scholarly journals Central values of additive twists of cuspidal L-functions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Modular Symbols ◽  
Central Values ◽  
Arithmetic Distribution ◽  
Holomorphic Cusp Forms ◽  
Normally Distributed

Abstract Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L-functions. In this paper we prove that central values of additive twists of the L-function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k ≥ 4 {k\geq 4} ) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L-functions consisting of central values of the form L ⁢ ( f ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

scholarly journals On Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

scholarly journals On the Distribution of Periods of Holomorphic Cusp Forms and Zeroes of Period Polynomials

2020 ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Behavior ◽  
Method Of Moments ◽  
Cusp Form ◽  
Joint Distribution ◽  
Random Variables ◽  
Kloosterman Sums ◽  
Limiting Distribution ◽  
Independent Random Variables ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

Abstract In this paper, we determine the limiting distribution of the image of the Eichler–Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore, we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

scholarly journals Quantum Variance for Eisenstein Series

2019 ◽  
Author(s):  
Bingrong Huang
Keyword(s):  
Quadratic Form ◽  
Asymptotic Formula ◽  
Eisenstein Series ◽  
Cusp Forms ◽  
Central Values ◽  
Quantum Variance

Abstract In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on $\operatorname{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions.

scholarly journals Cusp forms of weight 3/2

1980 ◽  
Vol 79 ◽  
pp. 111-122 ◽  
Author(s):  
Hisashi Kojima
Keyword(s):  
Vector Space ◽  
Cusp Form ◽  
Theta Series ◽  
Dirichlet Character ◽  
Cusp Forms

In this paper we deal with the problem (C) in § 4 of [4]. Let Ik be the Shimura mapping in [4] of Sk(4N, χ) into k-1(N′ χ2) (see p. 458). The problem (C) can be stated as follows: I3(f) is a cusp form if and only if ‹f, h› = 0 for all h ∈ U, where U is the vector space spanned by every theta series of S3(4N, χ) associated with some Dirichlet character.

scholarly journals Existence of Hilbert Cusp Forms with Non-vanishingL-values

2016 ◽  
Vol 68 (4) ◽  
pp. 908-960 ◽  
Author(s):  
Shingo Sugiyama ◽  
Masao Tsuzuki
Keyword(s):  
Asymptotic Formula ◽  
Trace Formula ◽  
Cusp Forms ◽  
Central Values ◽  
Relative Trace Formula ◽  
The Absolute ◽  
Derivatives Of

AbstractWe develop a derivative version of the relative trace formula on PGL(2) studied in our previous work, and derive an asymptotic formula of an average of central values (derivatives) of automorphicL-functions for Hilbert cusp forms. As an application, we prove the existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are arbitrarily large.

Central values of GL(2) ×GL(3) Rankin–Selberg L-functions

2019 ◽  
Vol 16 (05) ◽  
pp. 941-962
Author(s):  
Qinghua Pi
Keyword(s):  
Trace Formula ◽  
Cusp Form ◽  
Cusp Forms ◽  
Central Values ◽  
Kuznetsov Trace Formula ◽  
First Moment

Let [Formula: see text] be a normalized holomorphic cusp form for [Formula: see text] of weight [Formula: see text] with [Formula: see text]. By the Kuznetsov trace formula for [Formula: see text], we obtain the twisted first moment of the central values of [Formula: see text], where [Formula: see text] varies over Hecke–Maass cusp forms for [Formula: see text]. As an application, we show that such [Formula: see text] is determined by [Formula: see text] as [Formula: see text] varies.

scholarly journals Hecke structure of spaces of half-integral weight cusp forms

2000 ◽  
Vol 159 ◽  
pp. 53-85 ◽  
Author(s):  
Sharon M. Frechette
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Shimura Lift ◽  
Integral Weight ◽  
Structure Theorems ◽  
Space Of Cusp Forms ◽  
Shimura Correspondence

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp forms Sk/2(4N,χ) where k and N are odd nonnegative integers with k ≥ 3, and χ is an even quadratic Dirichlet character modulo 4N. We give complete results in the case where N is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on N and χ, by giving conditions under which a newform has an equivalent cusp form in Sk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such N and χ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

Non-vanishing of the first derivative of GL(3) ×GL(2) L-functions

2018 ◽  
Vol 14 (03) ◽  
pp. 847-869
Author(s):  
Guohua Chen ◽  
Xiaofei Yan
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Orthogonal Basis ◽  
Cusp Forms ◽  
First Derivative ◽  
Maass Cusp Form ◽  
Center Point

Let [Formula: see text] be a fixed self-dual Hecke–Maass cusp form for [Formula: see text] and [Formula: see text] be an orthogonal basis of odd Hecke–Maass cusp forms for [Formula: see text]. We prove an asymptotic formula for the average of the first derivative of the Rankin–Selberg [Formula: see text]-function of [Formula: see text] and [Formula: see text] at the center point [Formula: see text]. This implies the non-vanishing results for the first derivative of these [Formula: see text]-functions at the center point [Formula: see text].

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