Introduction to Elliptic Curves and Modular FormsSpringer Science & Business Media, 29. 4. 1993. - 248 страница This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments. At the same time I want to introduce topics which are at the forefront of current research. Down-to-earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book. With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra. Such students would learn applications of techniques from those courses. thereby solidifying their under standing of some basic tools used throughout mathematics. Graduate stu dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced under graduates could use the book for independent study projects, senior theses, and seminar work. |
Садржај
CHAPTER | 1 |
Congruent numbers | 3 |
A certain cubic equation | 6 |
Elliptic curves | 9 |
Doubly periodic functions 40221 | 14 |
The field of elliptic functions | 18 |
Elliptic curves in Weierstrass form | 22 |
The addition law | 29 |
The HasseWeil Lfunction and its functional equation | 79 |
The critical value | 90 |
CHAPTER III | 98 |
Modular forms for SL2Z | 108 |
Modular forms for congruence subgroups | 124 |
Transformation formula for the thetafunction | 147 |
The modular interpretation and Hecke operators | 153 |
CHAPTER IV | 176 |
Points of finite order | 36 |
Points over finite fields and the congruent number problem | 43 |
CHAPTER II | 51 |
The zetafunction of E | 57 |
Varying the prime p | 64 |
the Riemann zetafunction | 70 |
Eisenstein series of half integer weight for Ĩ4 | 185 |
Hecke operators on forms of half integer weight | 202 |
The theorems of Shimura Waldspurger Tunnell and the congruent | 212 |
Answers Hints and References for Selected Exercises | 223 |
240 | |
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a₁ algebraic b₁ change of variables compute congruence subgroup conjecture converges corresponding cusp form define definition denote Dirichlet character double coset e₁ eigenform Eisenstein series elements elliptic curve elliptic curve y² elliptic function equal Euler product example field finite follows form of weight formula Fourier functional equation fundamental domain geometric Hasse-Weil L-function Hecke operators holomorphic integer weight isomorphism L-series lattice Lemma linear m₁ Mellin transform modular forms modular function modular points modulo multiple n₁ n²x nontrivial nonzero Number Theory obtain P₁ P₂ point at infinity points of order pole polynomial positive integer prime proof of Proposition Prove q-expansion q-expansion coefficients quadratic rational numbers replace right coset right triangle root satisfies Shimura square squarefree suppose theorem w₁ w₂ x₁ y₁ Z/NZ z₁ zero zeta-function