Elliptic Curves, Modular Forms, and Their L-functions (Student Mathematical Library, Vol. 58)

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. Preface......Page 8 1.1. Elliptic Curves......Page 12 1.2. Modular Forms......Page 18 1.3. L-functions......Page 22 1.4. Exercises......Page 25 2.1. Why elliptic curves?......Page 28 2.2. Definition......Page 31 2.3. The group structure on E(Q)......Page 34 2.4. The torsion subgroup......Page 42 2.5. Elliptic curves over finite fields......Page 45 2.6. The rank and the free part of E(Q)......Page 52 2.7. Linear independence of rational points......Page 56 2.8. Descent and the weak Mordell-Weil theorem......Page 59 2.9. Homogeneous spaces......Page 69 2.10. Selmer and Sha......Page 76 2.11. Exercises......Page 78 3.1. Elliptic curves over C......Page 86 3.2. Functions on lattices and elliptic functions......Page 90 3.3. Elliptic curves and the upper-half plane......Page 93 3.4. The modular curve X(1)......Page 96 3.5. Congruence subgroups......Page 98 3.6. Modular curves......Page 99 3.7. Exercises......Page 102 4.1. Modular forms for the modular group......Page 106 4.2. Modular forms for congruence subgroups......Page 111 4.3. The Petersson inner product......Page 115 4.4. Hecke operators acting on cusp forms......Page 117 4.5. Exercises......Page 123 5.1. The L-function of an elliptic curve......Page 128 5.2. The Birch and Swinnerton-Dyer conjecture......Page 132 5.3. The L-function of a modular (cusp) form......Page 140 5.4. The Taniyama-Shimura-Weil conjecture......Page 141 5.5. Fermat's last theorem......Page 145 5.6. Exercises......Page 146 Appendix A. PARI/GP and SAGE......Page 150 A.1. Elliptic Curves......Page 151 A.2. Modular Forms......Page 156 A.3. L-functions......Page 158 A.4. Other SAGE commands......Page 160 Appendix B. The complex exponential function......Page 162 C.1. The projective line......Page 164 C.2. The projective plane......Page 166 C.4. Curves in the projective plane......Page 167 C.5. Singular and smooth curves......Page 169 Appendix D. The p-adic numbers......Page 172 D.1. Hensel's Lemma......Page 174 D.2. exercises......Page 175 Bibliography......Page 178 Index......Page 180 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.