## Families of Automorphic Forms by Roelof W. Bruggeman

By Roelof W. Bruggeman

Automorphic varieties at the top part aircraft were studied for a very long time. such a lot recognition has long gone to the holomorphic automorphic types, with quite a few purposes to quantity idea. Maass, [34], begun a scientific research of actual analytic automorphic kinds. He prolonged Hecke’s relation among automorphic kinds and Dirichlet sequence to actual analytic automorphic varieties. The names Selberg and Roelcke are hooked up to the spectral conception of genuine analytic automorphic varieties, see, e. g. , [50], [51]. This culminates within the hint formulation of Selberg, see, e. g. , Hejhal, [21]. Automorphicformsarefunctionsontheupperhalfplanewithaspecialtra- formation habit below a discontinuous crew of non-euclidean motions within the higher part airplane. One may possibly ask how automorphic varieties swap if one perturbs this crew of motions. this question is mentioned by means of, e. g. , Hejhal, [22], and Phillips and Sarnak, [46]. Hejhal additionally discusses the e?ect of version of the multiplier s- tem (a functionality at the discontinuous team that happens within the description of the transformation habit of automorphic forms). In [5]–[7] I thought of version of automorphic varieties for the complete modular workforce below perturbation of the m- tiplier method. a style in line with rules of Colin de Verdi` ere, [11], [12], gave the meromorphic continuation of Eisenstein and Poincar´ e sequence as services of the eigenvalue and the multiplier approach together. the current learn arose from a plan to increase those effects to even more common teams (discrete co?nite subgroups of SL (R)).

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Each f' -orbit of cuspidal points corresponds to at least two boundary segments of a given fundamental domain, hence the number of f' -orbits in the cuspidal points is finite. We call such a f'-orbit a cusp. 1 Connected neighborhoods of f' mod · ( ~ + 2i) and f' mod · ( ~ + ~ VJ) in f' mod \SJ, as represented in the standard fundamental domain. 7, describes how to put a topology on SJ* suchthat f'\SJ* is a Hausdorff topological space. 1 in [38]. 3 Notation. Y = f'\SJ ~ f'\(Jjk and X= f'\SJ*. X is a compact space, Y an open subspace.

The more strict the growth condition, the less automorphic forms satisfying it exist. 9). 5. Even unusual functions like eJ, with J the modular invariant, come under this definition. In fact, eJ is an automorphic form for r mod, with weight 0, trivial multiplier system, and eigenvalue 0. 4 Analyticity. Condition ii) imposes an elliptic differential equation with analytic coefficients. , [29], App. 4, §5 and [3], p. 207-210. 5 Holamorphie automorphic forms. The well known holomorphic automorphic forms on 5) of weight l, with multiplier system v satisfy i)' F('y · z) = v('y)(cz ii)' ßzF + d) 1F(z) for all I'= ( ~ ~) E f; = 0.

Characters with values in { t E IC* : ltl = 1 }. We say that a character x E X belongs to the weight l if x( () = e 1ril. The set of weights to which a given character belongs has the form l 0 + 2Z. 3. 4 Definition. An automorphic form for the group satisfying i) f(rg) = x(r)f(g) for all g E ii) f G, has weight l, for some l E IC. iii) wf =V for some >. E IC. f' is a function f E C 00 ( G) I Er, for some XE X. 32 CHAPTER 2 UNIVERSAL COVERING GROUP We call f an automorphic form of weight l, for the character x, with eigenvalue >..