CHAPTER 1 Introduction Let τ be in the complex upper half-plane and let q = exp(2πiτ). Fix a prime number . In the year 1954 Martin Eichler associated to the modular form f = n≥1 anqn = q 2q2 q3 + 2q4 + q5 + · · · of weight 2 and level Γ0(11) a very different type of object: A group representation ρf, : Gal(Q/Q) −→ GL2(Q ) such that for all primes p {11, } the representation ρf, is unramified at p and Trace ρf,(Frp) = ap and det ρf,(Frp) = p where Frp denotes a lift of Frobenius at p. A priori, this association makes no connection between the local properties at p = 11 of the modular form and the group representation. It is then natural to attempt to strengthen Eichler’s correspondence between the analytic object f and the algebraic object ρf,, and many results for generalizations of Eichler’s correspondence to more general settings have been obtained in the last half a century in this direction. Such questions are now often called the problem of local-global compatibility of Langlands correspondences. The current memoir is meant to contribute to these ongoing efforts. Traditionally, the main tool in this area is the detailed study of the geometry of Shimura varieties. This should not be too surprising: The very construction by Eichler of ρf, uses modular curves. For modular forms a reference for the traditional approach to local-global compatibility is [LAN], for Hilbert modular forms there is for example [CAR], and a more recent work in this tradition is the work [HT] by Harris-Taylor. The methods of our work are different, we use the deformation theory of au- tomorphic forms and Galois representations instead: The description of f above as an element of Z[[q]] lends itself to an obvious notion of congruences to other such formal power series in q. In many situations one knows the existence of an abundance of such congruences to other modular forms. We will exploit this to develop an approach to local-global compatibility questions that largely avoids the study of bad reduction of Shimura varieties. Phrased in terms of Eichler’s modular form f we would like to advertise the following maxim: The local behavior at 11 of the associated Galois representation is governed in a direct way by unramified local behavior of modular forms, not necessarily of the individual modular form f itself but of families of modular forms deforming f. This approach is very different than the standard approach to such problems via studying singularities of Shimura varieties and therefore in particular is useful in the following situations: 1
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