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To give a rigorous course in elliptic curves.
This course is about elliptic curves and modular forms. Topics on elliptic curves include a general discussion of algebraic curves and maps between curves, the degree of a map, divisors and the Riemann-Roch Theorem, elliptic curves and the group law, isogenies. Elliptic curves over finite fields, proof of Hasse's Theorem, the Weil conjectures, zeta functions. The Tate module and Galois representations. Topics on modular forms include SL2(Z) and its congruence subgroups, modular forms for the congruence subgroups, cusp forms, the dimension of the space of modular forms of weight k, Hecke operators, theta functions. The proof of the conjecture of Taniyama-Shimura-Weil and its famour corollary (Fermat's Last Theorem) will be discussed.