Abelian Varieties Theta Functions And The Fourier Transform

Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.

This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Răzvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology. Theta Functions and Knots can be read in two perspectives. Readers with an interest in theta functions or knot theory can learn how the two are related. Those interested in Chern–Simons theory will find here an introduction using the simplest case, that of abelian Chern–Simons theory. Moreover, the construction of abelian Chern–Simons theory is based entirely on quantum mechanics and not on quantum field theory as it is usually done. Both the theory of theta functions and low dimensional topology are presented in detail, in order to underline how deep the connection between these two fundamental mathematical subjects is. Hence the book is self-contained with a unified presentation. It is suitable for an advanced graduate course, as well as for self-study. Contents:PrologueA Quantum Mechanical PrototypeSurfaces and CurvesThe Theta Functions Associated to a Riemann SurfaceFrom Theta Functions to KnotsSome Results About 3- and 4-Dimensional ManifoldsThe Discrete Fourier Transform and Topological Quantum Field TheoryTheta Functions in the Quantum Group PerspectiveAn Epilogue — Abelian Chern–Simons Theory Readership: Graduate students and young researchers with an interest in complex analysis, mathematical physics, algebra geometry and low dimensional topology. Keywords:Theta Functions;Chern–Simons Theory;Knots;Skein Modules;Linking Number;Topological Quantum Field TheoryKey Features:A detailed study of the skein modules of the linking number, which provide the simplest example of a skein module (skein modules have become a major object of study in combinatorial topology)A complete discussion of the facts from low dimensional topology (Kirby's theorem, the Lickorish–Walace theorem, Wall's non-additivity of the signature) which are fundamental in Chern–Simons theoryReviews: “It looks like a really good book, presenting its many themes in a very accessible and clear fashion, replete with plenty of pictures and lots of wonderful theorems and proofs from representation theory as well as differential geometry and the kind of functional analysis needed to do quantum physics.” Mathematical Association of America

Originally published: New York: Rinehart and Winston, 1961.

The theory of theta functions has a long history; for this, we refer A. Krazer and W. Wirtinger the reader to an encyclopedia article by ("Sources" [9]). We shall restrict ourselves to postwar, i. e. , after 1945, periods. Around 1948/49, F. Conforto, c. L. Siegel, A. Well reconsidered the main existence theorems of theta functions and found natural proofs for them. These are contained in Conforto: Abelsche Funktionen und algebraische Geometrie, Springer (1956); Siegel: Analytic functions of several complex variables, Lect. Notes, I. A. S. (1948/49); Well: Theoremes fondamentaux de la theorie des fonctions theta, Sem. Bourbaki, No. 16 (1949). The complete account of Weil's method appeared in his book of 1958 [20]. The next important achievement was the theory of compacti­ fication of the quotient variety of Siegel's upper-half space by a modular group. There are many ways to compactify the quotient variety; we are talking about what might be called a standard compactification. Such a compactification was obtained first as a Hausdorff space by I. Satake in "On the compactification of the Siegel space", J. Ind. Math. Soc. 20, 259-281 (1956), and as a normal projective variety by W. L. Baily in 1958 [1]. In 1957/58, H. Cartan took up this theory in his seminar [3]; it was shown that the graded ring of modular forms relative to the given modular group is a normal integral domain which is finitely generated over C.

This book contains lectures on theta functions written by experts well known for excellence in exposition. The lectures represent the content of four courses given at the Centre de Recherches Mathematiques in Montreal during the academic year 1991-1992, which was devoted to the study of automorphic forms. Aimed at graduate students, the book synthesizes the classical and modern points of view in theta functions, concentrating on connections to number theory and representation theory. An excellent introduction to this important subject of current research, this book is suitable as a text in advanced graduate courses.