Elliptic curves, modular forms, and their L-functions

Elliptic curves, modular forms, and their L-functions /

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 un...

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Bibliographic Details
Main Author: Lozano-Robledo, Alvaro, 1978-
Format: Book
Language:English
Published: Providence, R.I. : Princeton, N.J. : American Mathematical Society ; Institute for Advanced Study, [2011]
Series:Student mathematical library ; v. 58.
Student mathematical library. IAS/Park City mathematical subseries.
Subjects:
Curves, Elliptic.
Forms, Modular.
L-functions.
Number theory.
Table of Contents:
  • 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.

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