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

ISBN-10: 376438283X

ISBN-13: 9783764382834

ISBN-10: 3764382848

ISBN-13: 9783764382841

This quantity includes lecture notes, survey and examine articles originating from the CIMPA summer season university mathematics and Geometry round Hypergeometric services held at Galatasaray collage, Istanbul, June 13-25, 2005. It covers quite a lot of themes with regards to hypergeometric services, therefore giving a huge standpoint of the cutting-edge within the box.

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CA/0310436. O. nl Progress in Mathematics, Vol. 260, 43–100 c 2007 Birkh¨ auser Verlag Basel/Switzerland Moduli of K3 Surfaces and Complex Ball Quotients Igor V. Dolgachev and Shigeyuki Kond¯o Abstract. 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.

Then F (a, b, c|1) = Γ(c − a)Γ(c − b) . Γ(c)Γ(c − a − b) This can be proven by evaluation of Euler’s integral using the Euler Betafunction. To study the analytic continuation of D(z) we use Schwarz’ reﬂection principle. Hopefully, the following picture illustrates how this works. 38 Frits Beukers D(1) D(z) 0 1 ∝ D(∝) D(0) The monodromy group modulo scalars arises as follows. Let W be the group generated by the reﬂections in the edges of the curvilinear triangle. The monodromy group is the subgroup of W consisting of all elements which are product of an even number of reﬂections.

40 Frits Beukers Let α, β, γ be the edges of Δ and rα , rβ , rγ the corresponding reﬂections. Suppose that the vertex angle between α and β is of the form mπ/n with gcd(m, n) = 1, but m > 1. Let δ be the geodesic between α and β whose angle with α is π/n. Let rδ be the reﬂection in δ. Then the dihedral group generated by rα and rβ is the same as the one generated by rα and rδ . Let Δ be the triangle with edges α, δ, γ. Then, clearly, W (Δ) = W (Δ ). If the volume of Δ is larger than half the volume of Δ we simply perform the above construction with α and β interchanged.

### Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005 by Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida

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