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Hypergeometric functions, my love. Modular interpretations of configuration spaces. (English) Zbl 0889.33008

Aspects of Mathematics E32. Wiesbaden: Vieweg (ISBN 978-3-528-06925-4/hbk; 978-3-322-90168-2/pbk; 978-3-322-90166-8/ebook). xvi, 292 p. (1997).
In this delightfully exuberant book, the author tells the story of the modular interpretation of the configuration space of six lines on the projective plane. The projective line minus \(\{0,1,\infty\}\) can be thought of as the configuration space for four points on the projective line, \(X(2,4)\). In this case, the story consists of explaining why the ratio of two linearly independent solutions of the hypergeometric equation, \(4x(1-x)u'' - 4xu' - u\) defines an isomorphism between this configuration space and a fundamental region of the upper half-plane under \(\Gamma(2)\). After excursions through elliptic curves, theta functions, modular forms, and hypergeometric integrals, the author extends his story to \(n\) points on the projective line, \(X(2,n)\), and then to the configuration space of \(X(3,6)\), six lines on the projective plane, with a complete description of the modular interpretation of this space.

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
11F03 Modular and automorphic functions
14J10 Families, moduli, classification: algebraic theory
20F29 Representations of groups as automorphism groups of algebraic systems
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
51F15 Reflection groups, reflection geometries
53B10 Projective connections
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