


Elliptic Curves. (MN40) 
An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjectselliptic curves and modular formscome together in EichlerShimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the TaniyamaWeil Conjecture and is known to imply Fermat's Last Theorem. Elliptic curves and the modeular forms in the Eichler Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematicsincluding class field theory, arithmetic algebraic geometry, and group representationsin which the concidence of L functions relates analysis and algebra in the most fundamental ways. Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern EichlerShimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains EichlerShimura theory in such an accessible manner. Series:
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