Mathematics 573(02) -- Topics in Number Theory -- Fall 1995
MTH 9:50-11:10 --- Hill 425 --- J. Tunnell
This course will cover topics in number theory which
lead to the proof of Fermat's Last Theorem. Several themes drawn from
number theory of the last 35 years will be considered, including those
relating to elliptic curves, modular forms and representations of Galois
groups. The starting point will be an overview of the proof to indicate
the main ideas, followed by
detailed examination of the relevant methods. Many of the most
technical portions of the proof will be treated in a concurrent seminar.
Topics will include some of the following :
- Overview of the proof of Fermat's Last Theorem
- Relation of integer solutions of $x^n+y^n=z^n$ to the
existence of elliptic curves with special properties
- Congruences for modular forms and
Ribet's descent argument
- Modular elliptic curves and Wiles' proof that sufficiently
many elliptic curves are modular
- Galois group representations attached to elliptic curves and
modular forms
- Deformations of modular forms and Galois representations
- Ramification properties of Galois representations
- Group cohomology and Selmer groups
- Geometry and number theory of modular curves
- Congruence properties of modular forms
- Open problems in elliptic curves and Galois representations
Prerequisites: For the first third of the semester only the usual first
year graduate courses will be assumed. Examination of the details of the
proof will require knowledge of basic algebraic number theory and elliptic
curves. At any point this knowledge could be replaced by a willing
suspension of disbelief.
Course Format: There will be an optional seminar where members of
the class will discuss details not covered in the main lectures. As
usual in my graduate courses there will be
assignments covering the methods introduced in lectures. No student will
be required to do more than they are capable of.
More Information: Contact J. Tunnell in Hill 546, email to
tunnell@math or examine
http://www.math.rutgers.edu/~tunnell/math573.html with any WWW browser.