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MATH 732: The Arithmetic of Elliptic Curves
The following is a tentative schedule for the course:
| Week |
Day |
Ch. |
Sec. |
Topics |
| 1/21-1/25 |
1 |
|
|
Introduction |
|
2 |
1&2 |
|
Algebraic Curves |
|
3 |
3 |
1,2 |
Weierstrass Equations and the Group Law |
| 1/28 – 2/1 |
4 |
|
3 |
Elliptic Curves |
|
5 |
|
4 |
Isogenies |
|
6 |
|
5 |
The invariant differential |
| 2/4 – 2/8 |
7 |
|
6 |
The dual isogeny |
|
8 |
|
7 |
The Tate module |
|
9 |
|
8 |
The Weil Pairing |
| 2/11 – 2/15 |
10 |
4 |
1 |
Formal groups, expansion around O |
|
11 |
|
2,3 |
Formal groups, and groups associated to
formal groups |
|
12 |
|
4,5 |
The invariant differential and the formal
logarithm |
| 2/18 – 2/22 |
13 |
|
6 |
Formal groups over DVR's |
|
14 |
5 |
1 |
Elliptic Curves over Finite Fields, Number
of Rational Points |
|
15 |
6 |
1,...,5 |
Elliptic Curves over C; the Uniformization
Theorem |
| 2/25 – 2/29 |
16 |
7 |
1,2 |
Minimal Weierstrass equations, reduction
modulo p |
|
17 |
|
3 |
Points of finite order |
|
18 |
|
4,5 |
Action of Inertia, good and bad reduction |
| 3/3 – 3/7 |
19 |
|
6,7 |
The group E/E_0 and the Criterion of
Neron-Ogg-Shafarevich |
|
20 |
8 |
1 |
The weak Mordell-Weil theorem |
|
21 |
|
2 |
The Kummer pairing via Cohomology |
| 3/10 – 3/14 |
22 |
|
3 |
The Descent procedure |
|
23 |
|
4 |
The Mordell-Weil theorem over Q |
|
24 |
|
5 |
Heights on projective space |
| 3/17 – 3/21 |
|
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|
SPRING BREAK – 3/15 – 3/23 |
|
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|
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| 3/24 – 3/28 |
25 |
8 |
6 |
Heights on elliptic curves |
|
26 |
|
7 |
Torsion Points |
|
27 |
|
9 |
The canonical height |
| 3/31 – 4/4 |
28 |
|
10 |
The rank of an elliptic curve |
|
29 |
9 |
1,2 |
Diophantine approximation, distance
functions |
|
30 |
|
3 |
Siegel's theorem |
| 4/7 – 4/11 |
31 |
|
4 |
The S-Unit equation |
|
32 |
|
5 |
Effective methods |
|
33 |
|
6 |
Shafarevich's theorem |
| 4/14 – 4/18 |
34 |
|
7 |
The curve Y^2 = X^3 + D |
|
35 |
10 |
1 |
Computing the Mordell-Weil group |
|
36 |
|
2 |
Twisting |
| 4/21 – 4/25 |
37 |
|
3 |
Homogeneous spaces |
|
38 |
|
4 |
The Selmer and Shafarevich-Tate groups |
|
39 |
|
5 |
Twisting of elliptic curves |
| 4/28 – 5/2 |
40 |
|
6 |
The curve Y^2 = X^3 + DX |
|
41 |
|
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Other topics |
|
42 |
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Other topics |
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