ELLIPTIC CURVES AND MODULAR FORMS Contents
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1. January 21, 2010 2 1.1. Why define a curve to be f rather than V (f) ⊂ P(k)? 3 1.2. Cubic plane curves 3 2. January 26, 2010 4 2.1. A little bit about smoothness 4 2.2. Weierstrass form 5 3. January 28, 2010 6 3.1. An algebro-geometric description of the group law in terms of divisors 6 3.2. Why are the two group laws the same? 7 4. February 2, 2010 7 4.1. Overview 7 4.2. Uniqueness of Weierstrass form 8 5. February 4, 2010 9 5.1. The invariant differential 9 5.2. First glimpse of modular forms and modular functions 11 6. February 9, 2010 11 6.1. Isogenies 11 7. February 11, 2010 13 7.1. Galois theory of isogenies 13 7.2. Isogenies and the invariant differential 15 7.3. The dual isogeny 15 8. February 16, 2010 15 8.1. Tate module and Galois representations 17 9. February 18, 2010 17 9.1. The Weil pairing 17 9.2. The Tate module 19 10. February 23, 2010 19 11. February 25, 2010 21 11.1. The endomorphism ring 21 11.2. Elliptic curves over finite fields 23 12. March 2, 2010 23 12.1. The Weil conjectures 24 13. March 9, 2010 25 13.1. Ordinariness and supersingularity 25 14. March 11, 2010 27 14.1. Elliptic curves over C (What is E(C)?) 27 14.2. The space of lattices 29 15. March 16, 2010 29 15.1. Level structures 30
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