Eberhard Freitag

Etale cohomology and the Weil conjecture

I. The Essentials of Etale Cohomology Theory.- II. Rationality of Weil ?-Functions.- III. The Monodromy Theory of Lefschetz Pencils.- IV. Delignes Proof of the Weil Conjecture.- Appendices.- A I. The Fundamental Group.- A II. Derived Categories.- A III. Descent.

Hilbert Modular Forms

A discrete subgroup Γ ⊂ SL, (2ℝ) acts discontinuously on the upper half-plane H. The parabolic elements of Γ give rise to a natural extension of H/Γ by the so-called cusp classes. We are mainly interested in the case where this extension is compact. Our basic example is Γ = SL(2, Z). The method of construction is such that it can easily be generalized to the case of several variables, i.e. discrete subgroups of SL(2, ℝ) n acting on the product of n upper half-planes. This will be done in the next section (§2).

Singular Modular Forms and Theta Relations

Siegel modular forms.- Theta series with polynomial coefficients.- Singular weights.- Singular modular forms and theta series.- The fundamental lemma.- The results.

Algebraische Eigenschaften der lokalen Ringe in den Spitzen der Hilbertschen Modulgruppen

Einleitung In dieser Arbeit wird die Untersuchung der Spltzen Hilbertscher Modulgruppen in mehr als zwei Variablen weitergeftihrt. Wir bauen auf der Arbeit [3] auf. Im Vordergrund werden algebraische Eigenschaften der lokalen analytischen Ringe in den Spitzen stehen. Wir wollen uns in der Einleitung damit begntigen, die Komplettierung dieser Ringe zu beschreiben. Gegeben seien: t) Ein total reeller algebraischer ZahlkSrper L. 2) Ein Gitter t ~ L vom Rang n. 3) Eine Untergruppe A yon endlichem Index in der Gruppe aller total positiven Einheiten, welche auf i operiert

Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper

Die genaue Struktur des Korpers der Modulfunktionen oder allgemeiner, die genaue Struktur des graduierten Ringes der Modulformen ist in mehreren Veranderlichen nur in wenigen Spezialfallen bekannt. So bewies Igusa [7], das alle Siegeischen Modulformen zweiten Grades geraden Gewichts als isobare Polynome in den Eisensteinreihen vom Gewicht 4, 6, 10, 12 darstellbar sind. Hieraus ergibt sich, das der Korper der Siegeischen Modulfunktionen zweiten Grades rational ist. In [4] wurde mit Hilfe von Thetareihen ein elementarer Beweis dieses Satzes gegeben. Mit der hierbei verwendeten Methode war es schon Gundlach [5] gelungen, das entsprechende Problem im Falle der Hilbertschen Modulgruppe zum Zahlkorper \(Q\left( {\sqrt 5 } \right)\) zu losen. Der Korper der symmetrischen Modulfunktionen zu dieser Gruppe erwies sich ebenfalls als rational.

The modular variety of hyperelliptic curves of genus three

The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi-projective variety which admits several standard compactifications. The first one realizes this variety as a subvariety of the Siegel modular variety of level two and genus three. It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components. Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with 8 branch points. There are two important compactifications of this configuration space. The first one, Y, uses the semistable degenerated point configurations in (P 1 ) 8 . This variety also can be identified with a Baily-Borel compactified ball-quotient Y = B/Γ[1 - i]. We will describe these results in some detail and obtain new proofs including some finer results for them. The other compactification uses the fact that families of marked projective lines can degenerate to stable marked curves of genus 0. We use the standard notation M 0,8 for this compactification. We have a diagram The horizontal arrow is only birational but not everywhere regular. In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) A, B such that X = proj(A), Y = proj(B).

Some Siegel Threefolds with a Calabi-Yau Model II

In the following we describe some examples of Calabi-Yau manifolds, which arise as desingularizations of certain Siegel threefolds. The first Siegel modular variety with a Calabi-Yau model and the essentially only one up to now has been discovered by Barth and Nieto.

A theta relation in genus 4

The 2 g theta constants of second kind of genus g generate a graded ring of dimension g(g + 1)/2. In the case g ≥ 3 there must exist algebraic relations. In genus g = 3 it is known that there is one defining relation. In this paper we give a relation in the case g = 4. It is of degree 24 and has the remarkable property that it is invariant under the full Siegel modular group and whose Φ-image is not zero. Our relation is obtained as a linear combination of code polynomials of the 9 self-dual doubly-even codes of length 24.

Lattices with many Borcherds products

We prove that there are only nitely 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 classication results in greater generality for nite quadratic modules.

The geometry and arithmetic of a Calabi-Yau siegel threefold

In this paper we treat in details a modular variety