Introduction to Elliptic Curves and Modular Forms
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.
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algebraic analytic change of variables completes the proof complex numbers compute congruence subgroup congruent number problem conjecture constant converges coordinates cusp form define definition denote derivative Dirichlet character dividing double coset eigenform Eisenstein series elements elliptic curve elliptic curve y2 elliptic function equal Euler product example expansion coefficient factor fixed follows form of weight formula Fourier functional equation fundamental domain Gauss sum geometric gives Hasse-Weil L-function Hecke operators hence holomorphic inner sum isomorphism L-series L(En lattice Lemma linear matrix Mellin transform Mk(N modular forms modular function modular points modulo nontrivial nonzero Note Number Theory obtain order of zero point at infinity points of order polynomial positive integer proof of Proposition Proposition 17 Prove rational numbers Recall replace residue right coset right triangle satisfies sides squarefree summation suppose theorem triple trivial Tunnell's vanishes write x-coordinate