Modular methods applied to Diophantine equations

Deep results about elliptic curves, modular forms and Galois representations have successfully been applied to solve FLT and other Diophantine equations. Most of such applications broadly proceed along the following lines. To a hypothetical solution is associated a certain elliptic curve, called a Frey curve, and it is shown that the mod l Galois representation ?l associated to the l-torsion points of the Frey curve (with l, say, an odd prime occurring as exponent in the Diophantine equation) is irreducible. Then by modularity and level lowering, one obtains that ?l is modular of some explicitly known level (weight 2 and trivial character). Finally, the modular forms of this level provide (possibly in a non trivial way) information about the original Diophantine equation. In this talk I will first describe the above mentioned method in some more detail and then focus on the problem of finding Frey curves and proving irreducibility of ?l for small primes l.