Lecture Notes

These are directions suggested to students on how to prepare their lectures. (PDF)

LEC # TOPICS LECTURE NOTES
1 The Projective Plane (PDF 1)

(PDF 2)

2 Curves in the Projective Plane (PDF)
3 Rational Points on Conics (PDF)
4 Geometry of Cubic Curves (PDF)
5 Weierstrass Normal Form (PDF)

(PDF)

6 Explicit Formulas for the Group Law (PDF)
7 Points of Order Two and Three (PDF)
8 The Discriminant

Points of Finite Order have Integer Coordinates - Part 1

(PDF)
9 Points of Finite Order have Integer Coordinates - Part 2 (PDF)
10 Points of Finite Order have Integer Coordinates - Part 3

The Nagell-Lutz Theorem

(PDF)
11 Real and Complex Points on Cubics (PDF)
12 Heights and Descent (PDF)
13 Height of P + P_0 (PDF)
14 Height of 2P (PDF)
15 A Useful Homomorphism - Part 1 (PDF)
16 A Useful Homomorphism - Part 2 (PDF)
17 Mordell’s Theorem - Part 1 (PDF)
18 Mordell’s Theorem - Part 2

Examples - Part 1

(PDF)
19 Examples - Part 2 (PDF)
20 Examples - Part 3 (PDF)
21 Singular Cubics (PDF)
22 Rational Points over Finite Fields (PDF)
23 Gauss’s Theorem - Part 1 (PDF)
24 Gauss’s Theorem - Part 2 (PDF)
25 Points of Finite Order Revisited (PDF)
26 Factorization using Elliptic Curves - Part 1 (PDF)
27 Factorization using Elliptic Curves - Part 2 (PDF)
28 Integer Points on Cubics

Taxicabs - Part 1

(PDF)
29 Taxicabs - Part 2

Thue’s Theorem - Part 1

(PDF)
30 Thue’s Theorem - Part 2 (PDF)
31 Construction of an Auxiliary Polynomial (PDF)
32 The Auxiliary Polynomial is Small (PDF)
33 The Auxiliary Polynomial Does Not Vanish (PDF)
34 Proof of the DAT

Further Developments

(PDF)
35 Congruent Numbers and Elliptic Curves I: Koblitz - Part 1 (PDF)
36 Congruent Numbers and Elliptic Curves II: Koblitz - Part 2 (PDF)

Course Info

Learning Resource Types

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