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Geometry and arithmetic. Around Euler partial differential equations. (English) Zbl 0595.14017

Mathematische Monographien, Bd. 20. Berlin: VEB Deutscher Verlag der Wissenschaften. 184 p. (1986).
Study of higher dimensional analogues to Euler’s hypergeometric differential equation on \({\mathbb{P}}_ 1({\mathbb{C}})\setminus \{0,1,\infty)\) and algebraic-geometric questions related to it have found new interest in recent years. In his introduction the author gives an interesting survey on historical developments in dimensions 1 and 2. The main part of his book is devoted to an extensive study of one specific example, a special case of the hypergeometric differential equation in two variables first considered by Appell and Picard. This system of second order linear partial differential equations (depending on parameters) locally has a three-dimensional solution space on \({\mathbb{P}}_ 2({\mathbb{C}})\setminus\) (6 lines), where the six lines are obtained by joining any two of four points in general position. For special values of parameters, the (projectivized) monodromy \(\pi_ 1({\mathbb{P}}_ 2\setminus \text{(6 lines)})\to PGL_ 2({\mathbb{C}})\) takes values in a lattice \(\Gamma \subset PU(2,1)=Aut({\mathbb{B}}_ 2)\). In the specific case considered by the author, the action of \(\Gamma\) on \({\mathbb{B}}_ 2\) exhibits the ball as a covering space of \({\mathbb{P}}_ 2\setminus\) (4 points) with ramification of order 3 along the six lines. The author identifies \(\Gamma\) as the principal congruence subgroup of the Eisenstein lattice \({\tilde \Gamma}=U((2,1),{\mathfrak o}_{{\mathbb{Q}}(\sqrt{-3})})\) corresponding to the element \(1-e^{2\pi i/3}\). He shows that the ring of automorphic functions of \(\Gamma\) and \({\tilde \Gamma}\) are polynomial rings in three variables. The ratio of three generators for \(\Gamma\) (all of weight 1) gives the map \({\mathbb{B}}_ 2\to {\mathbb{P}}_ 2\setminus\) (4 points) inverse to the multivalued monodromy map of the differential equation. On the other hand, the (Baily-Borel-compactification of) \({\mathbb{B}}_ 2/{\tilde \Gamma}\) is the moduli space of Picard curves \(y^ 3=x^ 4+g_ 2x^ 2+g_ 3x+g_ 4\). The author investigates this situation in detail.
In the second chapter (”The Gauss-Manin connection of cycloelliptic curve families”) the author investigates more general ”Euler-Picard equations” in r variables \(t_ 1,...,t_ r\) by algebro-geometric methods. These equations are characterized by having solutions of the form \(t\mapsto \int_{\alpha_ t}y^{-\ell}dx\), where \(\alpha_ t\) is a 1-cycle on the curve \(y^ n=(x-1)\cdot x\cdot (x-t_ 1)\cdot...\cdot (x-t_ r).\)
[This book is published also as volume 11 in the series ”Mathematics and its applications (East European Series)” of the Reidel Publishing Company announced above.]
Reviewer: T.Höfer

MSC:

14H15 Families, moduli of curves (analytic)
33C05 Classical hypergeometric functions, \({}_2F_1\)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
11F27 Theta series; Weil representation; theta correspondences
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
32N05 General theory of automorphic functions of several complex variables
11-02 Research exposition (monographs, survey articles) pertaining to number theory

Citations:

Zbl 0595.14016