# Download PDF by Greg W. Anderson (auth.), Y. Ihara, K. Ribet, J.-P. Serre: Galois Groups over ℚ: Proceedings of a Workshop Held March

By Greg W. Anderson (auth.), Y. Ihara, K. Ribet, J.-P. Serre (eds.)

This quantity is the offspring of a week-long workshop on "Galois teams over Q and similar topics," which used to be held on the Mathematical Sciences examine Institute throughout the week March 23-27, 1987. The organizing committee consisted of Kenneth Ribet (chairman), Yasutaka Ihara, and Jean-Pierre Serre. The convention occupied with 3 primary topics: 1. Extensions of Q with finite uncomplicated Galois teams. 2. Galois activities on primary teams, nilpotent extensions of Q bobbing up from Fermat curves, and the interaction among Gauss sums and cyclotomic devices. three. Representations of Gal(Q/Q) with values in GL(2)j deformations and connections with modular kinds. here's a precis of the convention application: • G. Anderson: "Gauss sums, round devices and the simplex" • G. Anderson and Y. Ihara: "Galois activities on 11"1 ( ••• ) and better round devices" • D. Blasius: "Maass types and Galois representations" • P. Deligne: "Galois motion on 1I"1(P-{0, 1, oo}) and Hodge analogue" • W. Feit: "Some Galois teams over quantity fields" • Y. Ihara: "Arithmetic point of Galois activities on 1I"1(P - {O, 1, oo})" - survey speak • U. Jannsen: "Galois cohomology of i-adic representations" • B. Matzat: - "Rationality standards for Galois extensions" - "How to build polynomials with Galois workforce Mll over Q" • B. Mazur: "Deforming GL(2) Galois representations" • ok. Ribet: "Lowering the extent of modular representations of Gal( Q/ Q)" • J-P. Serre: - Introductory Lecture - "Degree 2 modular representations of Gal(Q/Q)" • J.

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**Example text**

Shimura has shown [Shl] that Sk(f 1(N), e) = Sk(f 1(N), Q) @ and that Sk(f 1(N), Q) has a basis of forms whose Fourier expansions at 00 have bounded denominators. It follows easily, from the definition of PROOF: e, the adelic action in [Shl], that the RCt) respect Sk(f 1(N), Q). Thus, let 91, ... ,9t be a basis of Sk(f 1(N),Q) with Fourier coefficients in 1. Choose symmetric matrices Sl,"" Sf so that linear map sending I to (a(J,Sl), ... ,a(J,S,)) is injective on Sk(f 1(N),Q). Let IE L, t I = L:>~i 9i i=l (ai E Q).

Section 7 introduces the hypothesis (HI) and derives first consequences. Section 8 establishes an analytic estimate which is basic to the method of Deligne and Serre. It also contains a conjecture about global Whittaker models which may be of independent interest. This section relies upon the results of [BHR]. Section 9 studies the hypothesis (H2). Section 10 shows that (HI) and (H2) imply that the sought Galois representation exists. 8) the Mathematical Sciences Research Institute, Berkeley, whose hospitality is warmly acknowledged.

These facts are easy consequences of the functorial properties of Mumford's construction. 5) Suppose now that R = C. Then each element f of Mk(C) defines an analytic modular form on analytic families of abelian varieties, and thus, by a simple construction on the Siegel upper-half space f)g, an element of Mk(f(l), C). 1) of f on f)g to that defined here, one simply multiplies by (27l"i)gk. 6) PROPOSITION. 1) satisfies (i) a(O, hp ) == 1 (mod p). (ii) a(s, hp ) == 0 (mod p) if s PROOF: Assume g #- o.