Number Theory – Mathematics 2540
Brown University – Spring, 2017
Professor Joseph Silverman

Topic We will study elliptic curves, primarily from a number theoretic viewpoint.
Text I'll be covering material from:
The Arithmetic of Elliptic Curves , Joseph H. Silverman, Springer-Verlag, ISBN: 978-0-387-09493-9 – 2nd edition, © 2009. ($59.95 hardcover, $47.36 electronic)
If time permits, we'll do some additional topics from the second volume.
Office Mathematics Department, Kassar House, Room 202
Phone 863-1124
Email jhs@math.brown.edu
Web Site www.math.brown.edu/~jhs/MA0254/MA0254HomePage.html
Office Hours By appointment, send me an email and I'll find a time we can meet. I'm generally on campus on MWF, seldom on TTh.)
Course Time MWF 10:00–10:50am (C hour)
Course Location Kassar 105
Homework Homework assignments are posted below.
Note on Using Computers in Math 254 I tend to use a computer program called PARI-GP to do number theory and elliptic curve calculations. The good news about PARI is that it is free and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly. You can download PARI by clicking here. Another way to use PARI to do short calculations is to use the SAGE web site. You'll need to create a (free) account. Then you'll be able to type one or more PARI commands and type Shift-Return to perform the computation. Or you can use SAGE itself, which has a large collection of algorithms for working with number fields, local fields, and elliptic curves.

Schedule, Reading Assignments, HW Assignments
Week Chapter Sections Topic HW to turn in / Due date Problems to look at
1 Jan 25–27 I §§1.1–1.3 Algebraic Varieties # 1.1(b), 1.4, 1.6. 1.7, 1.8, 1.10 / Mon Jan 30 # 1.2, 1.3, 1.11, 1.12
II §§2.1–2.2 Algebraic Curves
2 Jan 30–Feb 3 II §§2.3–2.5 Algebraic Curves # 2.3, 2.4, 2.8 / Mon Feb 6 # 2.1, 2.6, 2.7, 2.10, 2.11, 2.14
III §3.3 Geometry of Elliptic Curves
3 Feb 6–Feb 10 III §§3.1, 3.2 Geometry of Elliptic Curves # 3.4, 3.5, 3.6(a,b), 3.8 / Weds Feb 22 # 3.3, 3.7, 3.9, 3.10
4 Feb 13 VI §6.1, §6.2 (start) Elliptic Curves over C
Lecture by Minsik Han
Feb 15 VI §6.2 (finish), §6.3 Elliptic Curves over C
Lecture by Thomas Silverman
Feb 17 VI §6.4, §6.5 Elliptic Curves over C
Lecture by Seoyoung Kim
# 6.1, 6.2, 6.3(a,b), 6.6 / Weds Feb 22
5 Feb 22–Feb 24 III §§3.4, 3.5 Geometry of Elliptic Curves
6 Feb 27–Mar 3 III §§3.6, 3.7, 3.8 Geometry of Elliptic Curves # 3.12, 3.14, 3.15, 3.20, 3.24 / Mon Mar 6 # 3.16, 3.23
7 Mar 6–Mar 10 V §§5.1, 5.2 Elliptic Curves over Finite Fields # 5.1, 5.3, 5.4(a), 5.12, 5.13 / Mon Mar 13 # 5.2, 5.14
8 Mar 13–Mar 17 VI §§6.1–6.4 Elliptic Curves over Local Fields # 7.2, 7.5(c), 7.7(a,b), 7.8, 7.15(a,b) / Mon Mar 20
9 Mar 20–Mar 24 VIII §§8.1–8.3, 8.5 Elliptic Curves over Global Fields
10 Apr 3–Apr 7 VIII §§8.5, 8.6, 8.9 Elliptic Curves over Global Fields # 8.1, 8.2, 8.6, 8.7(b), 8.8(c,d) / Mon Apr 10
11 Apr 10–Apr 14 VIII, C, X §§ 8.9, C.16, 10.1 Elliptic Curves over Global Fields # 8.10, 8.13(a,b), 8.17, 8.19(a,c) / Mon Apr 17
11 Apr 17–Apr 21 X, B §§ 10.4, 10.5, B.1, B.2 Computing the Mordell-Weil Group # 10.13, 10.16, 10.19, 10.20 / Mon Apr 24

Course Goals: To learn the fundamentals of the arithmetic of elliptic curves, including their properties over finite fields, local fields, and algebraic number fields.

Learning Activities and Time Allocation: Learning activities include class attendance, frequent problem sets, and a take-home final exam. The time to complete these activities are (1) attending lectures, approximately 3 hours/week; (2) working on the problem sets and the final exam, approximately 9 hours/week.

Assessment: Course grades will be determined by the quantity and quality of problem sets submitted (80% of grade) and by their grade on the takehome final exam (20% of grade).

Expectations of Students: It is expected that students will attend all lectures and participate in class discussion in an appropriate manner. Assignments are due on the listed dates. All students are expected to abide by Brown's academic code, which may found here

Tentative Syllabus:

  1. Algebraic Geometry Overview
  2. The Geometry of Elliptic Curves
  3. The Formal Group of an Elliptic Curve
  4. Elliptic Curves over Finite Fields
  5. Elliptic Curves over Local Fields
  6. Elliptic Curves over Global Fields
  7. Introduction to Group Cohomology: H0 and H1
  8. Computing the Mordell–Weil Group
Further topics as time permits:
  • Integral Points on Elliptic Curves
  • Elliptic Curves and Elliptic Functions over C
  • Elliptic Modular Functions and Modular Curves
  • Tate Curves
  • Elliptic Surfaces and Neron Models

Go to Professor Silverman's Home Page.