Ebook: Lectures on the theory of automorphic functions v.2= 自守函数理论讲义

"Felix Klein's famous Erlangen program made the theory of group actions into a central part of mathematics. In the spirit of this program, Klein set out to write a grand series of books which unified many different subjects of mathematics, including number theory, geometry, complex analysis, and discrete subgroups. The first book on icosahedron and the solution of equations of the fifth degree showed closed relations between three seemingly different subjects: the symmetries of the icosahedron, the solution to fifth degree algebraic equations, and the differential equation of hypergeometric functions. It was translated into English in 1888, four years after its original German version was published in 1884. It was followed by two volumes on elliptic modular functions by Klein and Fricke and two more volumes on automorphic functions also by Klein and Fricke. These four classic books are vast generalizations of the first volume and contain the highly original works of Poincaré and Klein on automorphic forms. They have been very influential in the development of mathematics and are now available in English for the first time. These books contain many original ideas, striking examples, explicit computations, and details which are not available anywhere else. They will be very valuable references for people at all levels and allow the reader to see the unity of mathematics through the eyes of one of the most influential mathematicians with vision, Felix Klein." --;Second volume. Part I. Narrower theory of the single-valued automorphic functions of one variable -- Concept, existence, and fundamental properties of the automorphic functions -- Form-theoretic discussions for the automorphic models of genus zero -- Theory of Poincaré series with special discussions for the models of genus zero -- The automorphic forms and their analytic representations for models of arbitrary genus -- Part II. Fundamental theorems concerning the existence of polymorphic functions on Riemann surfaces -- Proof of the principal-circle and the boundary-circle theorem -- Proof of the reentrant cut theorem -- An addition to the transformation theory of automorphic functions -- Commentaries by Richard Borcherds on elliptic modular functions -- Commentary by Jeremy Gray -- Commentary by William Harvey on automorphic functions -- Commentary by Barry Mazur -- Commentary by Series-Mumford-Wright -- Commentary by Domingo Toledo -- Commentaries by other mathematicians.;First volume. Introduction. Developments concerning projective determinations of measure -- Part I. Foundations for the theory of the discontinuous groups of linear substitutions of one variable -- The discontinuity of groups with illustrations by simple examples -- The groups without infinitesimal substitutions and their normal discontinuity domans -- Further approaches to the geometrical theory of the properly discontinuous groups -- Part II. The geometrical theory of the polygon groups of zeta-substitutions -- Treatment of the rotation groups on the foundation of the normal discontinuity domains -- The canonical polygons and the moduli of the hyperbolic rotations groups -- Study of the circular-arc quadrangles without principal circle and remarks on other non-rotation groups -- Part III. Arithmetic methods of definition of properly discontinuous groups of zeta-substitutions -- The rotation subgroups inside the Picard groups and the associated binary quadratic forms -- The reproducing groups of ternary and quaternary quadratic forms -- A special kind of principal-circle and polyhedron groups with integral algebraic substitution coefficients -- Commentaries -- Commentary by Richard Borcherds on elliptic modular functions -- Commentary by Jeremy Gray -- Commentary by William Harvey on automorphic functions -- Commentary by Barry Mazur -- Commentary by Series-Mumford-Wright -- Commentary by Domingo Toledo -- Commentaries by other mathematicians.