CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINOR L-VALUES

2021 ◽  
pp. 1-22
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Recent Work ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Hecke Eigenvalues ◽  
Genus 2

Abstract Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

Symplectic Kloosterman sums

2013 ◽  
Vol 50 (2) ◽  
pp. 143-158
Author(s):  
Árpád Tóth
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Kloosterman Sums ◽  
Genus 2 ◽  
Optimal Bounds

We give optimal bounds for Kloosterman sums that arise in the estimation of Fourier coefficients of Siegel modular forms of genus 2.

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.

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 .

ALGEBRAIC INDEPENDENCE OF VALUES OF MODULAR FORMS

2011 ◽  
Vol 07 (04) ◽  
pp. 1065-1074 ◽  
Author(s):  
SANOLI GUN ◽  
M. RAM MURTY ◽  
PURUSOTTAM RATH
Keyword(s):  
Recent Work ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Independence ◽  
Singular Values ◽  
Critical Values ◽  
Cm Points

We investigate values of modular forms with algebraic Fourier coefficients at algebraic arguments. As a consequence, we conclude about the nature of zeros of such modular forms. In particular, the singular values of modular forms (that is, values at CM points) are related to the recent work of Nesterenko. As an application, we deduce the transcendence of critical values of certain Hecke L-series. We also discuss how these investigations generalize to the case of quasi-modular forms with algebraic Fourier coefficients.

scholarly journals Siegel modular forms of genus 2 and level 2

2015 ◽  
Vol 26 (05) ◽  
pp. 1550034 ◽  
Author(s):  
Fabien Cléry ◽  
Gerard van der Geer ◽  
Samuel Grushevsky
Keyword(s):  
Modular Forms ◽  
Generating Functions ◽  
Theta Functions ◽  
Siegel Modular Forms ◽  
Genus 2 ◽  
Vector Valued ◽  
Level 2

We study vector-valued Siegel modular forms of genus 2 on the three level 2 groups Γ[2] ◁ Γ1[2] ◁ Γ0[2] ⊂ Sp(4, ℤ). We give generating functions for the dimension of spaces of vector-valued modular forms, construct various vector-valued modular forms by using theta functions and describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued Siegel modular forms.

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