Graduate School of Mathematics and Applications
Simon Fraser University, Burnaby, Canada.
As a program under the Graduate School of Mathematics and Applications ('Forskerskolen for Matematik og Anvendelser'), Imin Chen from Simon Fraser Univ. (Canada) will give a lecture series with the title:
Galois representations and Diophantine equations
The lectures take place on the following dates:
Monday, 22 May, 2006, 15.15-16.00 in Aud. 9
Tuesday, 23 May, 2006, 15.15-17.00 in Aud. 9
Wednesday, 24 May, 2006, 15.15-17.00 in Aud. 10
Abstract: Since the proof of Fermat's Last Theorem, there has been ongoing work to resolve more cases of the generalized Fermat equation $A x^p + B y^q = C z^r$. The methods used consist of three components.
1. Frey constructions: Constructing families of $2$-dimensional Galois representations with bounded ramification which are parametrized by non-trivial primitive solutions.
2. Modularity information: Showing the $2$-dimensional Galois representations in the families come from modular forms of bounded weight, level, and character. This requires modularity results and as well as adjustment results on weight, level, and character.
3. Non-isotriviality results: For each possible modular form, showing there are only finitely-many (resp. no) non-trivial primitive solutions which give rise to it. This involves the study of rational points on modular curves (resp. techniques to eliminate congruences between modular forms).
In this lecture series, I will review some of the background necessary to understand these three components and then explain in detail how the method can be successfully applied to special cases of the generalized Fermat equation. I will also discuss some further cases which might be approachable by similiar methods as well as future directions of study.