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

notes Lecture Notes
group_work Projects with Examples
assignment_turned_in Problem Sets with Solutions