A First Course in Modular Forms

This book explains a result called the Modularity Theorem: All rational elliptic curves arise from modular forms. Taniyama?rstsuggestedinthe1950 sthatastatementalongtheselinesmight be true, and a precise conjecture was formulated by Shimura. A paper of Weil Wei67] provides strong theoretical eviden [...]
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This book explains a result called the Modularity Theorem: All rational elliptic curves arise from modular forms. Taniyama?rstsuggestedinthe1950 sthatastatementalongtheselinesmight be true, and a precise conjecture was formulated by Shimura. A paper of Weil Wei67] provides strong theoretical evidence for the conjecture. The theorem was proved for a large class of elliptic curves by Wiles Wil95] with a key ingredient supplied by joint work with Taylor TW95], completing the proof of Fermat s Last Theorem after some 350 years. The Modularity Theorem was proved completely by Breuil, Conrad, Taylor, and the ?rst author of this book BCDT01]. Di?erent forms of it are stated here in Chapters 2, 6, 7, 8, and 9. To describe the theorem very simply for now, ?rst consider a situation from elementary number theory. Take a quadratic equation 2 Q: x =d, d?Z, d =0, and for each prime numberp de?ne an integera (Q), p the number of solutionsx of equationQ a (Q)= ?1. p working modulop The valuesa (Q) extend multiplicatively to valuesa (Q) for all positive - p n tegersn, meaning thata (Q)=a (Q)a (Q) for allm andn."