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In this short note we give a new approach to proving modularity of -adic Galois representations using a method of -adic approximations. This recovers some of the well-known results of Wiles and Taylor in many, but not all, cases. A feature of the new approach is that it works directly with the -adic Galois representation whose modularity is sought to be established. The three main ingredients are a Galois cohomology technique of Ramakrishna, a level raising result due to Ribet, Diamond, Taylor,...
This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over ? Let be a prime and a positive integer. We show that that the finite simple groups of Lie type if and appear as Galois groups over , for some divisible by . In particular, for each of the two Lie types and fixed we construct infinitely many Galois groups but we do not have a precise control...
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