Jen Paulhus, Ph.D.

Elliptic Curves

email: paulhus@math.grinnell.edu     pgp key
Office Hours: M: 11:00 AM - 12:00 PM
                       T: 1:30 PM -3:00 PM
                       R: 9:30 AM-11:00 AM
Office: 2519 Noyce Science Center

Text: Rational Points on Elliptic Curves, 2nd edition, Joseph Silverman and John Tate
Material Covered: most of the book

Syllabus

Homework Assignments

Make sure you are familiar with the Grading Policies and Rubric for the class. Homework solutions will be posted on Pweb.

HW Due Date Problems
# 0 Jan. 25 Email me with answers from class 1/23, and read syllabus
# 1 Feb. 1 pg. 28-30: 1.1, 1.5, 1.7, 1.10, 1.11 (a)-(d)     pg. 305: A.2
# 2 Feb. 8 pg. 305-307: A.1, A.6, A.8 (class 2/4 might help with terminology)
# 3 Feb. 15 pg. 32-33: 1.17, 1.18 (a)-(c)*, 1.19 (a)-(b). 1.20 pg. 306-307: A.5, A.9
# 4 Feb. 22 pg. 58-62: 2.1, 2.5 (a)-(b), (c)**, 2.6, 2.9, 2.12* ***, Research Assignment #1***
Mar. 1 R.A. #2***
#5 Mar.4 pg.111: 3.1(a),(b)**,3.2(a)-(c), additional problems on PWeb
Mar.8
 R.A. #3***
#6 Mar. 15 2.3 (b), 4.1, 4.3 (a)-(b), 4.5(a)*, additional problems on PWeb, R.A. #4***
Apr. 12 R.A. #5***
#7a Apr. 19 3.6 (a), (b)**, R.A. #6***
#7b Apr. 26 6.8, 6.20 (a)-(b) R.A. #7***
May 3 R.A. #8***
#7c May 6
 TBA
May 10
 Final Project Due 5 PM
* to be done via Magma
** bonus
*** see PWeb for additional instructions

Daily Topics

Date Pages Topic(s)
1/23 xv-xxii, 1-4 (middle) Introduction and Rational Points on Conics
1/25 6 (middle) -12 Rational Points on Cubics, Intro to Group Structure
1/28 13-16, 265-270 Prove Group Structure, Introduction to Projective Geometry*
1/30 Canceled Class
2/1 270-272 Projective Space, Homogeneous Polynomials*
2/4 273-278 Curves in the Projective Plane, Singularities
2/6 16-20 Weierstrass Normal Form
2/8 23-27 Explicit Formulas for Group Law
2/11 35-38 Points of Finite Order
2/13 45-47, 56-57 Discriminant, Nagell-Lutz Theorem (2)
2/15 48-50, 54-56 Nagell-Lutz Theorem (1)
2/18 51-53, 58 Nagell-Lutz finished **
2/20 65-68 Heights, 4 Lemmas
2/22 68-71 Proof of Mordell's Theorem
2/25 73-75 Lemma 3.2
2/27 71-72, 95-96 (sub)Lemma, Structure of Γ/2Γ
3/1 97 Structure of Γ[2], Complex Field and Complex Functions
3/4 41-42 Complex Poles, Lattices
3/6 43-45 Elliptic Curves over C, Finite Fields
3/8 117-121 Elliptic Curves over Finite Fields
3/11 133-138, 207-208 Using Finite Fields to Identify Points of Finite Order, Extensions of Q
3/13 209, 213-218 Cyclotomic Extensions, Embeddings, Algebraic Points on Cubic Curves
3/15 230-232 More on Algebraic Points on Cubic Curves, Complex Multiplication
4/1 75-76, 78-80 Lemma 3.3
4/3 78-80 Lemma 3.3 continued
4/5 No class: Midterm due
4/8 94, 80-83 Start Lemma 3.4
4/10 Research project discussion
4/12 86-88 More Lemma 3.4
4/15 89-94 Finish Lemma 3.4!!!!!!!!!!
4/17 Citation in math and bibtex
4/19 230-231 j-invariant and complex multiplication
4/22 232-235, 245-247 complex multiplication
4/24 How to give a bad math talk
4/26 Fermat's Last Theorem
4/29 Fermat's Last Theorem

* Some material from Chapter 8 of Ideals, Varieties, and Algorithms by Cox, Little, O'Shea.
** See PWeb for additional notes.