scholarly journals On Siegel modular forms part II

1995 ◽  
Vol 138 ◽  
pp. 179-197 ◽  
Author(s):  
Bernhard Runge
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Graded Ring

In this paper we compute dimension formulas for rings of Siegel modular forms of genus g = 3. Let denote the main congruence subgroup of level two, the Hecke subgroup of level two and the full modular group. We give the dimension formulas for genus g = 3 for the above mentioned groups and determine the graded ring of modular forms with respect to .

scholarly journals Projective varieties and rings of Thetanullwerte

1990 ◽  
Vol 118 ◽  
pp. 165-176
Author(s):  
Riccardo Salvati Manni
Keyword(s):  
Positive Integer ◽  
Holomorphic Function ◽  
Half Space ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Integral Closure ◽  
Graded Ring ◽  
Projective Varieties ◽  
Entry Vector

Let r denote an even positive integer, m an element of Q2g such that r·m ≡ 0 mod 1 and ϑm the holomorphic function on the Siegel upper-half space Hg defined by(1) ,in which e(t) stands for exp and m′ and m″ are the first and the second entry vector of m. Let Θg(r) denote the graded ring generated over C by such Thetanullwerte; then it is a well known fact that the integral closure of Θg(r) is the ring of all modular forms relative to Igusa’s congruence subgroup Γg(r2, 2r2) cf. [6]. We shall denote this ring by A(Γg(r2, 2r2)).

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 .

scholarly journals On the representation of a number as a sum of squares

1978 ◽  
Vol 19 (2) ◽  
pp. 173-197 ◽  
Author(s):  
Karl-Bernhard Gundlach
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Remainder Term ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Hilbert Modular Group ◽  
Totally Positive ◽  
Hilbert Modular ◽  
Image Position ◽  
Totally Real

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the formwhere Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact thatis a modular form for a certain congruence subgroup of the modular group, andwithwhere Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

scholarly journals ON THE GRADED RING OF SIEGEL MODULAR FORMS OF DEGREE TWO WITH RESPECT TO A NON-SPLIT SYMPLECTIC GROUP

2012 ◽  
Vol 23 (02) ◽  
pp. 1250032 ◽  
Author(s):  
HIDETAKA KITAYAMA
Keyword(s):  
Modular Forms ◽  
Symplectic Group ◽  
Discrete Subgroup ◽  
Ring Structure ◽  
Siegel Modular Forms ◽  
Graded Ring

We will give an explicit ring structure of the graded ring of Siegel modular forms of degree two with respect to a certain discrete subgroup of Sp(2;ℝ) contained in a non-split ℚ-form. We will also give the fundamental relations among the generators.

On the algebra of modular forms on a congruence subgroup

1979 ◽  
Vol 86 (3) ◽  
pp. 461-466 ◽  
Author(s):  
A. J. Scholl
Keyword(s):  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Fourier Expansion ◽  
Congruence Subgroup ◽  
Complex Numbers ◽  
Graded Algebra

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.

Vector-valued Siegel modular forms of weight detk ⊗ Sym(8)

2015 ◽  
Vol 26 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Tomoya Kiyuna
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Siegel Modular Forms ◽  
Vector Valued

We determine the structures of modules of vector-valued Siegel modular forms of weight det k ⊗ Sym (8) with respect to the full Siegel modular group of degree two.

On Hecke eigenvalues of Siegel modular forms in the Maass space

2018 ◽  
Vol 30 (3) ◽  
pp. 775-783 ◽  
Author(s):  
Sanoli Gun ◽  
Biplab Paul ◽  
Jyoti Sengupta
Keyword(s):  
Modular Forms ◽  
Absolute Constant ◽  
Modular Group ◽  
Siegel Modular Forms ◽  
Maass Forms ◽  
Formula One ◽  
Hecke Eigenvalues ◽  
Hecke Eigenforms ◽  
The Third ◽  
Limit Points

AbstractIn this article, we prove an Omega result for the Hecke eigenvalues {\lambda_{F}(n)} of Maass forms F which are Hecke eigenforms in the space of Siegel modular forms of weight k, genus two for the Siegel modular group {Sp_{2}({\mathbb{Z}})}. In particular, we prove\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\exp\biggl{(}c\frac{\sqrt{\log n}}{\log% \log n}\biggr{)}\biggr{)},when {c>0} is an absolute constant. This improves the earlier result\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\biggl{(}\frac{\sqrt{\log n}}{\log\log n}% \biggr{)}\biggr{)}of Das and the third author. We also show that for any {n\geq 3}, one has\lambda_{F}(n)\leq n^{k-1}\exp\biggl{(}c_{1}\sqrt{\frac{\log n}{\log\log n}}% \biggr{)},where {c_{1}>0} is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence {\{\lambda_{F}(n)/n^{k-1}\}_{n\in{\mathbb{N}}}} and show that it has infinitely many limit points. Finally, we show that {\lambda_{F}(n)>0} for all n, a result proved earlier by Breulmann by a different technique.

On Siegel paramodular forms of degree 2 with small levels

2016 ◽  
Vol 27 (02) ◽  
pp. 1650011 ◽  
Author(s):  
Hiroki Aoki
Keyword(s):  
Differential Operator ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Graded Ring

In this paper, we show that the graded ring of Siegel paramodular forms of degree [Formula: see text] with level [Formula: see text] has a very simple unified structure, taking with character. All are generated by six modular forms. The first five are obtained by a kind of Maass lift. The last one is obtained by a kind of Rankin–Cohen–Ibukiyama differential operator from the first five. This result is similar to the case of the graded ring of Siegel modular forms of degree [Formula: see text] with respect to [Formula: see text].

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