18.783 - Elliptic Curves


# Date Topic (references) Materials
22/11The group law, Weierstrass and Edwards equations (Washington 2.1-3, 2.6.3, Bernstein-Lange)notes, worksheet 1, worksheet 2
32/13Finite fields and integer arithmetic (Modern Computer Algebra Ch. 8)notes
42/19Finite field arithmetic (Modern Computer Algebra: Sec. 3.2, 9.1, 11.1, HEHCC Ch. 9, Rabin)notes
52/20Isogenies (Washington 2.9, Silverman III.4)notes
62/25Isogeny kernels and division polynomials (Washington 3.2, 12.3, Silverman III.4))notes, worksheet
72/27Endomorphism rings (Washington 4.2, , Silverman III.6)notes
83/4Hasse's Theorem, point counting (Washington 4.3)notes
93/6Schoof's algorithm (Washington 4.2, 4.5, Schoof)notes, worksheet
103/11Generic algorithms for discrete logarithms (Washington 5.2, Pohlig-Hellman, Pollard, Shoup)notes
113/13Index calculus, smooth numbers, factoring integers (Washington 5.1, 7.1, Granville, Lenstra)notes, worksheet 1, worksheet 2, worksheet 3
123/15Elliptic curve primality proving (ECPP) (Washington 7.2, Goldwasser-Kilian, Pomerance)notes
133/22Endomorphism algebras (Silverman III.9)notes
144/1Ordinary and supersingular curves (Silverman III.1,V, Washington 2.7, 4.6)notes
154/3Elliptic curves over C (part I) (Cox Sec. 10, Silverman VI.2-3, Washington 9.1-2)notes
164/8Elliptic curves over C (part II) (Cox Sec. 10-11, Silverman VI.4-5, Washington 9.2-3)notes
174/10Complex multiplication (CM) (Cox Sec. 11, Silverman VI.5, Washington 9.3)notes
184/17The CM torsor (Cox Sec. 7, Silverman (advanced topics) II.1.1)notes
194/22Riemann surfaces and modular curves (Silverman (advanced topics) I.2, Milne V.1)notes
204/24The modular equation (Cox Sec. 11, Milne V.2, Washington pp. 273-274)notes
214/29The Hilbert class polynomial (Cox Sec. 8, 11)notes
225/1Ring class fields and the CM method (Cox 8, 11)notes
235/6Isogeny volcanoes (Sutherland)notes
245/8The Weil pairing (Miller, Washington 11, Silverman III.8)notes
255/13Modular forms and L-series (Milne V.3-4)notes
265/15Fermat's last theorem (Milne V.7-9, Washington 15, Cornell-Stevens-Silverman I)notes