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

Heights and L-Series

1990 ◽  
Vol 42 (3) ◽  
pp. 533-560 ◽  
Author(s):  
Rhonda Lee Hatcher
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Ideal Class ◽  
Quadratic Character ◽  
Trivial Character ◽  
Image Position

Let be a cusp form of weight 2k and trivial character for Γ0(N), where N is prime, which is orthogonal with respect to the Petersson product to all forms g(dz), where g is of level L < N, dL\N. Let K be an imaginary quadratic field of discriminant — D where the prime N is inert. Denote by ∊ the quadratic character of determined by ∊(p) = (—D/p) for primes p not dividing D. For A an ideal class in K, let rA(m) be the number of integral ideals of norm m in A. We will be interested in the Dirichlet series L(f,A,s) defined by

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.

On the Sizes of Gaps in the Fourier Expansion of Modular Forms

2005 ◽  
Vol 57 (3) ◽  
pp. 449-470 ◽  
Author(s):  
Emre Alkan
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
The Riemann Zeta Function ◽  
Classical Paper ◽  
Riemann Zeta ◽  
Image Position ◽  
Almost All ◽  
Integer Weight

AbstractLet be a cusp form with integer weight k ≥ 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1, letConcerning bounded values of i f (n) we prove that for ∊ > 0 there exists M = M(∊, f ) such that Using results of Wu, we show that if f is a weight 2 cusp form for an elliptic curve without complex multiplication, then . Using a result of David and Pappalardi, we improve the exponent to for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate i f (n) on the average using well known bounds on the Riemann Zeta function.

scholarly journals Eta-products which are simultaneous eigenforms of Hecke operators

1993 ◽  
Vol 35 (3) ◽  
pp. 307-323 ◽  
Author(s):  
Anthony J. F. Biagioli
Keyword(s):  
Fourier Series ◽  
Half Plane ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Dedekind Eta Function ◽  
Jacobi Symbol ◽  
Hecke Operator ◽  
Upper Half Plane ◽  
Image Position ◽  
Eta Products

The Dedekind eta-function is defined for any τ in the upper half-plane bywhere x = exp(2πiτ) and x1/24 = exp(2πiτ/24). By an eta-product we shall mean a functionwhere N ≥ 1 and eachrδ ∈ ℤ. In addition, we shall always assume that is an integer. Using the Legendre-Jacobi symbol (—), we define a Dirichlet character ∈ bywhen a is odd. If p is a prime for which ∈(p) ≠ 0and if F is a function with a Fourier seriesthen we define a Hecke operator Tp bywhereand

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 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.

scholarly journals on l-adic representations attached to modular forms II

1985 ◽  
Vol 27 ◽  
pp. 185-194 ◽  
Author(s):  
Kenneth A. Ribet
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Dirichlet Character ◽  
Image Position

Suppose that is a newform of weight k on Г1(N). Thus f is in particular a cusp form on Г1(N), satisfyingfor all n≥1. Associated with f is a Dirichlet charactersuch thatfor all, .

scholarly journals Congruences between cusp forms and linear representations of the Galois group

1976 ◽  
Vol 64 ◽  
pp. 63-85 ◽  
Author(s):  
Masao Koike
Keyword(s):  
Galois Group ◽  
Cusp Form ◽  
Linear Representation ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Linear Representations

Let f(z) be a cusp form of type (l,ε) on Γ0(N) which is a common eigenfunction of all Hecke operators. For such f(z), Deligne and Serre [1] proved that there exists a linear representationsuch that the Artin L-function for p is equal to the L-function associated to f(z).

scholarly journals CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

2010 ◽  
Vol 06 (05) ◽  
pp. 1117-1137 ◽  
Author(s):  
T. SHEMANSKE ◽  
S. TRENEER ◽  
L. WALLING
Keyword(s):  
Hecke Operators ◽  
Cusp Forms ◽  
Hecke Eigenforms ◽  
Linearly Independent ◽  
Integral Weight ◽  
Trivial Character ◽  
Free Level ◽  
Fixed Space ◽  
Space Of Cusp Forms

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

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