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Overview"These notes constitute a lucid introduction to ""Elliptic Curves"", one of the central and vigorous areas of current mathematical research. The subject has been studied from diverse viewpoints --- analytic, algebraic and arithmetical; It offers the reader glimpses of all three aspects and presents some of the basic important theorems in all of them. The first part introduces a little of the theory of Riemann surfaces and goes on to the study of tori and their projective embeddings as cubics. This part ends with a discussion of the identification of the moduli space of complex tori with the quotient of the upper half plane by the modular groups. The second part handles the algebraic geometry of elliptic curves. It begins with a rapid introduction to some basic algebraic geometry and then focuses on elliptic curves. The Riemann-Roch theorem and the Riemann hypothesis for elliptic curves are proved and the structure of endomorphism ring of an elliptic curve are described. The third and last part is on the arithmetic of elliptic curves over $Q$. The Mordell-Weil theorem, Mazur's theorem on torsion in rational points of an elliptic curve over $Q$, theorems of Thue and Siegel are among the results which are presented. There is a brief discussion of theta functions, Eisenstein series and cusp forms with an application to representation of natural numbers as sums of squares. The notes end with the formulation of the Birch and Swinnerton-Dyer conjectures. There is an additional brief chapter (Appendix C) written in July 2004 by Kirti Joshi describing some developments." Full Product DetailsAuthor: R. V. Gurjar , Kirti Joshi , N. Mohan Kumar , et al.Publisher: Narosa Publishing House Imprint: Narosa Publishing House ISBN: 9788173195020ISBN 10: 8173195021 Pages: 370 Publication Date: 18 September 2006 Audience: College/higher education , Professional and scholarly , Undergraduate , Postgraduate, Research & Scholarly Format: Hardback Publisher's Status: Active Availability: Out of stock  The supplier is temporarily out of stock of this item. It will be ordered for you on backorder and shipped when it becomes available. Table of ContentsForeword / Preface / Part I: Analytic Theory of Elliptic Curves: Doubly Periodic Functions / Riemann Surfaces / Tori / Isomorphism of Tori and the j invariant / All Smooth Cubics are Complex Tori / Moduli / Part II: Algebraic Geometry of Elliptic Curves: Some Results From Commutative Algebra / Varieties / Further Properties of Varieties / Intersection Theory for Plane Curves / Geometry of Curves / Geometry of Elliptic Curves / Structure of Endomorphisms of Elliptic Curves / Part III: Arithmetic of Elliptic Curves: Rational Points on Curves / The Mordell-Weil Theorem / Computing the Mordell-Weil Group / The Theorems of Thue and Siegel / Representation of Numbers by Squares / The Conjecture of Birch and Swinnerton-Dyer / Part IV: Appendices / Bibliography / Index.ReviewsAuthor InformationKapil H. Paranjape.: Department of Mathematics The Institute of Mathematical Sciences, C.I.T. Campus, Taramani Chennai - 600113, India T. N. Shorey.: School of Mathematics Tata Institute of Fundamental Research, Colaba, Mumbai - 400005, India R. R. Simha.: Retired Professor, School of Mathematics Tata Institute of Fundamental Research, Colaba, Mumbai - 400005, India V. Srinivas.: School of Mathematics Tata Institute of Fundamental Research, Colaba, Mumbai - 400005, India R. V. Gurjar.: School of Mathematics Tata Institute of Fundamental Research, Colaba, Mumbai - 400005, India Kirti Joshi.: Department of Mathematics, University of Arizona 617 N. Santa Rita Ave., P.O. Box 210089, Tueson, AZ 85721 - 0089, USA N. Mohan Kumar.: Department of Mathematics Washington University in St. Louis, St. Louis, Missouri - 63130, USA Tab Content 6Author Website:Countries AvailableAll regions |
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