(15 minutes) Presentation - Perimeters of right triangles?
Patterns in congruent numbers modulo
?
(30 minutes) Definition of a group.
Definition 4.1
An abelian group
is a set
equipped with a binary operation
and an element
such that for all
,
(closure)
,
(identity)
,
(associativity)
,
(inverses) for every
in
there is
such that
,
(commutativity)
.
Examples:
The integers
under addition.
The rational numbers
under addition.
The integers
under addition
modulo
.
Let
be a prime. The integers
under multiplication modulo
.
This is called
.
(15 minutes) Experiment with some abelian groups in SAGE .
(10 minutes) Break.
(20 minutes) Definition of elliptic curve groups.
Definition 4.2
Fix integers
and
.
Let
be the set of solutions to
along with one ``extra point'' which we call
which is
the additive 0
element. This is an
abelian group (note: the associative law takes a lot of work to prove!).
(30 minutes) Participants: Graph elliptic curves.
Then derive an algebraic formula (by hand) for the
group operation.
(15 minutes) Elliptic curves modulo
.
Fix integers
and
and a prime
.
Let
be the set of solutions to
with
and
along with a formal extra
point
. This group is central in both cryptography (in making
and cracking cryptosystems) and the Birch and Swinnerton-Dyer
conjecture! I will explain how in both cases next week.
(15 minutes) Participants:
Graph and compute with some elliptic curves modulo
.