Automorphic Forms, Representation Theory and Arithmetic: by S. Gelbart, G. Harder, K. Iwasawa, H. Jaquet, N.M. Katz, I.

By S. Gelbart, G. Harder, K. Iwasawa, H. Jaquet, N.M. Katz, I. Piatetski-Shapiro, S. Raghavan, T. Shintani, H.M. Stark, D. Zagier

On Shimura’s correspondence for modular varieties of half-integral weight.- interval integrals of cohomology sessions that are represented through Eisenstein series.- Wave entrance units of representations of Lie groups.- On p-adic representations linked to ?p-extensions.- Dirichlet sequence for the crowd GL(n).- Crystalline cohomology, Dieudonné modules and Jacobi sums.- Estimates of coefficients of modular varieties and generalized modular relations.- A comment on zeta features of algebraic quantity fields.- Derivatives of L-series at s = 0.- Eisenstein sequence and the Riemann zeta function.- Eisenstein sequence and the Selberg hint formulation I.

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Extra info for Automorphic Forms, Representation Theory and Arithmetic: Papers presented at the Bombay Colloquium 1979

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15. 1 THEOREM. Suppose w= ® iiv is a unitary cuspidal repre- sentation of half-integral weight. Then: (i) S ( ii) = ® S ( ii v) exists; (ii) S ( 71) is automorphic, and is cuspidal form r. 2 Proof any v. F'\ AX . 3. Fixing I/J, let T be the set of places where S(71 v) = 'lfv may not be defined. According to Section 7, T is precisely the set of finite places where 71 v is supercuspidal but not a theta-representation. For almost all v1T, iffy is a class 1 representation of the form ir( 1-'1' 1-'2) (possibly of the form 71(vt' 2 1Iv -1/\ v I/2 1I v1/4).

Every element ~ e Go (A) can be written withk f e Kf (0). PROOF: We represent ~ e G o( A) by an element i e GLi A). e. det (i) = 1 then the assertion follows from strong approximation for SL 2 • We may modify i by an element ~ e I which we consider as an element of the center of GL 2 (A), then det(~) gets multiplied by Z2 • So the obstruction to get the determinantj equal to one sits in lie. We may modify i by an element in GL 2 (F) and by an element in the inverse image of Kf(O) in GL 2 (A). This means that the obstruction against writing.!

GELBART AND I. PIATETSKI-SHAPIRO CoROLLARY. , for at least one archimedean place vo' wVo is the "even" piece of the Wei! representation. Then there exists a grossencharacter X of F such that In particular, taking F = Q we obtain an alternate proof of the fact that linear combinations of the theta-series L ao 8 x (tt) = X(n) e2win2tl n= -00 exhaust the modular forms of weight 1/2 (as X runs through the set of primitive "even" Dirichlet characters); this result is the principal theorem of [Serre-Stark].

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