Modularity of Fibres in Rigid Local Systems

Abstract

(K), as is explained, for example, in [W1] and [W2].Fontaine and Mazur [FM] conjectured that this is always the case. Significantprogress on this conjecture was achieved [W3] by proving particular instancesof the following “lifting conjecture”:Conjecture 1.1. Suppose that l is odd and that the residual represen-tation ρ¯ attached to ρ is modular. Then ρ itself is modular.Conjecture 1.1 is proved in [W3] and [TW] when K = Q and the restriction ofρ to the decomposition groups at the primes above l are semistable in the senseof [DDT, §2.4]. This is enough (using the primes l = 3 and 5) to establish theShimura-Taniyama conjecture for semistable elliptic curves, thanks to a keyresult of Langlands and Tunnell. Progressively stronger cases of Conjecture 1.1were subsequently proved by [Di], [CDT], [Fu], and [SW1]; in [SW1], Skinnerand Wiles obtain quite general results in the context where K is any totallyreal field, the principal assumption being that ρ is ordinary at the primesabove l.In this note we consider Galois representations which occur in “rigid fam-ilies”, and establish their modularity under Conjecture 1.1. This implies themodularity (over suitable real abelian extensions) of the Galois representationsoccurring in the cohomology of the curvesy