Universal Fourier expansions of modular forms

1994 ◽  
pp. 59-94 ◽  
Author(s):  
Loïc Merel
Keyword(s):  
Modular Forms ◽  
Fourier Expansions

scholarly journals On the Fourier expansions of Jacobi forms of half-integral weight

2006 ◽  
Vol 2006 ◽  
pp. 1-11
Author(s):  
B. Ramakrishnan ◽  
Brundaban Sahu
Keyword(s):  
Modular Forms ◽  
Jacobi Form ◽  
Jacobi Forms ◽  
Number Of Components ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Fourier Expansions ◽  
The Given ◽  
The Relationship ◽  
Vector Valued

Using the relationship between Jacobi forms of half-integral weight and vector valued modular forms, we obtain the number of components which determine the given Jacobi form of indexp,p2orpq, wherepandqare odd primes.

scholarly journals EFFICIENTLY GENERATED SPACES OF CLASSICAL SIEGEL MODULAR FORMS AND THE BÖCHERER CONJECTURE

2010 ◽  
Vol 89 (3) ◽  
pp. 393-405 ◽  
Author(s):  
MARTIN RAUM
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Siegel Modular Forms ◽  
Generating Set ◽  
Arithmetic Properties ◽  
Fourier Expansions

AbstractWe state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set, we verify the Böcherer conjecture for nonrational eigenforms and discriminants with class number greater than one. As a further application we verify another conjecture for weights up to 150 and investigate an analog of the Victor–Miller basis. Additionally, we describe some arithmetic properties of the basis we found.

scholarly journals On the Fourier expansions of Jacobi forms

2004 ◽  
Vol 2004 (48) ◽  
pp. 2583-2594 ◽  
Author(s):  
Howard Skogman
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Jacobi Form ◽  
The Other ◽  
Jacobi Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Fourier Expansions ◽  
The Relationship ◽  
Vector Valued

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexesp,p2, andpqfor distinct odd primesp,q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the formpqorp2, there are restrictions on which of the components will uniquely determine the form. Moreover, for indexes of the formp, this note gives an explicit reconstruction of the entire Jacobi form from a single associated vector-valued modular form component. That is, we show how to start with a single associated vector component and use specific matrices fromSl2(ℤ)to find the other components and hence the entire Jacobi form. These results are used to discuss the possible modular forms of half-integral weight associated to the Jacobi form for different subgroups.

scholarly journals Universal Fourier expansions of Bianchi modular forms

2019 ◽  
Vol 16 (04) ◽  
pp. 731-746
Author(s):  
Tian An Wong
Keyword(s):  
Modular Forms ◽  
Hecke Operators ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Modular Symbols ◽  
Fourier Expansions

We generalize Merel’s work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols. Among the key inputs is a computation of the action of Hecke operators on Manin symbols, building upon the Heilbronn–Merel matrices constructed by Mohamed.

Elementary proofs of relations between Eisenstein series

1977 ◽  
Vol 76 (2) ◽  
pp. 107-117 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Diophantine Equations ◽  
Vector Spaces ◽  
Integral Weight ◽  
Fourier Expansions ◽  
Elementary Methods

SynopsisEisenstein series are entire modular forms Ek of even integral weight k≧ 4 with Fourier expansions given by (1.1). There are numerous identities, such as E8 = , relating these series. These are usually proved by arguments making use of the dimensions of vector spaces of modular forms, and not directly. The paper shows how such identities can be proved by elementary methods by studying chains of solutions of Diophantine equations of the form xξ+yη = n.

scholarly journals Fourier expansions of complex-valued Eisenstein series on finite upper half planes

2006 ◽  
Vol 2006 ◽  
pp. 1-17 ◽  
Author(s):  
Anthony Shaheen ◽  
Audrey Terras
Keyword(s):  
Eisenstein Series ◽  
Half Plane ◽  
Modular Forms ◽  
Bessel Functions ◽  
Kloosterman Sums ◽  
Fourier Expansions ◽  
Wave Forms ◽  
Upper Half Plane ◽  
Complex Valued

We consider complex-valued modular forms on finite upper half planesHqand obtain Fourier expansions of Eisenstein series invariant under the groupsΓ=SL(2,Fp)andGL(2,Fp). The expansions are analogous to those of Maass wave forms on the ordinary Poincaré upper half plane —theK-Bessel functions being replaced by Kloosterman sums.

SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II

2009 ◽  
Vol 05 (07) ◽  
pp. 1321-1345 ◽  
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Modular Forms ◽  
Prime Order ◽  
Galois Representations ◽  
Critical Value ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Fourier Expansions ◽  
Symmetric Square ◽  
Vector Valued ◽  
Siegel Cusp Form

We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.

FOURIER EXPANSIONS WITH MODULAR FORM COEFFICIENTS

2009 ◽  
Vol 05 (08) ◽  
pp. 1433-1446 ◽  
Author(s):  
AHMAD EL-GUINDY
Keyword(s):  
Fixed Point ◽  
Modular Form ◽  
Half Plane ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Genus Zero ◽  
Modular Curve ◽  
Hyperelliptic Involution ◽  
Fourier Expansions ◽  
Upper Half Plane

In this paper, we study the Fourier expansion where the coefficients are given as the evaluation of a sequence of modular forms at a fixed point in the upper half-plane. We show that for prime levels l for which the modular curve X0(l) is hyperelliptic (with hyperelliptic involution of the Atkin–Lehner type) then one can choose a sequence of weight k (any even integer) forms so that the resulting Fourier expansion is itself a meromorphic modular form of weight 2-k. These sequences have many interesting properties, for instance, the sequence of their first nonzero next-to-leading coefficient is equal to the terms in the Fourier expansion of a certain weight 2-k form. The results in the paper generalizes earlier work by Asai, Kaneko, and Ninomiya (for level one), and Ahlgren (for the cases where X0(l) has genus zero).

scholarly journals Rational functions, cotangent sums and Eichler integrals

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Johann Franke
Keyword(s):  
Eisenstein Series ◽  
Half Plane ◽  
Modular Forms ◽  
Holomorphic Functions ◽  
Rational Functions ◽  
Close Connection ◽  
General Transformation ◽  
Fourier Expansions ◽  
Upper Half Plane ◽  
Eichler Integrals

AbstractWith the help of so called pre-weak functions, we formulate a very general transformation law for some holomorphic functions on the upper half plane and motivate the term of a generalized Eisenstein series with real-exponent Fourier expansions. Using the transformation law in the case of negative integers k, we verify a close connection between finite cotangent sums of a specific type and generalized L-functions at integer arguments. Finally, we expand this idea to Eichler integrals and period polynomials for some types of modular forms.

Sign in / Sign up

Export Citation Format

Share Document