By Stephen S. Gelbart

This quantity investigates the interaction among the classical conception of automorphic types and the trendy thought of representations of adele teams. studying very important fresh contributions of Jacquet and Langlands, the writer offers new and formerly inaccessible effects, and systematically develops specific results and connections with the classical concept. The underlying subject is the decomposition of the usual illustration of the adele crew of GL(2). an in depth evidence of the distinguished hint formulation of Selberg is integrated, with a dialogue of the potential diversity of applicability of this formulation. during the paintings the writer emphasizes new examples and difficulties that stay open in the normal theory.

TABLE OF CONTENTS: 1. The Classical idea 2. Automorphic varieties and the Decomposition of L^{2}(PSL(2,R) three. Automorphic varieties as capabilities at the Adele team of GL(2) four. The Representations of GL(2) over neighborhood and worldwide Fields five. Cusp types and Representations of the Adele crew of GL(2) 6. Hecke thought for GL(2) 7. the development of a distinct classification of Automorphic types eight. Eisenstein sequence and the continual Spectrum nine. The hint formulation for GL(2) 10. Automorphic types on a Quaternion Algebra

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Vl 2 . 2s , , 2 and p,,(P)= Y modulo H:. I n order t o complete the proof of the lemma if suffices t o show that pn takes H! into zero. This is ininiediatc tor IZ = I , since HP is trivially zero by Lemma 1 . T h ~ i sthe p r e x n t lemma will follow by induction on y1 if we show that 0, takes H! into HnOp x, 2 . vyzM2 is in R,, . We distinguish 12 cases depending on the location of the two occurrences of x n in ill, h ( . ~y , z)ill,. , denote indeterminates distinct from s,, but not neccssarily distinct from each other.

Be the subspace of H , spanned by generators of H , of the form M , h(x,y , z)M2 where x,y , and z are indeterminates. In the first step we define a Collection Algorithm which enables us t o use induction on n. x, * . In particular, H; and x1 ... x, span R,, so that Theorem 1' holds with H: in place of H,. In the final step (Lemma 3 ) we show that H , = H:. vzM2 is in R , . The relation h(x,y , z ) = 0 modulo H implies many others. We indicate below six of the simplest ones. 1 ) is equivalent t o 0 modulo H for any x,y , z in L .

Ser. Mat. Nauk. 10 (1940) 166-170. (Russian) MR2, 212. , Solution of the Burnside problem for exponent 6, Proc. Nat. Acad. Sci. USA 4 3 (1957) 751-753. MR 19, 728. S. I. Adjan, (Infinite periodic groups) 0 b e s k o n e h y h periodiEeskih gruppah, I , 11,111, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 212-244, 251-524, 709-731 = Math. USSR Izv. 2 (1968) 209-236, 241-479, 665-685. MR 39 # 1532 a-b-c. S. I. Adjan, (Defining relations and the world problem for free periodic groups of odd order) Opredeljajdtie sootnosenija i problema totdestva dlja svobodnyh perioditeskih grupp netetnogo porjadka, Izv.