This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter,…mehr
This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.
I. Periods of meromorphic functions.- 1. Meromorphic functions.- 2. Periodic meromorphic functions.- 3. Jacobi's lemma.- 4. Elliptic functions.- 5. The modular group and modular functions.- Notes on Chapter I.- II. General properties of elliptic functions.- 1. The period parallelogram.- 2. Elementary properties of elliptic functions.- Notes on Chapter II.- III. Weierstrass's elliptic function ?(z).- 1. The convergence of a double series.- 2. The elliptic function ?(z).- 3. The differential equation associated with ?(z).- 4. The addition-theorem.- 5. The generation of elliptic functions.- Appendix I. The cubic equation.- Appendix II. The biquadratic equation.- Notes on Chapter III.- IV. The zeta-function and the sigma-function of Weierstrass.- 1. The function ?(z).- 2. The function ?(z).- 3. An expression for elliptic functions.- Notes on Chapter IV.- V. The theta-functions.- 1. The function ?(?, ?).- 2. The four sigma-functions.- 3. The four theta-functions.- 4. The differential equation.- 5. Jacobi's formula for ?' (0, ?).- 6. The infinite products for the theta-functions.- 7. Theta-functions as solutions of functional equations.- 8. The transformation formula connecting ?3(v, ?) and ?3(?, ?1/?) ..- Notes on Chapter V.- VI. The modular function J(?).- 1. Definition of J(?).- 2. The functions g2(?) and g3(?).- 3. Expansion of the function J(?) and the connexion with theta-functions.- 4. The function J(?) in a fundamental domain of the modular group ..- 5. Relations between the periods and the invariants of ?(u).- 6. Elliptic integrals of the first kind.- Notes on Chapter VI.- VII. The Jacobian elliptic functions and the modular function ?(?).- 1.The functions sn u, en u, dn u of Jacobi.- 2. Definition by theta-functions.- 3. Connexion with the sigma-functions.- 4. The differential equation.- 5. Infinite products for the Jacobian elliptic functions.- 6. Addition-theorems for sn u, cn u, dn u.- 7. The modular function ?(?).- 8. Mapping properties of ?(?) and Picard's theorem.- Notes on Chapter VII.- VIII. Dedekind's ?-function and Euler's theorem on pentagonal numbers.- 1. Connexion with the invariants of the ?-function and with the theta-functions.- 2. Euler's theorem and Jacobi's proof.- 3. The transformation formula connecting ?(z) and ?(?½).- 4. Siegel's proof of Theorem 1.- 5. Connexion between ?(z) and the modular functions J(z), ?(z).- Notes on Chapter VIII.- IX. The law of quadratic reciprocity.- 1. Reciprocity of generalized Gaussian sums.- 2. Quadratic residues.- 3. The law of quadratic reciprocity.- Notes on Chapter IX.- X. The representation of a number as a sum of four squares ..- 1. The theorems of Lagrange and of Jacobi.- 2. Proof of Jacobi's theorem by means of theta-functions.- 3. Siegel's proof of Jacobi's theorem.- Notes on Chapter X.- XI. The representation of a number by a quadratic form.- 1. Positive-definite quadratic forms.- 2. Multiple theta-series and quadratic forms.- 3. Theta-functions associated to positive-definite forms.- 4. Representation of an even integer by a positive-definite form.- Notes on Chapter XI.- Chronological table.
I. Periods of meromorphic functions.- 1. Meromorphic functions.- 2. Periodic meromorphic functions.- 3. Jacobi's lemma.- 4. Elliptic functions.- 5. The modular group and modular functions.- Notes on Chapter I.- II. General properties of elliptic functions.- 1. The period parallelogram.- 2. Elementary properties of elliptic functions.- Notes on Chapter II.- III. Weierstrass's elliptic function ?(z).- 1. The convergence of a double series.- 2. The elliptic function ?(z).- 3. The differential equation associated with ?(z).- 4. The addition-theorem.- 5. The generation of elliptic functions.- Appendix I. The cubic equation.- Appendix II. The biquadratic equation.- Notes on Chapter III.- IV. The zeta-function and the sigma-function of Weierstrass.- 1. The function ?(z).- 2. The function ?(z).- 3. An expression for elliptic functions.- Notes on Chapter IV.- V. The theta-functions.- 1. The function ?(?, ?).- 2. The four sigma-functions.- 3. The four theta-functions.- 4. The differential equation.- 5. Jacobi's formula for ?' (0, ?).- 6. The infinite products for the theta-functions.- 7. Theta-functions as solutions of functional equations.- 8. The transformation formula connecting ?3(v, ?) and ?3(?, ?1/?) ..- Notes on Chapter V.- VI. The modular function J(?).- 1. Definition of J(?).- 2. The functions g2(?) and g3(?).- 3. Expansion of the function J(?) and the connexion with theta-functions.- 4. The function J(?) in a fundamental domain of the modular group ..- 5. Relations between the periods and the invariants of ?(u).- 6. Elliptic integrals of the first kind.- Notes on Chapter VI.- VII. The Jacobian elliptic functions and the modular function ?(?).- 1.The functions sn u, en u, dn u of Jacobi.- 2. Definition by theta-functions.- 3. Connexion with the sigma-functions.- 4. The differential equation.- 5. Infinite products for the Jacobian elliptic functions.- 6. Addition-theorems for sn u, cn u, dn u.- 7. The modular function ?(?).- 8. Mapping properties of ?(?) and Picard's theorem.- Notes on Chapter VII.- VIII. Dedekind's ?-function and Euler's theorem on pentagonal numbers.- 1. Connexion with the invariants of the ?-function and with the theta-functions.- 2. Euler's theorem and Jacobi's proof.- 3. The transformation formula connecting ?(z) and ?(?½).- 4. Siegel's proof of Theorem 1.- 5. Connexion between ?(z) and the modular functions J(z), ?(z).- Notes on Chapter VIII.- IX. The law of quadratic reciprocity.- 1. Reciprocity of generalized Gaussian sums.- 2. Quadratic residues.- 3. The law of quadratic reciprocity.- Notes on Chapter IX.- X. The representation of a number as a sum of four squares ..- 1. The theorems of Lagrange and of Jacobi.- 2. Proof of Jacobi's theorem by means of theta-functions.- 3. Siegel's proof of Jacobi's theorem.- Notes on Chapter X.- XI. The representation of a number by a quadratic form.- 1. Positive-definite quadratic forms.- 2. Multiple theta-series and quadratic forms.- 3. Theta-functions associated to positive-definite forms.- 4. Representation of an even integer by a positive-definite form.- Notes on Chapter XI.- Chronological table.
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"...In the breadth, depth and inevitability of treatment of this beautiful material, the author has made a contribution to the mathematical community consistent with the distinction of his career. That he has succeeded in compressing this treatment into a succinct monograph of fewer than 190 pages is a testament to his taste, discipline and powers of exposition."-- MATHEMATICAL REVIEWS
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