This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

Higher Topos Theory

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

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The Geometry and Cohomology of Some Simple Shimura Varieties.

Linear differential equations form the central topic of this volume, Galois theory being the unifying theme.
A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used.
This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

Galois theory of linear differential equations

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On the Kodaira Dimension of the Moduli Space of Curves

The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.

Analytic Functions of Several Complex Variables

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

Etale Cohomology and the Weil Conjecture

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).

Hilbert Modular Forms

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

Eberhard Freitag

Papers

Analytic Functions of Several Complex Variables

Etale Cohomology and the Weil Conjecture

Hilbert Modular Forms

Singular Modular Forms and Theta Relations

Some Modular Varieties of Low Dimension