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Hyperbolic Geometry and the Moduli Space of Real Binary Sextics

The moduli space of real 6-tuples in ℂP 1 is modeled on a quotient of hyperbolic 3-space by a nonarithmetic lattice in IsomH 3. This is partly an expository note; the first part of it is an introduction to orbifolds and hyperbolic reflection groups.
Daniel Allcock, James A. Carlson, Domingo Toledo

Gauss’ Hypergeometric Function

We give a basic introduction to the properties of Gauss’ hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hypergeometric equation.
Frits Beukers

Moduli of K3 Surfaces and Complex Ball Quotients

These notes are based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to recent work on the complex ball uniformization of the moduli spaces of del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. We discuss the relationship of these constructions with the Deligne-Mostow theory of periods of hypergeometric differential forms. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on algebraic varieties with emphasis on abelian varieties and K3 surfaces.
Igor V. Dolgachev, Shigeyuki Kondō

Macbeaths infinite series of Hurwitz groups

In the present paper we will construct an infinite series of so-called Hurwitz groups. One possible way to describe Hurwitz groups is to define them as finite homomorphic images of the Fuchsian triangle group with the signature (2, 3, 7). A reason why Hurwitz groups are interesting lies in the fact, that precisely these groups occur as the automorphism groups of compact Riemann surfaces of genus g > 1, which attain the upper bound 84(g − 1) for the order of the automorphism group. For a long time the only known Hurwitz group was the special linear group PSL2(\( \mathbb{F}_7 \)), with 168 elements, discovered by F. Klein in 1879, which is the automorphism group of the famous Kleinian quartic. In 1967 Macbeath found an infinite series of Hurwitz groups using group theoretic methods. In this paper we will give an alternative arithmetic construction of this series.
Amir Džambić

Relative Proportionality on Picard and Hilbert Modular Surfaces

We introduce “orbital categories”. The background objects are compactified quotient varieties of bounded symmetric domains \( \mathbb{B} \) by lattice subgroups of the complex automorphism group of \( \mathbb{B} \). Additionally, we endow some subvarieties of a given compact complex normal variety V with a natural weight > 1, imitating ramifications. They define an “orbital cycle” Z. The pairs V = (V,Z) are orbital varieties. These objects — also understood as an explicit approach to stacks — allow to introduce “orbital invariants” in a functorial manner. Typical are the orbital categories of Hilbert and Picard modular spaces. From the finite orbital data (e.g. the orbital Apollonius cycle on ℙ2) we read off “orbital Heegner series” as orbital invariants with the help of “orbital intersection theory”. We demonstrate for Hilbert and Picard surface F how their Fourier coefficients can be used to count Shimura curves of given norm on F. On recently discovered orbital projective planes the Shimura curves are joined with well-known classical elliptic modular forms.
Rolf-Peter Holzapfel

Hypergeometric Functions and Carlitz Differential Equations over Function Fields

The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur’s hypergeometric function. We consider basic notions and results of calculus, analytic theory of differential equations with Carlitz derivatives (including a counterpart of regular singularity), umbral calculus, holonomic modules over the Weyl-Carlitz ring.
Anatoly N. Kochubei

The Moduli Space of 5 Points on ℙ1 and K3 Surfaces

We show that the moduli space of 5 ordered points on ℙ1 is isomorphic to an arithmetic quotient of a complex ball by using the theory of periods of K3 surfaces. We also discuss a relation between our uniformization and the one given by Shimura [S], Terada [Te], Deligne-Mostow [DM].
Shigeyuki Kondō

Uniformization by Lauricella Functions — An Overview of the Theory of Deligne-Mostow

This is a survey of the Deligne-Mostow theory of Lauricella functions, or what almost amounts to the same, of the period map for cyclic coverings of the Riemann sphere.
Eduard Looijenga

Invariant Functions with Respect to the Whitehead-Link

We survey our construction of invariant functions on the real 3-dimensional hyperbolic space ℍ3 for the Whitehead-link-complement group WGL 2(ℤ[i]) and for a few groups commensurable with W. We make use of theta functions on the bounded symmetric domain \( \mathbb{D} \) of type I 2,2 and an embedding i : ℍ3\( \mathbb{D} \). The quotient spaces of ℍ3 by these groups are realized by these invariant functions. We review classical results on the λ-function, the j-function and theta constants on the upper half space; our construction is based on them.
Keiji Matsumoto

On the Construction of Class Fields by Picard Modular Forms

The goal of this article is to construct modular functions living on the complex ball of dimension two such that the values in special points — similar to the elliptic modular function — generate class fields. For this purpose we are well prepared by the papers [5] and [6]. The first one classifies the moduli space of abelian 3-folds with a multiplication by ℚ(i) of type (2, 1) as projective surface. Via Jacobians we connect this Shimura surface with the moduli space of a family of curves of Shimura equation type. Thus we are able to continue the construction of the inverse period map of the family by theta constants given in [6]. Knowing the action of the modular group we reach a modular function j by modular forms with respect to the congruence subgroup of level (1 + i) of the full Picard modular group of Gauß numbers. If τ is the period of a (Jacobian of a) curve with complex multiplication the corresponding moduli field is generated over the rational numbers by j(τ). Hence the values in CM points of this function generate abelian extensions of the associated reflex field.
Thorsten Riedel

Algebraic Values of Schwarz Triangle Functions

We consider Schwarz maps for triangles whose angles are rather general rational multiples of π. Under which conditions can they have algebraic values at algebraic arguments? The answer is based mainly on considerations of complex multiplication of certain Prym varieties in Jacobians of hypergeometric curves. The paper can serve as an introduction to transcendence techniques for hypergeometric functions, but contains also new results and examples.
Hironori Shiga, Jürgen Wolfart

GKZ Hypergeometric Structures

This text is based on lectures by the author in the Summer School Algebraic Geometry and Hypergeometric Functions in Istanbul in June 2005. It gives a review of some of the basic aspects of the theory of hypergeometric structures of Gelfand, Kapranov and Zelevinsky, including Differential Equations, Integrals and Series, with emphasis on the latter. The Secondary Fan is constructed and subsequently used to describe the ‘geography’ of the domains of convergence of the Γ-series. A solution to certain Resonance Problems is presented and applied in the context of Mirror Symmetry. Many examples and some exercises are given throughout the paper.
Jan Stienstra

Orbifolds and Their Uniformization

This is an introduction to complex orbifolds with an emphasis on orbifolds in dimension 2 and covering relations between them.
A. Muhammed Uludağ

From the Power Function to the Hypergeometric Function

The hypergeometric function is a slight generalization of the power fucntion. We will see this by the Schwarz map of the hypergeometric equation focussing on the behavior of this map when the local exponent-differences are purely imaginary
Masaaki Yoshida

Problem Session

This article contains the open problems discussed during the problem session of the CIMPA summer school “Arithmetic and Geometry Around Hypergeometric Functions” held at Galatasaray University, İstanbul, 2005.
Celal Cem Sarıoğlu
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