ON THE FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

2016 ◽  
Vol 234 ◽  
pp. 1-16
Author(s):  
SIEGFRIED BÖCHERER ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Main Tool ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Growth Properties

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

scholarly journals Restriction of Siegel modular forms to modular curves

2002 ◽  
Vol 65 (2) ◽  
pp. 239-252 ◽  
Author(s):  
Cris Poor ◽  
David S. Yuen
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Modular Curves ◽  
Linear Relations

We study homomorphisms form the ring of Siegel modular forms of a given degree to the ring of elliptic modular forms for a congruence subgroup. These homomorphisms essentially arise from the restriction of Siegel modular forms to modular curves. These homomorphisms give rise to linear relations among the Fourier coefficients of a Siegel modular form. We use this technique to prove that dim .

AN EQUIDISTRIBUTION THEOREM FOR HOLOMORPHIC SIEGEL MODULAR FORMS FOR AND ITS APPLICATIONS

2018 ◽  
Vol 19 (2) ◽  
pp. 351-419 ◽  
Author(s):  
Henry H. Kim ◽  
Satoshi Wakatsuki ◽  
Takuya Yamauchi
Keyword(s):  
Modular Forms ◽  
Discrete Series ◽  
Main Tool ◽  
Density Theorem ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Holomorphic Discrete Series ◽  
General Aspects ◽  
Explicit Study ◽  
Invent Math

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $\mathit{GSp}_{4}/\mathbb{Q}$ in various aspects. A main tool is Arthur’s invariant trace formula. While Shin [Automorphic Plancherel density theorem, Israel J. Math.192(1) (2012), 83–120] and Shin–Templier [Sato–Tate theorem for families and low-lying zeros of automorphic $L$-functions, Invent. Math.203(1) (2016) 1–177] used Euler–Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A,B_{1}$ in Theorem 1.1 which have not been studied and a mysterious second term $B_{2}$ also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato–Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.

A characterization of degree two Siegel cusp forms by the growth of their Fourier coefficients

2014 ◽  
Vol 26 (5) ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Jacobi Forms

AbstractWe characterize all cusp forms among the degree two Siegel modular forms by the growth of their Fourier coefficients. We also give a similar result for Jacobi forms over the group

scholarly journals Concomitants of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three

2020 ◽  
Vol 26 (4) ◽  
Author(s):  
Fabien Cléry ◽  
Carel Faber ◽  
Gerard van der Geer
Keyword(s):  
Representation Theory ◽  
Modular Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Local Systems ◽  
Hyperelliptic Locus ◽  
Vector Valued

Abstract We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on $$\overline{\mathcal {M}}_{g}$$ M ¯ g and the middle cohomology of symplectic local systems on $${\mathcal {M}}_{g}\,$$ M g . In genus 3, we make this explicit in a large number of cases.

scholarly journals On signs of Fourier coefficients of cusp forms

2011 ◽  
Vol 152 (2) ◽  
pp. 207-222 ◽  
Author(s):  
KAISA MATOMÄKI
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Upper Bound ◽  
Alternative Treatment ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Rearrangement Inequalities

AbstractWe consider two problems concerning signs of Fourier coefficients of classical modular forms, or equivalently Hecke eigenvalues: first, we give an upper bound for the size of the first sign-change of Hecke eigenvalues in terms of conductor and weight; second, we investigate to what extent the signs of Fourier coefficients determine an unique modular form. In both cases we improve recent results of Kowalski, Lau, Soundararajan and Wu. A part of the paper is also devoted to generalized rearrangement inequalities which are utilized in an alternative treatment of the second question.

scholarly journals Local spectral equidistribution for Siegel modular forms and applications

2012 ◽  
Vol 148 (2) ◽  
pp. 335-384 ◽  
Author(s):  
Emmanuel Kowalski ◽  
Abhishek Saha ◽  
Jacob Tsimerman
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Test Functions ◽  
Genus 2 ◽  
Satake Parameters ◽  
Global Evidence

AbstractWe study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

scholarly journals ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

2021 ◽  
pp. 1-41
Author(s):  
Siegfried Böcherer ◽  
Soumya Das
Keyword(s):  
Functional Equation ◽  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Induction Argument ◽  
Integral Matrices ◽  
Fundamental Discriminant ◽  
Vector Valued

Abstract We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree $3$ and level $1$ .

scholarly journals Modular descent of Siegel modular forms of half integral weight and an analogy of the Maass relation

1986 ◽  
Vol 102 ◽  
pp. 51-77 ◽  
Author(s):  
Yoshio Tanigawa
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Maass Relation

In [8], H. Maass introduced the ‘Spezialschar’ which is now called the Maass space. It is defined by the relation of the Fourier coefficients of modular forms as follows. Let f be a Siegel modular form on Sp(2,Z) of weight k, and let be its Fourier expansion, where . Then f belongs to the Maass space if and only if

Cuspidality and the growth of Fourier coefficients of modular forms

2018 ◽  
Vol 2018 (741) ◽  
pp. 161-178 ◽  
Author(s):  
Siegfried Böcherer ◽  
Soumya Das
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Local Approach ◽  
Large Weight ◽  
Vector Valued

Abstract We characterize Siegel cusp forms in the space of Siegel modular forms of large weight {k>2n} on any Siegel congruence subgroup Γ of any degree n and any level N, by a suitable growth of their Fourier coefficients (e.g., by the well-known Hecke bound) at any one of the cusps. For this, we use a ‘local’ approach as compared to our previous results on this topic. We also touch upon the question in the context of vector-valued modular forms.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document