منحنی‌های بیضوی، فرم‌های مدولار و تابع‌های L آن‌ها (کتابخانه ریاضی دانشجویی، جلد ۵۸)

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory. Many Problems In Number Theory Have Simple Statements, But Their Solutions Require A Deep Understanding Of Algebra, Algebraic Geometry, Complex Analysis, Group Representations, Or A Combination Of All Four. The Original Simply Stated Problem Can Be Obscured In The Depth Of The Theory Developed To Understand It. This Book Is An Introduction To Some Of These Problems, And An Overview Of The Theories Used Nowadays To Attack Them, Presented So That The Number Theory Is Always At The Forefront Of The Discussion. Machine Generated Contents Note: Ch. 1 Introduction -- 1.1. Elliptic Curves -- 1.2. Modular Forms -- 1.3. L-functions -- 1.4. Exercises -- Ch. 2 Elliptic Curves -- 2.1. Why Elliptic Curves? -- 2.2. Definition -- 2.3. Integral Points -- 2.4. The Group Structure On E(q) -- 2.5. The Torsion Subgroup -- 2.6. Elliptic Curves Over Finite Fields -- 2.7. The Rank And The Free Part Of E(q) -- 2.8. Linear Independence Of Rational Points -- 2.9. Descent And The Weak Mordell-weil Theorem -- 2.10. Homogeneous Spaces -- 2.11. Selmer And Sha -- 2.12. Exercises -- Ch. 3 Modular Curves -- 3.1. Elliptic Curves Over C -- 3.2. Functions On Lattices And Elliptic Functions -- 3.3. Elliptic Curves And The Upper Half-plane -- 3.4. The Modular Curve X(1) -- 3.5. Congruence Subgroups -- 3.6. Modular Curves -- 3.7. Exercises -- Ch. 4 Modular Forms -- 4.1. Modular Forms For The Modular Group -- 4.2. Modular Forms For Congruence Subgroups -- 4.3. The Petersson Inner Product. 4.4. Hecke Operators Acting On Cusp Forms -- 4.5. Exercises -- Ch. 5 L-functions -- 5.1. The L-function Of An Elliptic Curve -- 5.2. The Birch And Swinnerton-dyer Conjecture -- 5.3. The L-function Of A Modular (cusp) Form -- 5.4. The Taniyama-shimura-weil Conjecture -- 5.5. Fermat's Last Theorem -- 5.6. Looking Back And Looking Forward -- 5.7. Exercises -- Appendix A Pari/gp And Sage -- A.1. Elliptic Curves -- A.2. Modular Forms -- A.3. L-functions -- A.4. Other Sage Commands -- Appendix B Complex Analysis -- B.1. Complex Numbers -- B.2. Analytic Functions -- B.3. Meromorphic Functions -- B.4. The Complex Exponential Function -- B.5. Theorems In Complex Analysis -- B.6. Quotients Of The Complex Plane -- B.7. Exercises -- Appendix C Projective Space -- C.1. The Projective Line -- C.2. The Projective Plane -- C.3. Over An Arbitrary Field -- C.4. Curves In The Projective Plane -- C.5. Singular And Smooth Curves -- Appendix D The P-adic Numbers -- D.1. Hensel's Lemma -- D.2. Exercises -- Appendix E Parametrization Of Torsion Structures. Ávaro Lozano-robledo. Includes Bibliographical References (p. 189-192) And Index. Br> Elliptic Curves, Modular Forms, and Their L-Functions by Lozano-Robledo, Alvaro Terms of use The text grew out of lecture notes for a course Lozano-Robledo taught at an undergraduate summer school as part of the 2009 Park City Mathematics Institute. It is an introductory survey of the theory of elliptic curves, modular forms, and their Lfunctions, emphasizing examples rather than proofs. His goal is to provide a big picture of the surprising connections among these three types of mathematical objects, which seem so distinct. One theme is the statement of the modularity theorem (nee Taniyuama-Shimura-Weil conjecture), and one of its most renowned consequences, Fermat's last theorem. Annotation ©2011 Book News, Inc., Portland, OR (booknews.com) Descriptive content provided by Syndetics"! a Bowker service An introductory survey of elliptic curves, modular forms, and $L$-functions. The main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory.