By Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida
This quantity contains lecture notes, survey and learn articles originating from the CIMPA summer time institution mathematics and Geometry round Hypergeometric capabilities held at Galatasaray collage, Istanbul, June 13-25, 2005. It covers quite a lot of themes relating to hypergeometric features, therefore giving a huge point of view of the state-of-the-art within the box.
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Additional info for Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005
One easily veriﬁes that 1 d (z + a)F (a, b, c|z) a dz d 1 (z(1 − z) − bz + c − a)F (a, b, c|z) F (a − 1, b, c|z) = c−a dz and similarly for F (a, b + 1, c|z), F (a, b − 1, c|z). Furthermore, F (a + 1, b, c|z) = F (a, b, c + 1|z) = F (a, b, c − 1|z) = c d (z(1 − z) + c − a − b)F (a, b, c|z) (c − a)(c − b) dz 1 d (z + c − 1)F (a, b, c|z). c − 1 dz d Hence there exists a linear diﬀerential operator Lk,l,m ∈ C(z)[ dz ] such that F (a + k, b + l, c + m|z) = Lk,l,m F (a, b, c|z). Since F satisﬁes a second order linear differential equation, Lk,l,m F can be written as a C(z)-linear combination of F and F .
Since f and g cannot vanish at the same time in a regular point, we have f (z1 ) = 0. – The map D(z) maps the segments (∞, 0), (0, 1), (1, ∞) to segments of circles or straight lines. For example, since a, b, c ∈ R we have two real solutions on ˜ (0, 1) (see Kummer’s solutions). Call them f˜, g˜. Clearly, the function D(z) = ˜ f /˜ g maps (0, 1) on a segment of R. Since f, g are C-linear combinations of ˜ f˜, g˜ we see that D(z) is a M¨ obius transform of D(z). Hence D(z) maps (0, 1) to a segment of a circle or a straight line.
Some of its arithmetic quotients have a realization as periods of hypergeometric Moduli of K3 Surfaces 45 diﬀerential forms via the Deligne–Mostow theory. We refer for the details to Looijenga’s article in the same volume [Lo]. A hypergeometric diﬀerential form has an interpretation as a holomorphic 1-form on a certain algebraic curve on which a cyclic group acts by automorphisms and the form is transformed according to a character of this group. In Section 6 we discuss in a more general setting the theory of what we call eigenperiods of algebraic varieties.
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