Cusp Forms Of Weight 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.

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.

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.

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.

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 .

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 non-vanishing of Fourier coefficients of half-integral weight cuspforms

On the non-vanishing of Fourier coefficients of half-integral weight cuspforms

Simultaneous non-vanishing of central values of GL(2)×GL(3) and GL(3) L-functions

We study simultaneous non-vanishing of L(12,Ϝ) and L(12,g⊗Ϝ), when Ϝ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for SL(3,Z) and g is a fixed SL(2,Z) holomorphic Hecke cusp form of weight k≡0(mod4).

Fourier coefficients of automorphic L-functions

We prove two unconditional results on Fourier coefficients of modular forms: Let [Formula: see text] be a Hecke eigen cusp form of weight [Formula: see text] and level [Formula: see text] ([Formula: see text] square free), and [Formula: see text] be the normalized Hecke eigenvalues. Let [Formula: see text] be the least prime [Formula: see text] such that [Formula: see text]. Then we show that in a family of modular forms of weight [Formula: see text] and level [Formula: see text], except for a density zero set, [Formula: see text] for some [Formula: see text]. Second, we show that outside a density zero set, a modular form is determined by Fourier coefficients up to [Formula: see text] for some [Formula: see text].

On a family of Siegel Poincaré series

Let [Formula: see text] be a congruence subgroup of [Formula: see text]. Using Poincaré series of [Formula: see text]-finite matrix coefficients of integrable discrete series representations of [Formula: see text], we construct a spanning set for the space [Formula: see text] of Siegel cusp forms of weight [Formula: see text]. We prove the non-vanishing of certain elements of this spanning set using Muić’s integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of [Formula: see text], we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of [Formula: see text]. Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.

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.

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.

Estimates for the Fourier coefficients of the Duke–Imamoglu–Ikeda lift

Abstract Let k and n be positive even integers. For a Hecke eigenform h in the Kohnen plus subspace of weight k - n 2 + 1 2 {k-\frac{n}{2}+\frac{1}{2}} for Γ 0 ⁢ ( 4 ) {\Gamma_{0}(4)} , let I n ⁢ ( h ) {I_{n}(h)} be the Duke–Imamoglu–Ikeda lift of h to the space of cusp forms of weight k for Sp n ⁡ ( ℤ ) {\operatorname{Sp}_{n}({\mathbb{Z}})} . We then give an estimate of the Fourier coefficients of I n ⁢ ( h ) {I_{n}(h)} . It is better than the usual Hecke bound for the Fourier coefficients of a Siegel cusp form.

Koecher–Maass series have infinitely many critical zeros

AbstractIn this article, we prove the infinitude of the non‐trivial zeroes of Koecher–Maass series associated to a Siegel cusp form of weight and degree 2 on the critical line.

First sign changes of modular forms for general level N

Let f(z)=∑n>0a(n)qn be a cusp form of weight k on Γ0(N) with real Fourier coefficients a(n). When N is squarefree, Choie and Kohnen gave a bound that the first sign change of a(n) occurs, and recently it was improved by He and Zhao. In this paper, we compute a bound for the first sign change problem for arbitrary N.

On the Miller basis for the space of cusp forms

In this paper, we study the existence of the Miller basis for [Formula: see text], the space of cusp forms of weight [Formula: see text] and level [Formula: see text]. We seek an easy criterion to determine whether or not the space [Formula: see text] admits the Miller basis. In particular, we prove that if the level [Formula: see text] is sufficiently composite, then [Formula: see text] does not have the Miller basis. We also prove that there exists a constant [Formula: see text] such that [Formula: see text] admits the Miller basis if and only if [Formula: see text] admits the Miller basis when [Formula: see text].

Ramanujan-style congruences for prime level

We establish Ramanujan-style congruences modulo certain primes \(\ell \) between an Eisenstein series of weight k, prime level p and a cuspidal newform in the \(\varepsilon \)-eigenspace of the Atkin–Lehner operator inside the space of cusp forms of weight k for \(\Gamma _0(p)\). Under a mild assumption, this refines a result of Gaba–Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler–Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by \(\ell .\) The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.