arXiv subjects
Publications and source records attributed to Eberhard Freitag.
We consider the space of Siegel modular forms of genus $g$ of weight two relative to the main congruence subgroup of level 2 and to Igusa's group $\Gamma_g(4, 8)$ and $\Gamma_g(2,4)$.One of the main results of this paper is that in the case $g\ge 8$ the space $[\Gamma_g[4,8],2]$ is generated by the products of 4 theta nullwerte. Thus this note can be considered as a completion of the example at the end of [Fr].
In [ Ge], Bert van Geemen computed the dimension of the space of the fourth power of the theta nullwerte. In [SM2], it has been observe that all linear relations between the $\theta_m^4$ are consequences of the quartic Riemann relations. In this note, we want to give a new proof of these result and extend them. In a last section we treat the linear dependencies between arbitrary powers $\vartheta[m]^k$. We will show that $k=4$ is the only case where such dependencies can occur. For this reason, we give a slightly different title: Some remarks to a Theorem of van Geemen
In this note, we describe several new examples of holomorphic modular forms on the group SU(2,1). These forms are distinguished by having weight $\frac{1}{3}$. We also describe a method for determining the levels at which one should expect to find such fractional weight forms.
Let $Γ_{F,n}$ be the Hermitian modular group of degree $n>1$ in sense of Hel Braun with respect to an imaginary quadratic field $F$. Let $r$ be a natural number. There exists a multiplier system of weight $1/r$ (equivalently a Hermitian modular form of weight $k+1/r$, $k$ integral) on some congruence group if and only if $r=1$ or $2$. This follows from a much more general construction of Deligne [De] combining it with results of Hill [Hi], Prasad [P] and Prasad-Rapinchuk [PR]. As far as we know, the systems of weight $1/2$ have not yet been described explicitly. Remarkably Haowu Wang [Wa] gave an example of a modular form of half integral weight. Actually he constructs a Borcherds product of weight $23/2$ for a group of type $O(2,4)$. This group is isogenous to the group $U(2,2)$ that contains the Hermitian modular groups of degree two. In this paper we want to study such multiplier systems. If one restricts them to the unimodular group one obtains a usual character. Our main result states that the kernel of this character is a non-congruence subgroup. For sufficiently small $Γ$ it coincides with the group described by Kubota in the case $n=2$ and by Bass Milnor Serre in the case $n>2$.
Deligne proved that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g>1 must be integral or half integral. We give a different proof for this. It uses Mennicke's result that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from a paper of Bass-Milnor-Serre.
In this paper we will prove that the six-dimensional Göpel variety in $P^{134}$ is generated by 120 linear, 35 cubic and 35 quartic relations. This result was already obtained in [RS] , but the authors used a statement in [Co] saying that the Göpel variety set theoretically is generated by the linear and cubic relations alone. Unfortunately this statement is false. There are 120 extra points. Nevertheless the results stated in [RS] are correct. There are required several changes that we will illustrate in some detail
Due to fundamental results of Igusa and Mumford the $N=2^{g-1}(2^g+1)$ even theta constants define for each genus $g$ an injective holomorphic map of the Satake compactification $X_g(4,8)=H_g/Γ_g[4,8]$ into the projective space $P^{N-1}$. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\ge 6$. Igusa also proved that in the cases $g\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. So we show that the theta map $$X_3(4,8)\to P^{35}$$ is biholomorphic onto the image. This is equivalent to the statement that the image is a normal subvariety of $P^{35}$ .
We study a projective Calabi-Yau threefold which has been constructed in an earlier paper. It is rigid and has Picard number two. We construct a pair of divisors which give a basis of the Picard group and determine all intersection numbers of three divisors.
We prove that there are only finitely many isometry classes of even lattices $L$ of signature $(2,n)$ for which the space of cusp forms of weight $1+n/2$ for the Weil representation of the discriminant group of $L$ is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of $L$ can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.
Let L be the even unimodular lattice of signature (2,10), In the paper [FS] we considered the subgroup O(L)^+ of index two in the orthogonal group. It acts biholomorphically on a ten dimensional tube domain H_{10}. We found a 715 dimensional space of modular forms with respect to the principal congruence subgroup of level two O^+(L)[2]. It defines an everywhere regular birational embedding of the related modular variety into the 714 dimensional projective space. In this paper, we prove that this space of orthogonal modular forms is related to a space of theta series. The main tool is a modular embedding of H_{10} into the Siegel half space of degree 16. As a consequence the modular forms in the 715 dimensional space can be obtained as restrictions of the simplest among all theta series.
Cléry and van der Geer determined generators for some modules of vector valued Picard modular forms on the two dimensional ball. In this paper we consider the case of a three dimensional ball with the action of the Picard modular group $Γ_3[\sqrt{-3}]$. The corresponding modular variety of dimension 3 is a copy of the Segre cubic.
Extending the method of the paper [FS3] we prove three structure theorems for vector valued modular forms, where two correspond to 4-dimensional cases (two hermitian modular groups, one belonging to the field of Eisenstein numbers, the other to the field of Gaussian numbers.) and one to a 6-dimensional case (a quaternionic modular group).
We study over rings of scalar valued Siegel modular forms. modules of vector valued modular forms of degree two. For the two simplest representations, standard and Sym^2, appears rather natural consider the cases of the group $Γ[4,8] $ and $Γ[2,4].$ In these case we give the complete structure of the modules . The main tools are Theta functions and their derivatives evaluated at $z=0$
We describe a parametrization of the box variety (variety of cuboids) by theta functions. This will imply that the box variety is a modular variety. Actually this parametrization can be defined over the Gauss number field.
The aim of this note is to treat one distinguished example of a Calabi--Yau variety that appears as a small resolution of a Picard modular variety
In connection with our previous investigation about Siegel threefolds which admit a Calabi--Yau model, we consider ball quotients which belong to the unitary group $\U(1,3)$. In this paper we determine a very particular example of a Picard modular variety of general type. Really we determine the ring of modular forms. This algebra has 25 generators, 15 modular forms $B_i$ of weight one and ten modular forms $C_j$ of weight 2. Both will appear as Borcherds products. We determine the ideal of relations. The forms $C_i$ are cuspidal. Their squares define holomorphic differential forms on the non-singular models.
In two recent papers, we described some Siegel modular threefolds which admit a weak Calabi--Yau model. Not all of them admit a {\it projective} model. The purpose of this paper is to exhibit criterions for the projectivity, to treat several examples, and to compute their Hodge numbers.
In this paper we treat in details a modular variety $\cal Y$ that has a Calabi-Yau model, $\tilde{\cal Y}$. We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of $\cal Y$ as the quotient of another known Calabi-Yau variety. In this case we will get the Hodge numbers considering the action of the group on a crepant resolution $\tilde{\cal X}$ of $\cal X$. The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi-Yau model $\tilde{\cal Y}$ and computing the Picard group and the Euler characteristic.