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 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 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 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.

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

scholarly journals Value-distribution of twisted L-functions of normalized cusp forms

2010 ◽  
Vol 51 ◽  
Author(s):  
Alesia Kolupayeva
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Complex Plane ◽  
Dirichlet Character ◽  
Value Distribution ◽  
Cusp Forms ◽  
Probability Measures ◽  
Convergence Of Probability Measures

A limit theorem in the sense of weak convergence of probability measures on the complex plane for twisted with Dirichlet character L-functions of holomorphic normalized Hecke eigen cusp forms with an increasing modulus of the character is proved.

scholarly journals Fourier coefficients of Siegel cusp forms of degree two

1984 ◽  
Vol 93 ◽  
pp. 149-171 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Siegel Cusp Form

Our purpose is to prove the followingTheorem. Let k be an even integer ≥ 6. Letbe a Siegel cusp form of degree two, weight k. Then we have

scholarly journals Cusp forms of weight one, quartic reciprocity and elliptic curves

1985 ◽  
Vol 98 ◽  
pp. 117-137 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Positive Integer ◽  
Elliptic Curves ◽  
Galois Group ◽  
Cusp Form ◽  
Rational Number ◽  
Galois Extension ◽  
Dihedral Group ◽  
Cusp Forms ◽  
Two Dimensional ◽  
Weight One

Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

Cancellation of Cusp Forms Coefficients over Beatty Sequences on GL(m)

2011 ◽  
Vol 54 (4) ◽  
pp. 757-762
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Maass Cusp Form ◽  
Beatty Sequences

AbstractLet A(n1, n2, … , nm–1) be the normalized Fourier coefficients of a Maass cusp form on GL(m). In this paper, we study the cancellation of A(n1, n2, … , nm–1) over Beatty sequences.

scholarly journals On products of Fourier coefficients of cusp forms

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Eric Hofmann ◽  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms

AbstractThe purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form,

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.

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