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 types of half-integral weight.- interval integrals of cohomology periods that are represented via 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 types and generalized modular relations.- A comment on zeta capabilities 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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Gives us a direct realisation of a Sh and makes the commutativity of the diagram clear. e. M ® K is an irreducible a-module. Again we want to assume R that R contains enough roots of unity. HARDER We consider H\r o,Bo,M). We have fITr ~ ~ ~) ro,Bo uo (( ={(~ ~)ltE~;X'UE m}= ro,vo·w Iu. <9] and W is cyclic of orderfour generated by We always identify we r ={(~ ~) ItE (m/a)X, o,B o with its image in Do UEm/a}. Since we assume that IG Iis invertible in our ring R we see that the action of (j 0 on M is semisimple and it is obvious that H 1 ( r o,uo,M) = Hom ( r o,vo,M Uo ) where we have to take into account that M ii 0 = M r 0, v 0 • (The notation M Uo means of course that we take the invariants).

We call S: 'if -+ '" the Shimura map. Its "kernel"-those cuspidal 'iT which map to non-cuspidal '" -consists precisely of those 'iT which come from automorphic forms on GL(l). That a similar situation arises with the lifting of cusp forms from GL(2) to GL(3) (cf. [GeJa] ) cannot be coincidental. 7. Though we have not written down all the details, it seems likely we can prove that the Land E factors of 'iT v (and their twists by Xv) completely determine 'iT v' From this it follows that (i) the Shimura map S: 'if -+ '" is I-to-I ; and (ii) strong multiplicity one holds for A 0 (w).

We denote by Ep the completion at P, by mp C F p the ring of p-adic integers and by mE,p = mp the ring ofP-adic integers. We drop the index E if it is clear which field we refer to. m; m;. We put Up = and Up = The place ofF at infinity will be denoted by co and the completion F (() is canonically identified with (C. The ring of adeles of F is denoted by A and by the letter we denote the group ideles ofF. Ifwe refer to another field E we write AE ,I E . c, .... e. x p' Xq are the p, q components.

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