Papers with Abstracts of Eberhard Freitag

Es wird ein elementarer Beweis des Satzes von Igusa über dir Struktur der Algebra der Siegelschen Modulformen zweiten Grades gegeben.

Modular forms with respect to the Hermitean modular group of the Gauss number field and of genus two are investigated. The field of modular functions is a rational function field.

Introduction into the theory of linear algebraic groups.

Modular embeddings of Hilbert modular groups into Siegel paramodular groups are investigated. Every Hilbert modular group admits an embedding into a suitable paramodular group.

Let A be a Hermitean symmetric domain which is embedded into another Hermitean symmetric domain B and let G be an arithmetic group acting on B and H its "projection" to A. Under very weak conditions the map A/H-->B/G is generically injective which implies that every modular function on A/H is the restricion of a modular function on B.

Hilbert modular varieties of arbitrary dimension are investigated from an algebraic geometric point of view.

8. Lokale und globale Invarianten der Hilbertschen Modulgruppe. PDF
Inv.Math. 17, 106-134 (1972)

The arithmetic genus of a Hilbert modular variety is expressed in terms of Shimizu's dimension formulae for spaces of modular forms. The cohomology groups of the structure sheaf are computed and also the cohomology groups with support in the cusps. The dualizing complex in the sense of Hartshorne of the mininal compactification is determined.

Every scalar valued automorphy factor of a group commensurable with Hilberts modular group of a totally real number field of degree n>2 is a standard one.

Continuation of 8 concerning the properties of the local rings at the cusps of Hilbert modular varieties. An example is found where the local ring is factorial but not a Cohen Macaulay ring.

Local and global singular and coherent cohomology groups of Hilbert modular varieties (extending no 6,7,8,10)

Holomorphic differential forms of degree two on the Siegel upper half plane of genus two, which are invariant under certain congruence subgroups of the Siegel modular group, are constructed. It is proved that they extend holomorphically to a nonsingular projective model. Hence the corresponding function fields are not unirational. The construction uses a certain bilinear differential operator on the space of modular forms of weight 1/2. As application one obtains that all those forms are singular.

The previous paper is generalized to arbitrary genus. The differential operators are constructed by means of subdeterminants of the matrix of partial derivatives (suitably normalized). A linear algebra fomalism has to be developed to obtain the transformation formalism of theese operators.

Using the constructions of the previous paper in the case n=1 mod 24, n>1, non-vanishing differential foms of degree n-1 on the (desingularized) Siegel modular variety with respect to the full Siegel modular group are constructed. As a consequence, the field of modular functions is not rational. (The numerical calculation of a non-zero Fourier coefficient contains an error which can be corrected if one takes instead of the unit matrix the matrix with entries 1 in the diagonal and 1/2 outside the diagonal).

Introduction to the theory of Siegel modular varieties (surview).

The space of Siegel modular forms of integral weight and with respect to a character is invariant under a certain Hecke algebra. In an ammendment to the paper (no 21) it is pointed that only in the case of the trivial character the full Hecke algebra can be taken. The proof contains a gap in the case of a non-trivial multiplier which forces to take a proper sub-algebra.

The main result of the paper states that the projective limit of the algebras of Siegel modular forms with respect to the full Siegel modular group (and varying genus) is a polynomial ring whose variables can be identified with classes of even self dual lattices. A modular form is called stable if it lifts into this projective limit. Stable modular forms turn out to be theta series. The paper contains a very short proof for this fact in case of the full Siegel modular group.

Using theta series with harmonic coefficients it can be shown that the (desingularized) Siegel modular variety of genus g=0 mod 24 admits non trivial canonical forms and hence is not uni-rational. By the way, the argument shows that it is of general type in theese cases (which has later been proved by Tai in the cases n>8, in my Springer book on Siegel modular forms it has been improved to n>7, Mumford settled the case n=7. As far as I know the case n=6 is still open whereas in the cases n<6 the variety is unirational.)

Vector valued holomorphic Siegel modular forms of the transformation typy f(MZ)=v(CZ+D)f(Z) for a polynomial representation v of the general linear group are constructed. If f does not vanish, one irreducible component of v has to be trivial (and the corresponding component of f is constant) or divisible by det(CZ+D). This result is obtained by restriction to Hilbert modular forms. The result implies that invariant holomorphic differential forms of degree 1 vanish but another application to differential forms of higher degree is false. A correct version is in my common paper with Pommerening (no 25).

The paper restricts to the full modular group. In this case it generalizes the results of no 16 to theta series with harmonic coefficients. The method rests on the fact that theta series with harmonic coefficients can be characterized as stable in the sense that they are images under Siegels specialization operator of modular forms of arbitrarily high degree. But one has to admit vector valued modular forms (otherwise one would obtain no non-trivial) harmonic coefficients.

It is proved that (compactified) Hilbert modular varities of sufficiently high level do not contain rational or elliptic curves. The idea is to consider Hilbert modular forms as symmetric tensors (and not as multicanonical forms as usual) and to investigate how a holomorphic curve can run into a cusp.

Using the theory of singular modular forms a very short proof of Andrianovs formulae for the action of Hecke operators on theta series is given. As application one obtains a very short proof of a special case of Siegels main theorem.

It is proved that in case of genus g>1 every holomorphic differential form on the regular locus of a Siegel modular variety extends holomorphically to a nonsingular model. As a consequence those forms are closed. (This generalizes no 12. Later Pommerening generalized this result to other Hermitean symmetric domains and Bauerman found a local version.)

Using no 22, explicit formulae for the action of Hecke operators on theta series with harmonic coefficients are derived. As example an eigen-cusp-form of degree 24 and weight 13 is constructed by means of the Leech lattice.

The locus of curves of genus four in the Siegel modular variety of genus four can be described as zero set of a cusp form of weight 8. The point is to prove the irreducibility of this zero locus. A first convincing proof has been given by Igusa. Here we give a very short cohomological proof.

Holomomorphic tensors on the regular locus of Siegel modular varieties can be considered as holomorphic vector valued modular forms. Such forms can be constructucted as products of scalar valued modular forms and of constant vector valued forms. The extension to non singular models is investigated. The main application states that for sufficiently large genus every subvariety od codimension one of the Siegel modular variety is of general type. As a consequence the field of Siegel modular functions (with respect to the full Siegel modular group) admits no automorphism besides the identity. (Theese results have been generalized later by Weissauer who also derived concrete bounds.)

The transformation formalism of vector valued theta series with harmonic coefficients is developed using Eichlers embedding trick. To do this it is necessary to admit polynomial coefficients, which are not harmonic. Most of the results of this paper are contained also in my Springer lecture notes "Singular modular Forms"

Singular modular forms with respect to arbitrary level are represented as theta series with harmonic coefficients and the linear relations between them are described. So in principle one can write down dimension formulae. This paper is a short version of my Springer lectore notes on this topic. The bound for the singular weight r is r <n/2 but we need r<n because a certain elementary lemma could be proved only under this assumption. (Meanwhile I know how to prove the general case but the proof is long and complicated and nobody seems to be interested in it, so I gave up to produce a readable version and to publish it.)

Some cases of the lemma mentioned in the previous article but still not yet the most general case.

As no 30. Forschungsschwerpunkt Geometrie, Heidelberg Nr. 45 (1989)

The ring of elliptic modular forms of level (4,8) and with respect to the theta multiplier system is generated by the three Jacobi thetas and the Jacobi relation is the defining one. We give a very short and elementary proof which can be reprocduced in one or two hours of an introductory lecture into the theory of modular forms.

Some results about the local coherent cohomology at the zero dimensional cusps of modular varieties are described without proofs. Birational invariants like the dimensions of spaces of holomorphic alternating differential forms are expressed in computable terms.

Singular modular forms with respect to the Hilbert-Siegel modular group are represented as linear combinations of theta serie and the linear relations between the generators are described.

Linear combinations of Siegel Eisensteinseries of integral weight r>g+1 can be characterized by the fact that they don't vanish at all zero-dimensional cusp and that they are eigen forms of one non-trivial Hecke operator. (This is a generaization of a result of Elstrodt). As a consequence one can derive the analytic version of Siegels main theorem. In a second part of the paper the values of theta series at zero dimensional cusps are investigated. Theta series do not seperate the values at the zero dimensional cups. A precise description of what is possible is given. (The paper contains a minor gap which is corrected in a subsequent paper of Salvati-Manni.)

The ring of Siegel modular forms of genus 3 and level (2,4) is generated by the 8 classical theta constants of second kind and there is one defining relation of degree 16 (which corresponds to the Schottky-relation). This result is due to Runge. We give a short proof which rests on a computer calculation.

A Siegel cusp form of degree 12 and weight 12 is constructed. Upto a constant factor it can be characterized as the unique linear combination of the 24 theta series corresponding to the Niemeier lattices, which represents a cusp form. The Hecke eigen value of T(2) is computed.

There is a unique quintic hypersurface in the 5-dimensional projective space which is invariant under the action of the Weyl group of the root lattice E6. The dual hypersurfece is explicitely deteremined. Its degree is 32.

The quaternionic modular group in the sense of Krieg with respect to the Gauss number field and of level (1+i) acts on a 6-dimensional domain. The Baily-Borel compactification of the quotient is a covering of degree 24 of a projective space. Inside this modular variety several interesting varieties of dimensions 5,4,3 are investigated. The modular forms, which provide the coordinates, are constructed as theta series. For the determination of the zero locus, Borcherds products are used.

A relation of degree 24 between the 16 theta constants of second kind, which is invariant under the full Siegel modular group is constructed as linear combination of the code polynomials of the 9 self-dual doubly even binary codes of length 24.

The moduli space of marked cubic surfaces can be described as quotient of a 4-ball by an artithmetic subgroup. Using Borcherds liftings for the group O(2,8) one can construct an embedding of the Satake compactification into the nine dimensional projective space which is equivariant with respect to actions of the Weyl group of the root lattice E6. The image variety is described as intersection of one W(E6)-orbit of cubic eightfolds.

The moduli space of marked cubic surfaces can be described as a ball quotient by an arithmetic subgroup of the unitary group U(1,4) . The unitary group can be considered as subgroup of O(2,8). It is natural to use the lifting constructions of R. Borcherds for the group O(2,n) to construct projective models for the moduli space of marked cubic surfaces. In the paper above such a model has been constructed by means of the singular Borcherds lift. In this paper modular forms are consturcted in a more systematic way and several interesting spaces of modular forms are obtained .We will not discuss applications to the moduli space of cubics in this paper.

The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a hermitean symmetric quotient of type O(2,n) is computed. The main ingredient is a local version of Borcherd's products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. In some cases they are the same as Borcherd's global obstructions.

The algebra of modular forms on the moduli space od marked cubic surfaces is determined. It is generated by ten forms. The ideal of relations is generated by an W(E6)-orbit of one cubic relations.

The classical models of the moduli space of marked cubic surfaces coming form 1) Cayley's cross ratios and 2) Coble-Yoshida polynomials are identified with the model which has recently been obtained using Picard modular forms (see the three previous papers).

The minimality conjecture for the Hilbert modular surfaces for the full Hilber modular group acting on two half planes (upper or lower) is proved if the discriminant od the corresponding quadratic field is not kongruent 1 mod 8.

The structure of the ring of Siegel modular forms of genus two and level three is determined. It is generated by 5 forms of weight 1 and 5 forms of weight three. There are 20 relations.

The structure of the ring of Siegel modular forms for a subgroup of index two of the principal congruence subgroup of genus two and level three is determined. It is generated by 5 forms of weight one, two and three. The relations between the 15 forms are determined.

Connections between results of Dern and Krieg about Hermitian modular forms for the Eisenstein field and the work of the authors about the Burkhardt quartic are discussed. It tuns out that not only the Burkhardt quartic but also the embedding projective space is a modular variety.

This paper is a continuation of the paper 40. The structure of a certain 6-dimensional variety, which belongs to the group O(2,6) has been determined completely. It turns out to be a weighted projective space. The Weyl group of the E6-lattice acts on this variety and the quotient of the variety by this group is a weighted projective space as well. The latter result is due to Krieg who obtained it in a different way using the quaternionic symplectic group of degree 2 instead of the orthogonal group. His paper will appear in Math. Z.

Up to isomorphism there is a unique even unimodular lattice L of signature (2,10). We investigate the modular variety which belongs to the principal congruence subgroup of level 2 of the orthogonal group of this lattice.

This paper is a continuation of the paper 51. The structure of a certain 5-dimensional variety, which belongs to the group O(2,5) has been determined completely. This paper reproves and extends some results of the Phd thesis of Klöcker.

Several compactifications of the modular variety of hyperelliptic curves of genus three are studied.

Macdonald generalized a formula of Weyl, which is valid for reduced root systems, to affine root systems. This identity can be considered as an identity of a power of the Dekekind Eta-function and a certain theta series. By means of the theta inversion formula we give a direct short proof.

Some Siegel modular varieties with respect to some subgroups of the Siegel modular group of genus two containing the principal congruence subgroup of level 4 lead to Calabi-Yau varieties.

Extended list of Siegel Calabi-Yau manifolds.

Distinguished example of a Siegel Calabi-Yau threefold. Computation of Hodge numbers and of the related elliptic modular form (proof of modularity).

59. On Siegel three folds with a projective Calabi--Yau model PDF
(joint work with Riccardo Salvati Manni)
Commun. Number Theory Phys. 5, No. 3, 713-750 (2011

Extension of papers 56-58. The projectivity of the constructed weak Calabi-Yau models is investigated. A theory of local Borcherds products for 0-dimensional cusps of Siegel threefolds has to be developped.

We consider the prinicpal congruence subgroup of level 3 of thePicard modular group with respect to the ring of Eisenstein numbers acting on the three dimensional ball. We determine the structure of the ring of modular forms. It is generated by 15 forms of weight 1 and 10 cusp forms of weight 2. Defining relations are described. The associated modular variety is a covering of the Segre cubic.

Using the Riemann-Roch theorem we derive the dimension formulae for spaces of vector valued automorphic forms in one variableof arbitrary rational weight. The case of weight 2 is included.

62. Some ball quotients with a Calabi-Yau model PDF
(joint work with Riccardo Salvati Manni
To appear on Proc. Am. Math. Soc

We recover a known Calabi-Yau variety) given by the equations X_0X_1X_2=X_3X_4X_5, X_0^3+X_1^3+X_2^3=X_3^3+X_4^3+X_5 2 as a Picard modular variety with respect to a certain Picard modular group.

A structure theorem for vector valued Siegel modular forms of genus three with respect to the representation Sym^2det^k and for the Igusa group Gamma[2,4] is proved.

The box variety (variety of cuboids) is recovered as a modular surface for a subgroup of SL(2,Z)xSL(2,Z)

65. Some vector valued Siegel modular forms of genus two PDF
(joint work with Riccardo Salvati Manni)
Osaka Journal of Math. 52(3) (2013)

A structure theorem for vector valued Siegel modular forms of genus two with respect to the representation Sym^2det^k and for the Igusa group Gamma[4,8] is proved.

66. Vector valued Siegel modular forms of level [2,4,8] PDF
(joint work with Riccardo Salvati Manni and Thomas Wieber)
preprint (2013)

A structure theorem for vector valued Siegel modular forms of genus two with respect to the representation Sym^2det^k and for the group Gamma[2,4,8] which has been introduced by van Geemen and van Straten is proved.

67. Basic vector valued Siegel modular forms of genus two PDF
(joint work with Riccardo Salvati Manni)
Osaka J. Math. Bolume 52, 879-895 (2015)

We give a new proof of Wieber's structure result on vector valued Siegel modular forms for Igusa'as group of genus 2 and level (2,4) and with respect to the representation Sym^2. We also obtain the structure theorem for level (4,8) and the standard representation.

68. Vector valued hermitian and quaternionic modular forms PDF
(joint work with Riccardo Salvati Manni)
Kyoto J. Math. Vol. 55, Number 4, 819-836 (2015)

We prove structure theorems for vector valued hermitian modular forms of degree in two cases which belong to the fields of Eisenstein resp. Gauss numbers and we treat a case of a quaternionic modular group of degree two which belongs to the Hurwitz integers.

69. Vector valued modular forms on three dimensional ball PDF
(joint work with Riccardo Salvati Manni)
Transaction of the Am. Math. Soc. 371(8) (2014)

We determine the structure of a module of modular forms on a three-dimensional ball. The associated modular variety is a copy of the Segre cubic.

70. Octavic theta series PDF
(joint work with Riccardo Salvati Manni)
Asian Journal of Mathematics, Volume 21 (2017)

This is a continuation of the paper no 52. There has been studied a 10-dimensional tube domain related to the even unimodular lattice of signature (2,10). A basic 715-dimensional space of modular forms of weight 4 has been constructed. In this paper we construct a modular embedding into the Siegel half plane of genus 16 and we obtain the elements of this 715-dimensional space as restrictions of the theta series of second kind.

71. Lattices with many Borcherds products PDF
(joint work with Jan Hendrik Bruinier and Stephan Ehlen)
Math. Comp. 85, 1953-1981 (2016)

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.

72. A rigid Calabi-Yau manifold with Picard number two PDF
Communications in Number Theory and Physics 10(3), 2015

A certain three dimensional Siegel modular variety which admits a projective model as Calabi-Yau manifold is studied. It is rigid and has Picard number two. The trilinear intersection form on the Picard group is determined.

73. On the variety associated to the ring of theta constants in genus 3 PDF
(joint work with Riccardo Salvati Manni)
American J. Math. 141(3) (2016)

The 36 classical theta nullwerte of genus 3 define a biholomorphic map from the Satake compactification with resect to Igusa's congruence group of level (4,8) onto a (normal) subvariety in the 35-dimensional space.

74. On the Göpel variety PDF
(joint work with Riccardo Salvati Manni)
Experimental Mathematics, Volume 28, Issue 3 (2019)

The Göpel variety is a 6-dimensional variety that is birational equivalent to the Siegel modular variety of genus 3 and level 3. It has an embedding into the projective space of dimension 134 and satifies 120 linear and 35 cubic relations. Coble stated 1929 that these equations cut out the Göpel variety. Unfortunately this is false. Since this result has been used in subsequent papers, we correct this and prove that the known additional quartic relations give defining relations.

75. Multiplier systems for Siegel modular groups PDF
(joint work with Adrian Hauffe-Waschbüsch)
http://arxiv.org/abs/2009.06455

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.

76. Multiplier systems for Hermitian modular groups PDF
http://arxiv.org/abs/2009.06455

If one restricts a multiplier system of a subgroup of finite index of a Hermitian modular group to the unimodular group, one obtains a usual character. It can be described by means of a Mennicke symbol. Its kernel is a well-known non congruence subgroup.