Holomorphic Cusp Forms Research Articles

On the effective determination of Maass cusp forms by L-values

Let f be a Hecke–Maass cusp form of Laplace eigenvalue 1/4+μ 2 with |μ|≤Λ for $\mathit{SL}_{2}(\mathbb{Z})$ . We show that f is uniquely determined by the central values of Rankin–Selberg L-functions L(s,f⊗g), where g runs over the set of holomorphic cusp forms of weight k≍ ϵ Λ 1+3θ+ϵ for any ϵ>0 for $\mathit{SL}_{2}(\mathbb{Z})$ .

Sums of absolute values of cusp form coefficients and their application

Let Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let λf(n), λg(n) and λh(n) be the nth normalized Fourier coefficients of three holomorphic Hecke cusp forms f(z),g(z),h(z)∈Sk(Γ), Sk1(Γ) and Sk2(Γ) respectively. In this paper we consider several averages of absolute values of Fourier coefficients of holomorphic Hecke cusp forms. In particular, we have (under suitable conditions)∑n⩽x|λf(n)λg(n)|≪x(logx)−2θ3,∑n⩽x|λf(n)λg(n)λh(n)|≪x(logx)−θ3, where and throughout this paperθ=1−83π=0.1512…. As an application, we consider sums of coefficients of the Rankin–Selberg L-function and the triple product L-function, and prove that∑n⩽xλf(n)λg(n)≪x35(logx)−2θ3,∑n⩽xλf(n)λg(n)λh(n)≪x79(logx)−θ3, and other similar results.

Distribution of mass of holomorphic cusp forms

We prove an upper bound for the L 4 -norm and for the L 2 -norm restricted to the vertical geodesic of a holomorphic Hecke cusp form f of large weight k . The method is based on Watson’s formula and estimating a mean value of certain L -functions of degree 6. Further applications to restriction problems of Siegel modular forms and subconvexity bounds of degree 8 L -functions are given.

Average behavior of Fourier coefficients of cusp forms over sum of two squares

Let λ f ( n ) denote the n th normalized Fourier coefficient of the classical holomorphic cusp form of even integral weight k ≥ 2 for the full modular group SL ( 2 , Z ) . In this paper, we investigate the average behavior of the power sum ∑ a 2 + b 2 ≤ x λ f ( a 2 + b 2 ) ℓ for x ≥ 1 , 2 ≤ ℓ ≤ 8 and a , b , ℓ ∈ Z .

The circle method and non-lacunarity of meromorphic modular forms

Abstract Serre proved that any holomorphic cusp form of weight one for Γ1(N) is lacunary while a holomorphic modular form for Γ1(N) of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see [Publ. Math. I.H.E.S. 54 (1981), 323–401, Sections 7.6 and 7.7]). In this paper, we show that when a non-zero meromorphic modular form of arbitrary real weight for any finite index subgroup of the modular group SL 2 ( ℤ ) ${\operatorname{SL}}_2(\mathbb {Z})$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.

Transition mean values of shifted convolution sums

Let f be a classical holomorphic cusp form for SL_2(Z) of weight k which is a normalized eigenfunction for the Hecke algebra, and let \lambda(n) be its eigenvalues. In this paper we study "shifted convolution sums" of the eigenvalues \lambda(n) after averaging over many shifts h and obtain asymptotic estimates. The result is somewhat surprising: one encounters a transition region depending on the ratio of the square of the length of the average over h to the length of the shifted convolution sum. The phenomenon is similar to that encountered by Conrey, Farmer and Soundararajan in their 2000 paper Transition Mean Values of Real Characters, and the connection of both results to Eisenstein series and multiple Dirichlet series is discussed.

On Hecke eigenvalues at primes of the form [g(n)

We study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form [g(n)] .

On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms

On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms

Additive twists of Fourier coefficients of modular forms

We study sums of the form ∑ n ⩽ N a ( n ) e 2 π i α n , where α is any real number and the a ( n ) are the Fourier coefficients of either a holomorphic cusp form, a Maass cusp form, or the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform in both α and the form itself. We also improve a bound on a sum of the form ∑ n ⩽ N a ( n ) e 2 π i ( α n + β n θ ) , where the a ( n ) are the Fourier coefficients of a holomorphic cusp form, α and β are any real numbers, and 0 ⩽ θ < 1 . This last bound is uniform in α , but not with respect to the form.

On summatory functions for automorphic L-functions

Let λ f (n) denote the nth normalized Fourier coefficient of a primitive holomorphic cusp form f for the full modular group. Let Δ(x, f ⨂ f) be the error term in the asymptotic formula of Rankin and Selberg for $$ \sum\limits_{n \leqslant x} {{\lambda_f}{{(n)}^2}.} $$ It is proved that Δ(x, f ⨂ f) = Ω (x 3/8) and $$ \sum\limits_{n \leqslant x} {{\lambda_f}\left( {{n^2}} \right) = \Omega \left( {{x^{{{1} \left/ {3} \right.}}}} \right).} $$ Other summatory functions associated with automorphic L-functions also are studied. Bibliography: 22 titles.

Spectral average of central values of automorphic L-functions for holomorphic cusp forms onSO0(m,2), I

Given a maximal even integral lattice L of signature ( m + , 2 − ) ( m ≥ 3 ), we consider an orthonormal Hecke eigen basis B l of the holomorphic cusp forms of weight l on a tube domain with respect to the discriminant subgroup of the orthogonal group O ( L ) . We construct a certain adèlic holomorphic Poincaré series whose spectral expansion is described by the central standard L -value of F ∈ B l and the square of a Fourier–Bessel coefficient of F .

Eichler–Shimura theory for mock modular forms

We use mock modular forms to compute generating functions for the critical values of modular \(L\)-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler–Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler–Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on \(M_{k}^{!}\) in terms of periods.

On the shifted convolution problem in mean

We study a mean value of the shifted convolution problem over the Hecke eigenvalues of a fixed non-holomorphic cusp form. We attain a result also for a weighted case. Furthermore, we point out that the proof yields analogous upper bounds for the shifted convolution problem over the Fourier coefficients of a fixed holomorphic cusp form in mean.

Determination of cusp forms by central values of Rankin–Selberg L-functions*

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as uj runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).

On exponential sums involving Fourier coefficients of cusp forms

Let a ( n ) be the normalized Fourier coefficient of a holomorphic cusp form of weight k or a Maass cusp form with Laplacian eigenvalue 1 4 + r 2 for SL 2 ( Z ) . We consider exponential sum of the form ∑ n ⩽ X a ( n ) e ( α n 2 + β n ) and obtain the upper bound X 7 8 + ε which is uniformly in α , β ∈ R .

On the mean square of short exponential sums related to cusp forms

The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form.

Quantitative version of the joint distribution of eigenvalues of the Hecke operators

Recently, Murty and Sinha proved an effective/quantitative version of Serreʼs equidistribution theorem for eigenvalues of Hecke operators on the space of primitive holomorphic cusp forms. In the context of primitive Maass forms, Sarnak figured out an analogous joint distribution. In this paper, we prove a quantitative version of Sarnakʼs theorem that gives explicitly estimate on the rate of convergence. The same result also holds for the case of holomorphic cusp forms.

On general divisor problems involving Hecke eigenvalues

We study two general divisor problems related to Hecke eigenvalues of classical holomorphic cusp forms, which have been considered by Fomenko, and by Kanemitsu, Sankaranarayanan and Tanigawa respectively. We improve previous results.

Non-vanishing of derivatives of 𝐺𝐿(3)×𝐺𝐿(2) 𝐿-functions

Let f f be a fixed self-dual Hecke-Maass cusp form for S L 3 ( Z ) SL_3(\mathbb {Z}) and let B k \mathcal {B}_k be an orthogonal basis of holomorphic cusp forms of weight k ≡ 2 ( m o d 4 ) k \equiv 2(\mathrm {mod} \,4) for S L 2 ( Z ) SL_2(\mathbb {Z}) . We prove an asymptotic formula for the first moment of the first derivative of L ( s , f × g ) L\left (s,f\times g\right ) at the central point s = 1 / 2 s=1/2 , where g g runs over B k \mathcal {B}_k , K ≤ k ≤ 2 K K\leq k\leq 2K , K K large enough. This implies that for each K K large enough there exists g ∈ B k g\in \mathcal {B}_k with K ≤ k ≤ 2 K K\leq k\leq 2K such that L ′ ( 1 / 2 , f × g ) ≠ 0 L’(1/2,f\times g)\neq 0 .

On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms

Abstract Let be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let and be the n-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms , respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.(i)For any , we have(ii)If , then for any , we haveIf , then for any , we haveIf and , then for any , we havewhere P(x) is a polynomial of degree 3.