Abstract
In §1, we introduce the notion of potential stable automorphy of modular galois representations, and state a general result on the ubiquity of such representations. In §2 we state some rather precise grouptheoretic results on the monodromy of the Dwork family, and use them to prove the general result of §1. In §3 we discuss variants and possible future applications of the general result. In §4 we prove the grouptheoretic results stated in §2, as well as some supplements to those results.
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