## Download Arithmetic and Geometry Around Hypergeometric Functions: by Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida PDF

By Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida

This quantity contains lecture notes, survey and learn articles originating from the CIMPA summer season tuition mathematics and Geometry round Hypergeometric capabilities held at Galatasaray college, Istanbul, June 13-25, 2005. It covers a variety of subject matters regarding hypergeometric services, hence giving a wide point of view of the cutting-edge within the box.

Read or Download Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005 PDF

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Extra resources for Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005

Sample text

This closely reﬂects the nature of the natural hermitian form on the monodromy group itself. 32. Let a, b, c ∈ R be such that 0 ≤ λ, μ, ν < 1, λ + μ + ν < 1 + 2 min(λ, μ, ν), where λ = |1 − c|, μ = |c − a − b|, ν = |a − b|. Let M be the monodromy group of (2). Then, M is spheric ⇐⇒ λ + μ + ν > 1 M is euclidean ⇐⇒ λ + μ + ν = 1 42 Frits Beukers M is hyperbolic ⇐⇒ λ + μ + ν < 1. Proof. In the case when none of the numbers a, b, c − a, c − b is integral, this statement can already be inferred from the proof of the previous lemma (we get only the hyperbolic and spheric case).

First we look at subgroups generated by reﬂection in two intersecting geodesics. 25. Let ρ, σ be two geodesics intersecting in a point P with an angle πλ. Let r, s be the reﬂections in ρ, σ respectively. Then the group D generated by r, s is a dihedral group consisting of rotations (rs)n around P with angles 2nπλ, n ∈ Z and reﬂections in the lines (rs)n (ρ), (rs)n (σ). In particular D is ﬁnite of order 2m if and only if λ = q/m for some q ∈ Z with q = 0 and gcd(m, q) = 1. Furthermore, D is discrete if and only if λ is either zero or a rational number.

Vinberg, Some arithmetical discrete groups in Lobacevskii spaces. In Discrete Subgroups of Lie Groups and Applications to Moduli, 328–348. Oxford, 1975. [19] M. Yoshida, The real loci of the conﬁguration space of six points on the projective line and a Picard modular 3-fold, Kumamoto J. Math. 11 (1998), 43–67. 22 Daniel Allcock, James A. edu/~allcock James A. edu/~toledo Progress in Mathematics, Vol. 260, 23–42 c 2007 Birkh¨ auser Verlag Basel/Switzerland Gauss’ Hypergeometric Function Frits Beukers Abstract.