By Haruzo Hida
Within the early years of the Eighties, whereas i used to be traveling the Institute for advert vanced learn (lAS) at Princeton as a postdoctoral member, I received a desirable view, learning congruence modulo a chief between elliptic modular kinds, that an automorphic L-function of a given algebraic crew G must have a canon ical p-adic counterpart of numerous variables. I instantly determined to determine the explanation at the back of this phenomenon and to increase the speculation of normal p-adic automorphic kinds, allocating 10 to fifteen years from that time, taking out the meant mathematics research of Shimura types through L-functions and Eisenstein sequence (for which I visited lAS). even though it took greater than 15 years, we now recognize (at least conjecturally) the precise variety of variables for a given G, and it's been proven that it is a common phenomenon legitimate for holomorphic automorphic types on Shimura kinds and likewise for extra normal (nonholomorphic) cohomological automorphic kinds on automorphic manifolds (in a markedly diverse way). whilst i used to be requested to provide a chain of lectures within the Automorphic Semester within the 12 months 2000 on the Emile Borel heart (Centre Emile Borel) on the Poincare Institute in Paris, I selected to offer an exposition of the speculation of p-adic (ordinary) households of such automorphic varieties p-adic analytically de pending on their weights, and this e-book is the outgrowth of the lectures given there.
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