By André Unterberger

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2002.

The topic of this publication is the research of automorphic distributions, wherein is intended distributions on R2 invariant below the linear motion of SL(2,Z), and of the operators linked to such distributions less than the Weyl rule of symbolic calculus.

Researchers and postgraduates drawn to pseudodifferential analyis, the idea of non-holomorphic modular kinds, and symbolic calculi will enjoy the transparent exposition and new effects and insights.

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Additional info for Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi

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2 ), (N- i7r £ T'Jjst It,o, h) = L ad=N,d>O b mod d L ad=N, d>O b mod d e2i7r~ h(dx - b, a) e 2i7rX dx 1 00 -00 h(dx, a) e2i7rX dx. 14) 5. 3 indeed justifies the convergence of both sides in the case when h lies in the image of S(JR. 2 ) by the operator 7[2 £2 . 3, in which the right-hand side is still meaningful when 1! 2). :~ or on cusp-distributions (l)1~ l)~ is easy to describe too. 4. 3) of the cusp-form JVji,,£" Proof. 1. 19) o Chapter 1. 5. 21) Proof. 6). l IT = (1 - p-sr;ist + p-2s)-1 .

8), [ . 20) and that 2in [ h = -iA h if h is a distribution on lR 2 homogeneous of degree -1 - iA. 21), together with the Q-invariance of W(uz,u z ) and the relation Q(in [) Q-I = -in [, makes it possible to check that, indeed, wigP(U z , u z ) is invariant under ~p. 16) is also meaningful whenever h E L~verJlR2): the result is just the same as opP( ~(h + ~ph)). 29) that applying the operator n 2 [2 to a function of q = 27r i~,~;~12, viewed as a function of x,~, was equivalent to applying it, viewed 58 Chapter 2.

61 ) o In the next section, we shall make the decomposition of the Bezout distribution into homogeneous components explicit, at the same time proving that not only ~£, with C ~ 1, but also ~ itself makes sense as a tempered distribution. Remark. 62) where the matrix (::':::;1) has determinant (m, n). The automorphic distribution «(1 + 2i7f £) ~, which bears the same relation to ~ as the relation of 1) to 1)prime (cf. 63) 4 The structure of automorphic distributions We now need to refresh the reader's memory on Maass cusp-forms and the RoelckeSelberg expansion - or, as the case may be, deliver him a crash course on the subject.

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