Description

This course will be an introduction to elliptic curves, which roughly speaking are smooth cubic curves in the projective plane (turns out they have a simple model of the form y^2=x^3+ax+b). The surprising feature of elliptic curves is that their points can be made into an abelian group, and this group is finitely generated when we focus on points with coordinates in the rational numbers lying on an elliptic curve with rational coefficients. Elliptic curves are central in modern number theory, e.g., they were essential in the proof of Fermat’s Last Theorem. The goal of the course will be to understand and calculate the group of all rational points on an elliptic curve (i.e. calculate its torsion and rank), and a number of more refined invariants (such as the order of the Shafarevich-Tate group).

The prerequisites for this course are the Abstract Algebra sequence (Math 5210 and 5211) and a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background as much as I can. Our textbook will be “The Arithmetic of Elliptic Curves,” by Silverman, which is the standard graduate-level textbook for the subject.

Prerequisites: Math 5210 and Math 5211.
Credits: 3
Meets: Tuesdays, and Thursdays, 2-3:15, at MSB 315.

Books

I will be following the main reference: Joseph H. Silverman, “The Arithmetic of Elliptic Curves”, Springer, 2nd Edition. However, there are other references that may be very useful.

1. J. H. Silverman, J. Tate, “Rational Points on Elliptic Curves”, Springer – This book is an elementary version of the main reference.

2. J. H. Silverman, “Advanced Topics in the Arithmetic of Elliptic Curves”, Springer.

3. J. S. Milne, “Elliptic Curves”, Kea Books – this book is freely available at Milne’s website.

4. N. Koblitz, “Introduction to Elliptic Curves and Modular Forms”.

5. J. Tate, “The arithmetic of elliptic curves”, Invent. Math. 23 (1974), 179-206 – The main reference was heavily influenced by this key survey article.

6. D. Husemöller, “Elliptic Curves“, Second Edition, Springer.

7. L. Washington, “Elliptic Curves: Number Theory and Cryptography“, Second Edition (Discrete Mathematics and Its Applications) 2nd Edition, CRC Press.

Homework

During the semester I will propose assignments to be handed in or presented in class. Silverman’s book contains an excellent collection of exercises. I will also propose exercises to be solved with Magma (or Sage, or PARI). You are encouraged to talk with your classmates about the homework. If you have difficulties, do not waste time — get help! Please come to office hours!

Homework assignments:

• Homework 1 (due Tuesday, September 15th) — 1.3, 1.6. In addition: find a 1-parameter parametrization of all the rational points on the conic x^2-3xy+y^2=5. Also, problems 2.3, 2.6.
• Homework 2.
• Homework 3.

 

A tentative schedule, day by day

Day	Ch.	Sec.	Topics (ideally)	Topics (reality)
1			Introduction, Algebraic Curves	Introduction to Diophantine Equations.
2	1&2	1.2	Algebraic Curves, Weierstrass Equations and the Group Law	Alg. Number Theory VS Elliptic Curves
3		3, 4	Elliptic Curves, Isogenies	[No class – Alvaro at a conference]
4		4, 5	Isogenies, The invariant differential	Intro. to algebraic varieties and curves
5		6, 7	The dual isogeny, the Tate module	Morphisms, algebraic curves
6		7, 8	The Tate module, the Weil pairing	Ramification, divisors
7	4	1, 2, 3	Formal groups, expansion around O, and groups associated to formal groups	Riemann-Roch and Weierstrass models
8		3, 4, 5	The invariant differential and the formal logarithm	The Group Law
9		6, 5.1	Formal groups over DVR’s; Elliptic curves over finite fields	Isogenies
10	5	1; 6.1,…,5	Elliptic Curves over Finite Fields, Number of Rational Points, number of rational points; Elliptic Curves over C; the Uniformization Theorem	Isogenies and the Dual Isogeny
11	7	1, 2, 3	Minimal Weierstrass equations, reduction modulo p, points of finite order	The Tate Module, the Weil Pairing.
12		3, 4, 5	Points of finite order, Action of Inertia, good and bad reduction	Properties of the Weil Pairing, Elliptic Curves over C.
13		6, 7	The group E/E_0 and the Criterion of Neron-Ogg-Shafarevich	Elliptic Curves over C, Formal Groups
14	8	1, 2	The weak Mordell-Weil theorem, The Kummer pairing via Cohomology	Formal groups in general, groups associated to formal groups
15		3, 4	The Descent procedure, The Mordell-Weil theorem over Q	The formal logarithm, and formal groups over DVR’s
16		4, 5	The Mordell-Weil theorem over Q, Heights on projective space	Elliptic curves over finite fields, and local fields, minimal models, the reduction map
17	8	6, 7	Heights on elliptic curves, Torsion points	E_1(K)=E(M), points of finite order over local fields
18		7, 9	Torsion Points, the canonical height	Points of finite order, action of inertia, criterion of Neron-Ogg-Shafarevich
19		10; 9.1, 2	The rank of an elliptic curve, Diophantine approximation, distance functions	Good and bad reduction, the group E/E_0
20	9	2, 3	Diophantine approximation, distance functions, Siegel’s theorem	The Weak Mordell-Weil theorem: the Kummer pairing
21		4, 5	The S-Unit equation, Effective methods	Finiteness of the maximal abelian extension of exponent m unramified outside a finite set of primes
22		5, 6	Effective methods, Shafarevich’s theorem	The Descent procedure
23		7; 10.1	The curve Y^2 = X^3 + D; Computing the Mordell-Weil group	Torsion points (Nagell-Lutz), and the canonical height
24	10	1, 2	Computing the Mordell-Weil group, twisting	The canonical height pairing, and Z-linear independence of rational points; computing the Mordell-Weil group: the method of complete 2-descent.
25		3, 4	Homogeneous spaces, The Selmer and Shafarevich-Tate groups	Complete 2-descent, and an example.
26		4, 5	The Selmer and Shafarevich-Tate groups, Twisting of elliptic curves	Descent via 2-isogeny
27		6	The curve Y^2 = X^3 + DX	Student presentation: 2-descent on hyperelliptic curves.
28			Other topics