# Arithmetic and Geometry around Hypergeomet by Holzapfel R.-P.

By Holzapfel R.-P.

This quantity contains lecture notes, survey and learn articles originating from the CIMPA summer season university mathematics and Geometry round Hypergeometric services held at Galatasaray college, Istanbul in the course of June 13-25, 2005. a variety of subject matters relating to hypergeometric features is roofed, therefore giving a wide standpoint of the cutting-edge within the box.

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In this case geodesic triangles are planar triangles in the euclidean geometry with ﬁnite area. The latter property is equivalent to positivity of all angles. Suppose that λ+μ+ν > 1. From spherical geometry it follows that a spherical triangle exists if and only if our condition is satisﬁed. We let W (Δ) be the group of isometries of S generated by the 3 reﬂections through the edges of a geodesic triangle Δ. 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.

Then, letting X 2 +a1 X +a2 and X 2 +b1 X +b2 be the characteristic polynomials of A, B, we have up to common conjugation, A= 0 −a2 , 1 −a1 B= 0 −b2 . 1 −b1 Proof. Choose v ∈ ker(A − B) and w = Av = Bv. Since A, B have disjoint eigenvalue sets, v is not an eigenvector of A and B. Hence w, v form a basis of C2 . With respect to this basis A, B automatically obtain the form given in our Lemma. 14. Suppose that (2) is irreducible. Then, up to conjugation, the monodromy group depends only on the values of a, b, c modulo Z.

Other examples relating the arithmetic complex ball quotients arising in the Deligne–Mostow theory to moduli spaces of Del Pezzo surfaces were found in [MT], [HL]. It turns out that all of these examples are intimately related to the moduli space of K3 surfaces with special structure of its Picard group of algebraic cycles and an action of a cyclic group. In Section 10 we develop a general theory of such moduli spaces. In Section 11 we brieﬂy discuss all the known examples of moduli spaces of Del Pezzo surfaces and curves of low genus which are isomorphic to the moduli space of such structures on K3 surfaces, and via this isomorphism admit a complex ball uniformization by an arithmetic group.