By Haruzo Hida

Within the early years of the Nineteen Eighties, whereas i used to be vacationing the Institute for advert­ vanced examine (lAS) at Princeton as a postdoctoral member, I bought a desirable view, learning congruence modulo a chief between elliptic modular varieties, that an automorphic L-function of a given algebraic team G must have a canon­ ical p-adic counterpart of a number of variables. I instantly determined to determine the explanation at the back of this phenomenon and to boost the idea of normal p-adic automorphic types, allocating 10 to fifteen years from that time, taking away the meant mathematics examine of Shimura kinds through L-functions and Eisenstein sequence (for which I visited lAS). even though it took greater than 15 years, we now understand (at least conjecturally) the precise variety of variables for a given G, and it's been proven that this can be a common phenomenon legitimate for holomorphic automorphic types on Shimura forms and likewise for extra common (nonholomorphic) cohomological automorphic kinds on automorphic manifolds (in a markedly varied 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 types p-adic analytically de­ pending on their weights, and this booklet is the outgrowth of the lectures given there.

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