scholarly journals HECKE OPERATORS ON HILBERT–SIEGEL MODULAR FORMS

2007 ◽  
Vol 03 (03) ◽  
pp. 391-420 ◽  
Author(s):  
SUZANNE CAULK ◽  
LYNNE H. WALLING
Keyword(s):  
Number Field ◽  
Modular Forms ◽  
Product Space ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Linear Operators ◽  
Hecke Operators ◽  
Hilbert Modular Forms ◽  
Linear Transformations ◽  
Hilbert Modular

We define Hilbert–Siegel modular forms and Hecke "operators" acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert–Siegel forms (i.e. with arbitrary Siegel degree) we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals).

scholarly journals Siegel modular forms of degree two attached to Hilbert modular forms

2012 ◽  
Vol 132 (4) ◽  
pp. 543-564 ◽  
Author(s):  
Jennifer Johnson-Leung ◽  
Brooks Roberts
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

scholarly journals Eisenstein series in the Kohnen plus space for Hilbert modular forms

2016 ◽  
Vol 12 (03) ◽  
pp. 691-723 ◽  
Author(s):  
Ren-He Su
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Hilbert Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
The One ◽  
Totally Real

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE

2020 ◽  
pp. 1-22
Author(s):  
JIM BROWN ◽  
HUIXI LI
Keyword(s):  
Functional Equation ◽  
Modular Forms ◽  
Class Number ◽  
Automorphic Forms ◽  
Siegel Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Free Level ◽  
Totally Real

Abstract It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

2016 ◽  
Vol 152 (6) ◽  
pp. 1159-1197
Author(s):  
Yingkun Li
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Maass Forms ◽  
Stark's Conjecture ◽  
Hilbert Modular ◽  
Harmonic Maass Forms ◽  
The Individual

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

scholarly journals On the Kohnen plus space for Hilbert modular forms of half-integral weight I

2013 ◽  
Vol 149 (12) ◽  
pp. 1963-2010 ◽  
Author(s):  
Kaoru Hiraga ◽  
Tamotsu Ikeda
Keyword(s):  
Modular Form ◽  
Fourier Coefficient ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Jacobi Forms ◽  
Hecke Operator ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular

AbstractIn this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

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