1,709 research outputs found
Modular elliptic curves over real abelian fields and the generalized Fermat equation
Using a combination of several powerful modularity theorems and class field
theory we derive a new modularity theorem for semistable elliptic curves over
certain real abelian fields. We deduce that if is a real abelian field of
conductor , with and , , , then every
semistable elliptic curve over is modular.
Let , , be prime, with , and .To a
putative non-trivial primitive solution of the generalized Fermat
we associate a Frey elliptic curve defined over
, and study its mod representation with the help
of level lowering and our modularity result. We deduce the non-existence of
non-trivial primitive solutions if , or if and , .Comment: Introduction rewritten to emphasise the new modularity theorem. Paper
revised in the light of referees' comment
- …