By Nikolsky S.M.

A textbook for college scholars (physicists and mathematicians) with designated supplementary fabric on mathematical physics. in line with the path learn through the writer on the Moscow Engineering Physics Institute. quantity 2 includes a number of integrals, box conception, Fourier sequence and Fourier crucial, differential manifolds and differential types, and the Lebesgue fundamental.

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Suppose if = r x , with X = nxv a grossencharacter of F. Then for each v, S(rx ) is defined, and for almost every v, S(rx ) is v v one-dimensional and class 1. The resulting representation Sew) = ®S(r x ) v is always automorphic, by the criterion of [Langlands]. Our purpose now is to show that any unitary cuspidal representation of half-integral weight has an automorphic Shimura image. and this image is actually cuspidal if 'If ~ r. for any v. § 15. 1 THEOREM. Suppose w= ® iiv is a unitary cuspidal repre- sentation of half-integral weight.

Indeed in this case, we have just shown that S("v) still exists for all v. Moreover, we have shown that 'lTv = S(7i v) must be supercuspidal for VE'T. 3 implies 'lTv is a quotient of p( P-v, "v) for all v). On the other hand, ifT is empty, then S("v) exists a priori, for all v, but 'IT = ® S(" v) is not a priori cuspidal. To complete the proof in this case it remains to note that now L(s, 'IT ® X) = L(s,,,, X) and "" L(s, 'IT ® X -1 ) = L(s,,,, X) I"IIIJ 33 S. GELBART AND l. PIATETSKI-SHAPIRO for all grossencharacters X.

13. 1. To relate L(s, 7i', X) to 1/I*(s,'P,fJ x,F) we need to express 1/1* as a product of local integrals of the form I/I(s, Wy, W Xy ,4»y). In greater generality, this Euler product decomposition is sketched in [PiatetskiShapiro]. To treat the explicit case at hand, we assume that the "first Fourier coefficients" W:',. "X of 'P(g) and 8 x (cf. 2» are of the form lJ Wy(g) W:',. "y) and Wx (g) IE "Ir(rx ,1/1-1). 2. '" as above, and Re(s)~O, 1/I*(s,'P,fJ x ,F"') = TI"I'(s, Wy, W ,4»y). ) Proof Replacing E by the series defining it, we have 1/I*(s,'P,fJ x,F"') = 2 f f 'P(g) fJ x (g) E(g,F,s)dg.

### A Course of Mathematical Analysis (Vol. 2) by Nikolsky S.M.

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