Modular independence and generator matrices for codes over $${\mathbb {Z}_m}$$

We introduce a new notion of modular independence to define bases and the generator matrices for the codes over the ring of integers of general modulus m. We define standard forms for such generator matrices, and discuss how to find such forms and the parity check matrices.      

On primitive elements of algebraic function fields and models of $$X_0(N)$$ X 0 ( N )

The Ramanujan Journal - This paper is a continuation of our previous works where we study maps from $$X_0(N)$$ , $$N\ge 1$$ , into $${\mathbb {P}}^2$$ constructed via modular forms of the same...     

Zagier duality for level p weakly holomorphic modular forms

The Ramanujan Journal - We prove Zagier duality between the Fourier coefficients of canonical bases for spaces of weakly holomorphic modular forms of prime level p with $$11 \le p \le 37$$ with...     

A simple evaluation of a theta value,the Kronecker limit formula and a formula of Ramanujan

The Ramanujan Journal - We evaluate the classic sum $$\sum _{n\in {\mathbb {Z}}} e^{-\pi n^2}$$ . The novelty of our approach is that it does not require any prior knowledge about modular forms,...     

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

Some modular varieties of low dimension II

The paper extends results obtained by Frieder Hermann and Eberhard Freitag about a six-dimensional modular variety related to the orthogonal group of signature (2.6). The ring of modular forms of this variety turns out to be a weighted polynomial ring in 7 variables.     

On special values at integers of L-functions of Jacobi theta products of weight 3

The Ramanujan Journal - In this paper, we consider L-functions of two modular forms of weight 3, which are products of the Jacobi theta series, and express their special values at $$s=3$$ , 4 in...     

On Picard Modular Forms

We study moduli spaces of principally polarized abelian varieties with an automorphism of finite order. After some examples (e. g. hermitian modular forms) we compute the ring of Picard modular forms in the case considered by Picard.     

Siegel modular forms of degree 2 over rings

Based on moduli theory of abelian varieties, extending Igusa's result on Siegel modular forms over C, we describe the ring of Siegel full modular forms of degree 2 over any Z-algebra in which 6 is invertible.     

The ring of differential Fourier expansions

For a fixed prime we prove structure theorems for the kernel and the image of the map that attaches to any differential modular function its differential Fourier expansion. The image of this map, which is the ring of differential Fourier expansions, plays the role of ring of functions on a “differential Igusa curve”. Our constructions are then used to perform an analytic continuation between isogeny covariant differential modular forms on the differential Igusa curves belonging to different primes.     

The Burkhardt Group and Modular Forms

In this paper we investigate the ring of Siegel modular forms of genus two and level 3. We determine the structure of this ring. It is generated by 10 modular forms (5 of weight 1 and 5 of weight 3) and there are 20 relations (5 in weight 5 and 15 in weight 6). The proof consists of two steps. In a first step we prove that the Satake compactification of the modular variety of genus 2 and level 3 is the normalization of the dual of the Burkhardt quartic. The second part consists in the normalization of the Burkhardt dual. Our basic tool is the representation theory of the Burkhardt group G = G_{25 920}, which acts on our varieties.     

RINGS OF HILBERT MODULAR FORMS ON TOTALLY REAL NUMBER FIELDS WITH ODD DEGREE

E. Thomas and A. T. Vasques proved the following result: For any totally real cubic number field K and subgroup $\[\Gamma \]$ of modular type of $\[PS{L_2}({O_K})\]$, the ring of Hilbert modular forms for $\[\Gamma \]$ over k s not Gorenstein ring. In thE present paper the author comes to the same conclusion for any totally real number field of odd degree     

Type II Codes over $$\mathbb{Z}/2k\mathbb{Z}$$, Invariant Rings and Theta Series

We consider type II codes over finite rings . It is well-known that their gth complete weight enumerator polynomials are invariant under the action of a certain finite subgroup of , which we denote H_k,g. We show that the invariant ring with respect to H_k,g is generated by such polynomials. This is carried out by using some closely related results concerning theta series and Siegel modular forms with respect to .     

A $$p$$ -arton model for modular cusp forms

Theoretical and Mathematical Physics - To a modular form, we propose to associate $$($$ an infinite number of $$)$$ complex-valued functions on $$p$$ -adic numbers $$\mathbb{Q}_p$$ for each prime...     

Quaternionic theta constants

We investigate the six quaternionic theta constants introduced by Freitag and Hermann. More precisely we investigate their restrictions to the Hermitian resp. Siegel half-space of degree 2. It turns out that these theta constants generate the graded ring of symmetric Hermitian modular forms for the principal congruence subgroup of level 1 + i over the Gaussian number field resp. of Siegel modular forms for the principal congruence subgroup of level 2 and even weight. As an application we obtain a simple construction of Igusa’s Siegel modular form of degree 2 and weight 30 with respect to the non-trivial character.     

Projections of modular forms on Eisenstein series and its application to Siegel’s formula

The Ramanujan Journal - Let $$k \ge 2$$ and N be positive integers and let $$\chi $$ be a Dirichlet character modulo N. Let f(z) be a modular form in $$M_k(\Gamma _0(N),\chi )$$ . Then we have a...