elliptic curves

math 4800, fall 2018

tues, thurs 2pm - 3:20pm

jwb 308

rational points on elliptic curves, second edition, by silverman and tate

algebraic number theory, by jarvis

syllabus: pdf

problem sets

homework 1 due thursday, august 30

exercises A.1, A.2, A.3, A.4, A.6, A.8 in appendix A

homework 2 due thursday, sep 6

exercises A.5, A.9 (a) and (b), A.10, A.12, A.16(a), in appendix A

optional supplementary exercise: A.11 (do not turn in)

homework 3 due thursday, sep 13

exercises A.16(a) and (b), 1.1, 1.4, 1.5

homework 4 due thursday, sep 20

exercises 1.7, 1.8, 1.12, 1.13

homework 5 due thursday, sep 27

exercises 1.17, 1.19(d), 2.1, 2.2

homework 6 due thursday, oct 4

2.5(a) and (b), 2.6, 2.10

optional: 2.5(c), 2.3 (a must-do if you are excited about complex analysis)

homework 7 due thursday, oct 18

2.7, 2.8, optional: 2.11 (strong Nagell-Lutz)

for basic terminology on rings and ideals:

Clark's notes: chapter 1, sections 1,3,6,8

homework 8 due thursday, oct 25

4.2, 4.10 (hint: recall that the multiplicative group of a finite field is cyclic, see p. 122 of Silverman-Tate)

homework 9 and final project proposals due thursday, nov 1

Exercise 2.5 and 2.6 on p.25 of Jarvis (Chapter 2)

homework 10 due thursday, nov 8

You should be working on your final project. But if your final project involves algebraic number theory, try Exercise 4.16 on p. 80 of Jarvis.

final presentation schedule

nov 15 - Wei, Chris

nov 20 - Catherine, Stockton

nov 22 - Thanksgiving

nov 27 - John, Sarah

nov 29 - Winston, Wei

dec 4 - AJ, Sriram

dec 6 - finish proof of Mordell's theorem

final projects

AJ Bull:

Galois representations and elliptic curves

Sriram Gopalakrishnan:

modular forms and elliptic curves

John Ludlum:

the Hasse–Minkowski theorem

Sarah Marshall:

Mordell's theorem for curves with 2-torsion

Chris Phillips:

several algorithms using elliptic curves

Winston Stucki:

no torsion points of order 11 on elliptic curves over the rationals

Catherine Warner:

p-adic numbers and the Skolem–Mahler–Lech theorem

Yuhui Yao:

Riemann-Roch theorem for curves