Mod pq Galois representations and Serre's conjecture
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Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial.
In this paper we investigate when a pair of mod p and mod q representations of the absolute Galois group of a number field K simultaneously arises from an {\it automorphic motive}: we do this in the 1-dimensional (Section 2) and 2-dimensional (Section 3: this time assuming $K={\mathbb Q}$) cases. In Section 3 we formulate a mod pq version of Serre's conjecture refining in part a question of Barry Mazur and Ken Ribet.
This is an older preprint that was made available elsewhere on Sep. 19, 2001
This is an older preprint that was made available elsewhere on Sep. 19, 2001