18.704 | Fall 2004 | Undergraduate

# Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves

## Calendar

LEC # TOPICS KEY DATES
1 The Projective Plane

2 Curves in the Projective Plane

3 Rational Points on Conics

4 Geometry of Cubic Curves Homework 1 due
5 Weierstrass Normal Form

6 Explicit Formulas for the Group Law

7 Points of Order Two and Three Homework 2 due
8 The Discriminant

Points of Finite Order have Integer Coordinates - Part 1

9 Points of Finite Order have Integer Coordinates - Part 2

10 Points of Finite Order have Integer Coordinates - Part 3

The Nagell-Lutz Theorem

Homework 3 due
11 Real and Complex Points on Cubics

12 Heights and Descent

13 Height of P + P_0 Homework 4 due
14 Height of 2P

15 A Useful Homomorphism - Part 1 Homework 5 due
16 A Useful Homomorphism - Part 2

17 Mordell’s Theorem - Part 1

18 Mordell’s Theorem - Part 2

Examples - Part 1

Homework 6 due
19 Examples - Part 2

20 Examples - Part 3

21 Singular Cubics

22 Rational Points over Finite Fields

23 Gauss’s Theorem - Part 1

24 Gauss’s Theorem - Part 2 Homework 7 due
25 Points of Finite Order Revisited

26 Factorization using Elliptic Curves - Part 1

27 Factorization using Elliptic Curves - Part 2 Homework 8 due
28 Integer Points on Cubics

Taxicabs - Part 1

29 Taxicabs - Part 2

Thue’s Theorem - Part 1

30 Thue’s Theorem - Part 2 Homework 9 due
31 Construction of an Auxiliary Polynomial

32 The Auxiliary Polynomial is Small

33 The Auxiliary Polynomial Does Not Vanish

34 Proof of the DAT

Further Developments

Homework 10 due
35 Congruent Numbers and Elliptic Curves I: Koblitz - Part 1

36 Congruent Numbers and Elliptic Curves II: Koblitz - Part 2

## Course Info

Fall 2004
##### Learning Resource Types
Lecture Notes
Projects with Examples
Problem Sets with Solutions