Author | Koblitz, Neal. author |
---|---|
Title | Introduction to Elliptic Curves and Modular Forms [electronic resource] / by Neal Koblitz |
Imprint | New York, NY : Springer US, 1984 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0255-1 |
Descript | online resource |
I From Congruent Numbers to Elliptic Curves -- 1. Congruent numbers -- 2. A certain cubic equation -- 3. Elliptic curves -- 4. Doubly periodic functions -- 5. The field of elliptic functions -- 6. Elliptic curves in Weierstrass form -- 7. The addition law -- 8. Points of finite order -- 9. Points over finite fields, and the congruent number problem -- II The Hasse-Weil L-Function of an Elliptic Curve -- 1. The congruence zeta-function -- 2. The zeta-function of En -- 3. Varying the prime p -- 4. The prototype: the Riemann zeta-function -- 5. The Hasse-Weil L-function and its functional equation -- 6. The critical value -- III Modular forms -- 1. $$S {{L}_{2}}(\mathbb{Z}) $$ and its congruence subgroups -- 2. Modular forms for $$ S{{L}_{2}}(\mathbb{Z}) $$ -- 3. Modular forms for congruence subgroups -- 4. Transformation formula for the theta-function -- 5. The modular interpretation, and Hecke operators -- IV Modular Forms of Half Integer Weight -- 1. Definitions and examples -- 2. Eisenstein series of half integer weight for $$ {{\tilde{\Gamma }}_{0}}(4) $$ -- 3. Hecke operators on forms of half integer weight -- 4. The theorems of Shimura, Waldspurger, Tunnell, and the congruent number problem -- Answers, Hints, and References for Selected Exercises