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Book review

Author(s) Holzapfel, R.-P.
Title Geometry and Arithmetic Around Euler Partial Differential Equations
Publisher D.Reidel Publ. Co.
Year of publication 1986
Reviewed by Valery Sanyuk

To avoid a possible misleading of the reader it should be noted, that there are several partial differental equations in pure & applied mathematics, and in mathematical physics named after L. Euler. To distinguish among them people usually put another name, e.g. Euler-Lagrange equations etc. In the refered case it looks like more appropriate to use in the title "Euler-Picard equation" (or Euler-Picard problem), since by and large the book deals with solutions of this long-standing problem.

In a narrow sense the monograph presents the author's solution of the Picard's problem on determination of hypergeometric functions and integrals as particular solutions of partial differential equations. The main results are the following: developed the general system of Euler-Picard type in arbitrary dimension; found its solutions in the form of integrals on the corresponding Riemann surfaces; established the connection of solutions with automorphic forms with respect to Eisenstein lattices in the 2-dimensional complex unit ball. Using the advanced techniques of algebraic geometry and Holzapfel's general theory of quotient manifolds of the complex ball, the main body of the book concerns with the classification of Picard Curves. In the rest part one will find the results on Gauss-Manin connection of cycloelliptic curve families. It should be stressed that all the listed results are of highest level achievments in the field.

The monograph is written in the theorem-definition-proof manner, though it is supplied with a brief historical outline of the subject, with quotations from such Grands as C.F. Gauss, F. Klein, B. Riemann. Most of the proofs are given in their full length, nevertheless it should be recognized that it could not be recomended as a text book, or for the initial study. It is just professionally written summing up specialistic research monograph, with a very dense and compact exposition of material, and with a good index and an exhaustive list of references.

Beyond any doubt the book will be interesting for specialists in algebraic geometry (in respect of "technical applications"), in the theory of functions, and in the number theory (most of given applications are related with this branch of mathematics). The book may be of great help for those who are interested in the similar study of another problem and who will find here a sort of "know-how" for their own research. From the pragmatic point it is a pity, that such a promissing field of applications for the developed mathematics as the contemporary string theory in physics turn out to be far from the topic of the author. But maybe it could be the subject for the next book.