups we are able to calculate \Omega(S) exactly if \Omega (S) is the orbit of a Fuchsian group. This technique also allows us to derive in a new way well known results on zeros of cusp forms and indeed calculate the whole algebra of modular forms for \operatorname{PLS}_2 (\mathbb{Z}) .

Dimension formulas for Siegel modular forms of level 4

AbstractWe prove several dimension formulas for spaces of scalar‐valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of whose local component at admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at , we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.

Zero-free regions for spectral averages of Hecke L-functions

We obtain an explicit zero-free region for a weighted sum of [Formula: see text]-functions over the orthogonal basis of Hecke eigen cusp forms of a large integral weight for the full modular group. We also estimate the number of such forms whose [Formula: see text] value does not vanish at a given point inside this zero-free region.

The average of the exponential sums related to cusp forms

In this paper, we investigate the average [Formula: see text] where [Formula: see text] for any [Formula: see text] and [Formula: see text], with [Formula: see text] running over primitive holomorphic cusp forms of weight [Formula: see text] and prime level P. As a result, we prove pointwise uniform bounds with respect to [Formula: see text], [Formula: see text] for the frequency of the P-values, and present that there exist strong oscillations in the exponential sum [Formula: see text], so long as the level of f varies. Concerning the applications of the main result, we consider the quadratic Waring problem associated to the coefficients of cusp forms exhibiting considerable cancellations.

Prime divisors of ℓ-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level ℓ

Let ℓ be any fixed prime number. We define the ℓ-Genocchi numbers by Gn:=ℓ(1−ℓn)Bn, with Bn the n-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime p is ℓ-Genocchi irregular if it divides at least one of the ℓ-Genocchi numbers G2,G4,…,Gp−3, and ℓ-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of ℓ-Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ=2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1−ℓn)B2n/2n and (1+ℓn)B2n/2n. This allows us to estimate the number of primes p≤x for which there exist modulo p Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level ℓ.

Additive twists of L2-norm of Fourier–Jacobi coefficients of Siegel cusp forms

Let F be a Siegel cusp form of weight k and degree n≥2 and having Fourier–Jacobi coefficients ϕm. We study estimates of L2-norms of these Fourier–Jacobi coefficients and their additive twists in two ways. One of our results generalizes a result of Kohnen and Sengupta.

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

AbstractLet f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{\lambda_f(n)\}_{n \in \mathcal{N}_K}$ . More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $\lambda_f(n)\lambda_f(n') < 0$ , where n′ is the first element of $\mathcal{N}_K$ greater than n for which $\lambda_f(n') \neq 0$ .For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 \;:\; m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author.In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,\varepsilon} X^{1/2-\varepsilon}$ sign changes for $\lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$ .

On squares of Fourier coefficients twist exponential functions with applications

Let f be a Hecke–Maass cusp form of weight zero for [Formula: see text] and [Formula: see text] be the nth Fourier coefficient. For almost all [Formula: see text] we have [Formula: see text] which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than [Formula: see text] For all [Formula: see text] of type [Formula: see text] we also show that [Formula: see text] where [Formula: see text] This result relies heavily on a generalized double sum (see Theorem 1.3).

On sign changes of Fourier coefficients of Hermitian cusp forms of degree two

We prove a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form of degree 2. In addition, we prove a quantitative result for the number of sign changes of the primitive Fourier coefficients. We give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2 over certain imaginary quadratic extensions.

Hecke nilpotency for modular forms mod 2 and an application to partition numbers

A well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on $${{\,\mathrm{\textrm{SL}}\,}}_2({{\,\mathrm{{\mathbb {Z}}}\,}})$$ . We give an algorithm for calculating the degree of Hecke nilpotency for cusp forms, and we obtain a formula for the total number of cusp forms mod 2 of any given degree of nilpotency. Using these results, we find that the degrees of Hecke nilpotency in spaces $$M_k$$ have no limiting distribution as $$k \rightarrow \infty $$ . As an application, we study the parity of the partition function using Hecke nilpotency.

The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

Suppose X is a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Let G be a finite group acting faithfully on X over k such that G has non-trivial, cyclic Sylow p-subgroups. If E is a G-invariant Weil divisor on X with deg(E)>2g(X)−2, we prove that the decomposition of H0(X,OX(E)) into a direct sum of indecomposable kG-modules is uniquely determined by the class of E modulo G-invariant principal divisors, together with the ramification data of the cover X→X/G. The latter is given by the lower ramification groups and the fundamental characters of the closed points of X that are ramified in the cover. As a consequence, we obtain that if m>1 and g(X)≥2, then the kG-module structure of H0(X,ΩX⊗m) is uniquely determined by the class of a canonical divisor on X/G modulo principal divisors, together with the ramification data of X→X/G. This extends to arbitrary m>1 the m=1 case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss applications to the tangent space of the global deformation functor associated to (X,G) and to congruences between prime level cusp forms in characteristic 0. In particular, we complete the description of the precise kPSL(2,Fℓ)-module structure of all prime level ℓ cusp forms of even weight in characteristic p=3.

On the first simultaneous sign change and non-vanishing of Fourier coefficients of cusp forms

On the first simultaneous sign change and non-vanishing of Fourier coefficients of cusp forms