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Lozano-Robledo, Alvaro, 1978-
Elliptic curves, modular forms, and their L-functions / Ávaro Lozano-Robledo
Providence, R.I. : American Mathematical Society ; Princeton, N.J. : Institute for Advanced Study, c2011
book jacket
Location Call Number Status
 4th Floor  QA567.2.E44 L69 2011    AVAILABLE
Subject(s) Curves, Elliptic
Forms, Modular
L-functions
Number theory
Physical Description xiv, 195 p. : ill. ; 22 cm
Note Includes bibliographical references (p.189-192) and index
Contents 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
Series Student mathematical library ; v. 58
Student mathematical library. IAS/Park City mathematical subseries

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