*"Diophantine equations via Galois representations"*

Imin Chen, Simon Fraser University

Recently, the use of Galois representations attached to elliptic
curves has been used to resolve several cases of the generalized Fermat
equation. In this talk, I will discuss the method and some further cases which
can be analyzed at least partially, including the equation a^2+ b^2p = c^r,
where r = 3 or 5. Although a complete resolution is not yet possible, a
computational criterion can be obtained for r = 3, based on previous work by
Bennett-Skinner and Kraus. For r = 5, I outline a possible strategy using
a combination of quadratic Q-curves and elliptic curves over Q.