Download or read book Theta Functions written by Jun-ichi Igusa and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book Algebraic Geometry and Theta Functions written by Arthur B. Coble and published by American Mathematical Soc.. This book was released on 1929-12-31 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the result of extending and deepening all questions from algebraic geometry that are connected to the central problem of this book: the determination of the tritangent planes of a space curve of order six and genus four, which the author treated in his Colloquium Lecture in 1928 at Amherst. The first two chapters recall fundamental ideas of algebraic geometry and theta functions in such fashion as will be most helpful in later applications. In order to clearly present the state of the central problem, the author first presents the better-known cases of genus two (Chapter III) and genus three (Chapter IV). The case of genus four is discussed in the last chapter. The exposition is concise with a rich variety of details and references.
Download or read book Theta Functions on Riemann Surfaces written by J. D. Fay and published by Springer. This book was released on 2006-11-15 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes present new as well as classical results from the theory of theta functions on Riemann surfaces, a subject of renewed interest in recent years. Topics discussed here include: the relations between theta functions and Abelian differentials, theta functions on degenerate Riemann surfaces, Schottky relations for surfaces of special moduli, and theta functions on finite bordered Riemann surfaces.
Download or read book Riemann Surfaces and Generalized Theta Functions written by Robert C. Gunning and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: The investigation of the relationships between compact Riemann surfaces (al gebraic curves) and their associated complex tori (Jacobi varieties) has long been basic to the study both of Riemann surfaces and of complex tori. A Riemann surface is naturally imbedded as an analytic submanifold in its associated torus; and various spaces of linear equivalence elasses of divisors on the surface (or equivalently spaces of analytic equivalence elasses of complex line bundies over the surface), elassified according to the dimensions of the associated linear series (or the dimensions of the spaces of analytic cross-sections), are naturally realized as analytic subvarieties of the associated torus. One of the most fruitful of the elassical approaches to this investigation has been by way of theta functions. The space of linear equivalence elasses of positive divisors of order g -1 on a compact connected Riemann surface M of genus g is realized by an irreducible (g -1)-dimensional analytic subvariety, an irreducible hypersurface, of the associated g-dimensional complex torus J(M); this hyper 1 surface W- r;;;, J(M) is the image of the natural mapping Mg- -+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold symmetric product Mg- jSg-l of the Riemann surface M.
Download or read book Introduction to Algebraic and Abelian Functions written by Serge Lang and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to Algebraic and Abelian Functions is a self-contained presentation of a fundamental subject in algebraic geometry and number theory. For this revised edition, the material on theta functions has been expanded, and the example of the Fermat curves is carried throughout the text. This volume is geared toward a second-year graduate course, but it leads naturally to the study of more advanced books listed in the bibliography.
Download or read book A Treatise on the Theory of Functions written by James Harkness and published by . This book was released on 1893 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Theta Functions Bowdoin 1987 Part 2 written by Leon Ehrenpreis and published by American Mathematical Soc.. This book was released on 1989 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Introduction to Algebraic Independence Theory written by Yuri V. Nesterenko and published by Springer. This book was released on 2003-07-01 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.
Download or read book Algebraic and Geometric Methods in Mathematical Physics written by Anne Boutet de Monvel and published by Springer Science & Business Media. This book was released on 1996-01-31 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the Kaciveli Summer School, Crimea, Ukraine, 1993
Download or read book International Catalogue of Scientific Literature written by and published by . This book was released on 1907 with total page 2040 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Revue Semestrielle Des Publications Mathematiques written by and published by . This book was released on 1899 with total page 760 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Advances on Superelliptic Curves and Their Applications written by L. Beshaj and published by IOS Press. This book was released on 2015-07-16 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book had its origins in the NATO Advanced Study Institute (ASI) held in Ohrid, Macedonia, in 2014. The focus of this ASI was the arithmetic of superelliptic curves and their application in different scientific areas, including whether all the applications of hyperelliptic curves, such as cryptography, mathematical physics, quantum computation and diophantine geometry, can be carried over to the superelliptic curves. Additional papers have been added which provide some background for readers who were not at the conference, with the intention of making the book logically more complete and easier to read, but familiarity with the basic facts of algebraic geometry, commutative algebra and number theory are assumed. The book is divided into three sections. The first part deals with superelliptic curves with regard to complex numbers, the automorphisms group and the corresponding Hurwitz loci. The second part of the book focuses on the arithmetic of the subject, while the third addresses some of the applications of superelliptic curves.
Download or read book Modular Functions of One Variable VI written by J.-P. Serre and published by Springer. This book was released on 2006-11-15 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: The proceedings of the conference are being published in two parts, and the present volume is mostly algebraic (congruence properties of modular forms, modular curves and their rational points, etc.), whereas the second volume will be more analytic and also include some papers on modular forms in several variables.
Download or read book Explorations in Complex Functions written by Richard Beals and published by Springer Nature. This book was released on 2020-10-19 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook explores a selection of topics in complex analysis. From core material in the mainstream of complex analysis itself, to tools that are widely used in other areas of mathematics, this versatile compilation offers a selection of many different paths. Readers interested in complex analysis will appreciate the unique combination of topics and connections collected in this book. Beginning with a review of the main tools of complex analysis, harmonic analysis, and functional analysis, the authors go on to present multiple different, self-contained avenues to proceed. Chapters on linear fractional transformations, harmonic functions, and elliptic functions offer pathways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions lead into L-functions, while a chapter on entire functions opens pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give rise to Hilbert and Fourier transforms, with an emphasis on the connection to complex analysis. Valuable additional topics include Riemann surfaces, steepest descent, tauberian theorems, and the Wiener–Hopf method. Showcasing an array of accessible excursions, Explorations in Complex Functions is an ideal companion for graduate students and researchers in analysis and number theory. Instructors will appreciate the many options for constructing a second course in complex analysis that builds on a first course prerequisite; exercises complement the results throughout.
Download or read book On the Direct Numerical Calculation of Elliptic Functions and Integrals written by Louis Vessot King and published by . This book was released on 1924 with total page 60 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Current Topics in Complex Algebraic Geometry written by Charles Herbert Clemens and published by Cambridge University Press. This book was released on 1995 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 1992/93 academic year at the Mathematical Sciences Research Institute was devoted to complex algebraic geometry. This volume collects survey articles that arose from this event, which took place at a time when algebraic geometry was undergoing a major change. The editors of the volume, Herbert Clemens and János Kollár, chaired the organizing committee. This book gives a good idea of the intellectual content of the special year and of the workshops. Its articles represent very well the change of direction and branching out witnessed by algebraic geometry in the last few years.
Download or read book Infinite Families of Exact Sums of Squares Formulas Jacobi Elliptic Functions Continued Fractions and Schur Functions written by Stephen C. Milne and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of representing an integer as a sum of squares of integers is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. Here, the author employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or 4n(n+1) squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for 16 and 24 squares. These results depend upon new expansions for powers of various products of classical theta functions. This is the first time that infinite families of non-trivial exact explicit formulas for sums of squares have been found. The author derives his formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. His results (in Theorem 5.19) generalize to separate infinite families each of the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions in sections 40-42 of the Fundamental Nova. The author also uses a special case of his methods to give a derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of 4n2 or 4n(n+1) triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. George Andrews says in a preface of this book, `This impressive work will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.' Audience: This research monograph on sums of squares is distinguished by its diversity of methods and extensive bibliography. It contains both detailed proofs and numerous explicit examples of the theory. This readable work will appeal to both students and researchers in number theory, combinatorics, special functions, classical analysis, approximation theory, and mathematical physics.
