Let be a cuspidal automorphic representation of where is the ring of adeles of a totally real algebraic number field of degree over , of the same type as representations corresponding to Hilbert modular forms of weight , i.e. whose local components for each of the Archimedean places of are essentially square integrable representations of occurring in the induced representation (under unitary induction) with characters , of given by , for integral and , all depending on . For even, is taken to be a special or cuspidal representation of , for at least one non-Archimedean place of . Let be an algebraic closure of .
The main theorem proved is the following: there exists an algebraic number field depending on the given and a strictly compatible system of continuous 2-dimensional -adic representations of such that for every non-Archimedean place of and of with residue characteristic different from that of , the restriction of to the local Weil group is equivalent to
What is new here is that the author determines for every non-Archimedean place of . Moreover, according to Ribet, turns out to be irreducible and as such, is characterized entirely.
First, for a weaker version of the main theorem, the author gives a (“geometric”) proof and then he uses ”base change for ” to deduce the main theorem. As a corollary of the main theorem for and weight , the author shows an affirmative answer to a conjecture on Weil curves over .