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New PDF release: Arithmetic and Geometry Around Hypergeometric Functions:

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

ISBN-10: 376438283X

ISBN-13: 9783764382834

ISBN-10: 3764382848

ISBN-13: 9783764382841

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

Sample text

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.

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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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