Reading Course on
The Arithmetic of Elliptic Curves – MA1970 & MA2980
Brown University – Spring, 2015
Professor Joseph Silverman
Course Description: The theory of elliptic curves is a beautiful amalgamation of geometry, analysis, and algebra, in part due to the fact that the curve (a geometric object) has a natural structure as an abelian group (an algebraic object). The focus of this course will be to first study the geometry and algebra of elliptic curves, and then to use that information to analyze the arithmetic. A major goal will be the celebrated Mordell–Weil theorem, which says that the group of points on an elliptic curve with coordinates in a number field form a finitely generated group.
Registration This course will be run as a "Reading and Research" course.
Graduate students should sign up for Math 2980 - Section S19 (CRN: 22596).
Undergraduates should sign up for Math 1970 - Section S15 (CRN: 22570). If you need an override code, write to get one from me.
Prerequisites The primary prerequisite is a working knowledge of basic algebraic number theory, which includes number fields, rings of integers, factorization of ideals into primes, ramification, completions, and the two fundamental finiteness theorems: (1) Finiteness of the ideal class group. (2) Finite generation of the unit group. If you've taken Math 2530 or know this material from another source, that should be fine. We'll also use some algebraic geometry, but the material needed is summarized (mostly without proof) in the first two chapters of the book, which we'll cover in the first two weeks of the course.
Grade Options Undergraduates should sign up to take this course S/NC. Graduate students may choose either grade option, but this course will not be eligible for qual credit.
Text The Arithmetic of Elliptic Curves
Office Mathematics Department, Kassar House, Room 202
Phone 863-1124
Email jhs@math.brown.edu
Web Site www.math.brown.edu/~jhs/MA0272/MA0272ReadingCourse2015.html
Course Meeting Time Weds 9:00–9:50 (with me) plus additional meetings of participants at times to be determined.
Course Location Kassar 205 (for all 9:00–9:50 meetings)
Office Hours None scheduled specifically for this course, but if you want to ask about something, send me an email to make an appointment.
Schedule
Week Chapter Sections Topic —
1 Jan 21–23 I 1.1–1.3 Algebraic Varieties —
2 Jan 26–30 II 2.1–2.5 Algebraic Curves —
3 Feb 2–6 III 3.1–3.4 The Geometry of Elliptic Curves —
4 Feb 9–13 III 3.5–3.8 The Geometry of Elliptic Curves —
5 Feb 16–20 IV 4.1–4.3 The Formal Group of an Elliptic Curve —
6 Feb 23–27 IV 4.4–4.6 The Formal Group of an Elliptic Curve —
7 Mar 2–6 V 5.1–5.3 Elliptic Curves over Finite Fields —
8 Mar 9–13 VI 6.1–6.3 Ellipti Curve over C —
9 Mar 16–20 VII 7.1–7.3 Elliptic Curves over Local Fields —
— Mar 23–27 — — Brown closed —
10 Mar 30–Apr 3 VII 7.4–7.6 Elliptic Curves over Local Fields —
11 Apr 6–10 VIII 8.1–8.4 Elliptic Curves over Global Fields —
12 Apr 13–17 VIII 8.5–8.9 Elliptic Curves over Global Fields —
13 Apr 20–24 X 10.1–10.3 Computing the Mordell–Weil Group —
14 Apr 27–May 1 X 10.4–10.6 Computing the Mordell–Weil Group —
    Additional Lectures
  • Tate curves
  • Elliptic surfaces
  • Complex multiplication
Computer packages to use for Math 272 I tend to use a computer package called PARI-GP to do number theory 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 at this site. 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 number theoretic algorithms, including a large elliptic curve package.

Go to Professor Silverman's Home Page.