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MATH 5020: The Arithmetic of Elliptic Curves
The following is a tentative schedule for the course (in pdf format, ready to print):
| Week |
Day |
Ch. |
Sec. |
Topics |
| 1/19-1/23 |
1 |
|
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Introduction,
Algebraic Curves |
|
2 |
1&2 |
1.2 |
Algebraic
Curves, Weierstrass Equations and the Group Law |
| 1/26 – 1/30 |
3 |
|
3,
4 |
Elliptic
Curves, Isogenies |
|
4 |
|
4,
5 |
Isogenies,
The invariant differential |
| 2/2 – 2/6 |
5 |
|
6,
7 |
The
dual isogeny, the Tate module |
|
6 |
|
7,
8 |
The
Tate module, the Weil pairing |
| 2/9 – 2/13 |
7 |
4 |
1,
2, 3 |
Formal
groups, expansion around O, and groups associated to formal groups |
|
8 |
|
3,
4, 5 |
The
invariant differential and the formal logarithm |
| 2/16 – 2/20 |
9 |
|
6,
5.1 |
Formal
groups over DVR's; Elliptic curves over finite fields |
|
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 |
| 2/23 – 2/27 |
11 |
7 |
1,
2, 3 |
Minimal
Weierstrass equations, reduction modulo p, points of finite order |
|
12 |
|
3,
4, 5 |
Points
of finite order, Action of Inertia, good and bad reduction |
| 3/2 – 3/6 |
13 |
|
6,
7 |
The
group E/E_0 and the Criterion of Neron-Ogg-Shafarevich |
|
14 |
8 |
1,
2 |
The
weak Mordell-Weil theorem, The Kummer pairing via Cohomology |
| 3/9 – 3/13 |
|
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SPRING
BREAK |
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| 3/16 – 3/20 |
15 |
|
3,
4 |
The
Descent procedure, The Mordell-Weil theorem over Q |
|
16 |
|
4,
5 |
The
Mordell-Weil theorem over Q, Heights on projective space |
| 3/23 – 3/27 |
17 |
8 |
6,
7 |
Heights
on elliptic curves, Torsion points |
|
18 |
|
7,
9 |
Torsion
Points, the canonical height |
| 3/30 – 4/3 |
19 |
|
10;
9.1, 2 |
The
rank of an elliptic curve, Diophantine approximation, distance functions |
|
20 |
9 |
2,
3 |
Diophantine
approximation, distance functions, Siegel's theorem |
| 4/6 – 4/10 |
21 |
|
4,
5 |
The
S-Unit equation, Effective methods |
|
22 |
|
5,
6 |
Effective
methods, Shafarevich's theorem |
| 4/13 – 4/17 |
23 |
|
7;
10.1 |
The
curve Y^2 = X^3 + D; Computing the Mordell-Weil group |
|
24 |
10 |
1,
2 |
Computing
the Mordell-Weil group, twisting |
| 4/20 – 4/24 |
25 |
|
3,
4 |
Homogeneous
spaces, The Selmer and Shafarevich-Tate groups |
|
26 |
|
4,
5 |
The
Selmer and Shafarevich-Tate groups, Twisting of elliptic curves |
| 4/27 – 5/1 |
27 |
|
6 |
The
curve Y^2 = X^3 + DX |
|
28 |
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Other
topics |
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