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000437918 010__ $$z  2011044712
000437918 020__ $$a9781400841714$$q(electronic bk.)
000437918 020__ $$z9780691151199
000437918 035__ $$a(OCoLC)ocn773567194
000437918 035__ $$a(CaPaEBR)ebr10527170
000437918 035__ $$a437918
000437918 040__ $$aCaPaEBR$$beng$$cCaPaEBR
000437918 05014 $$aQA343$$b.A97 2012eb
000437918 08204 $$a515/.983$$223
000437918 1001_ $$aAsh, Avner,$$d1949-
000437918 24510 $$aElliptic tales$$h[electronic resource] :$$bcurves, counting, and number theory /$$cAvner Ash, Robert Gross.
000437918 260__ $$aPrinceton :$$bPrinceton University Press,$$c2012.
000437918 300__ $$a1 online resource (xvii, 253 p.) :$$bill.
000437918 504__ $$aIncludes bibliographical references and index.
000437918 5050_ $$aMachine generated contents note: pt. I DEGREE -- ch. 1 Degree of a Curve -- 1.Greek Mathematics -- 2.Degree -- 3.Parametric Equations -- 4.Our Two Definitions of Degree Clash -- ch. 2 Algebraic Closures -- 1.Square Roots of Minus One -- 2.Complex Arithmetic -- 3.Rings and Fields -- 4.Complex Numbers and Solving Equations -- 5.Congruences -- 6.Arithmetic Modulo a Prime -- 7.Algebraic Closure -- ch. 3 The Projective Plane -- 1.Points at Infinity -- 2.Projective Coordinates on a Line -- 3.Projective Coordinates on a Plane -- 4.Algebraic Curves and Points at Infinity -- 5.Homogenization of Projective Curves -- 6.Coordinate Patches -- ch. 4 Multiplicities and Degree -- 1.Curves as Varieties -- 2.Multiplicities -- 3.Intersection Multiplicities -- 4.Calculus for Dummies -- ch. 5 Bezout's Theorem -- 1.A Sketch of the Proof -- 2.An Illuminating Example -- pt. II ELLIPTIC CURVES AND ALGEBRA -- ch. 6 Transition to Elliptic Curves -- ch. 7 Abelian Groups -- 1.How Big Is Infinity? -- 2.What Is an Abelian Group? -- 3.Generations -- 4.Torsion -- 5.Pulling Rank -- Appendix: An Interesting Example of Rank and Torsion -- ch. 8 Nonsingular Cubic Equations -- 1.The Group Law -- 2.Transformations -- 3.The Discriminant -- 4.Algebraic Details of the Group Law -- 5.Numerical Examples -- 6.Topology -- 7.Other Important Facts about Elliptic Curves -- 5.Two Numerical Examples -- ch. 9 Singular Cubics -- 1.The Singular Point and the Group Law -- 2.The Coordinates of the Singular Point -- 3.Additive Reduction -- 4.Split Multiplicative Reduction -- 5.Nonsplit Multiplicative Reduction -- 6.Counting Points -- 7.Conclusion -- Appendix A Changing the Coordinates of the Singular Point -- Appendix B Additive Reduction in Detail -- Appendix C Split Multiplicative Reduction in Detail -- Appendix D Nonsplit Multiplicative Reduction in Detail -- ch. 10 Elliptic Curves over Q -- 1.The Basic Structure of the Group -- 2.Torsion Points -- 3.Points of Infinite Order -- 4.Examples -- pt. III ELLIPTIC CURVES AND ANALYSIS -- ch. 11 Building Functions -- 1.Generating Functions -- 2.Dirichlet Series -- 3.The Riemann Zeta-Function -- 4.Functional Equations -- 5.Euler Products -- 6.Build Your Own Zeta-Function -- ch. 12 Analytic Continuation -- 1.A Difference that Makes a Difference -- 2.Taylor Made -- 3.Analytic Functions -- 4.Analytic Continuation -- 5.Zeroes, Poles, and the Leading Coefficient -- ch. 13 L-functions -- 1.A Fertile Idea -- 2.The Hasse-Weil Zeta-Function -- 3.The L-Function of a Curve -- 4.The L-Function of an Elliptic Curve -- 5.Other L-Functions -- ch. 14 Surprising Properties of L-functions -- 1.Compare and Contrast -- 2.Analytic Continuation -- 3.Functional Equation -- ch. 15 The Conjecture of Birch and Swinnerton-Dyer -- 1.How Big Is Big? -- 2.Influences of the Rank on the Np's -- 3.How Small Is Zero? -- 4.The BSD Conjecture -- 5.Computational Evidence for BSD -- 6.The Congruent Number Problem -- EPILOGUE -- Retrospect -- Where Do We Go from Here?.
000437918 506__ $$aAccess limited to authorized users.
000437918 520__ $$aElliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.  The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from some very deep--and often very mystifying--mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and in the process venture to the very frontiers of modern mathematics. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.
000437918 588__ $$aDescription based on print version record.
000437918 650_0 $$aElliptic functions.
000437918 650_0 $$aCurves, Elliptic.
000437918 650_0 $$aNumber theory.
000437918 7001_ $$aGross, Robert,$$d1959-
000437918 77608 $$iPrint version:$$aAsh, Avner, 1949-$$tElliptic tales.$$dPrinceton : Princeton University Press, 2012$$z9780691151199$$w(DLC) 2011044712$$w(OCoLC)761850914
000437918 8520_ $$bacq
000437918 85280 $$bebk$$hProquest Ebook Central
000437918 85640 $$3ProQuest Ebook Central$$uhttps://univsouthin.idm.oclc.org/login?url=https://ebookcentral.proquest.com/lib/usiricelib-ebooks/detail.action?docID=843813$$zOnline Access
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000437918 980__ $$aEBOOK
000437918 980__ $$aBIB
000437918 982__ $$aEbook
000437918 983__ $$aOnline