This volume is the offspring of a week-long workshop on "Galois groups over Q and related topics," which was held at the Mathematical Sciences Research Institute during the week March 23-27, 1987. The organizing committee consisted of Kenneth Ribet (chairman), Yasutaka Ihara, and Jean-Pierre Serre. The conference focused on three principal themes: 1. Extensions of Q with finite simple Galois groups. 2. Galois actions on fundamental groups, nilpotent extensions of Q arising from Fermat curves, and the interplay between Gauss sums and cyclotomic units. 3. Representations of Gal(Q/Q) with values in GL(2)j deformations and connections with modular forms. Here is a summary of the conference program: . G. Anderson: "Gauss sums, circular units and the simplex" . G. Anderson and Y. Ihara: "Galois actions on 11"1 ( ... ) and higher circular units" . D. Blasius: "Maass forms and Galois representations" . P. Deligne: "Galois action on 1I"1(P-{0, 1, oo}) and Hodge analogue" . W. Feit: "Some Galois groups over number fields" . Y. Ihara: "Arithmetic aspect of Galois actions on 1I"1(P - {O, 1, oo})" - survey talk . U. Jannsen: "Galois cohomology of i-adic representations" . B. Matzat: - "Rationality criteria for Galois extensions" - "How to construct polynomials with Galois group Mll over Q" . B. Mazur: "Deforming GL(2) Galois representations" . K. Ribet: "Lowering the level of modular representations of Gal( Q/ Q)" . J-P. Serre: - Introductory Lecture - "Degree 2 modular representations of Gal(Q/Q)" . J.

Inhalt

Normalization of the Hyperadelic Gamma Function.- 1.Gaussian units.- 2.The structure of Mr,?.- 3.Normalization.- Maass Forms and Galois Representations.- 1.Automorphic forms via representations.- 2.Holomorphic automorphic forms for GSp(4).- 3.Geometric automorphic forms.- 4.Reductions.- 5.Heuristics and conjectures.- 6.Transfer of problem to GSp(4, A?).- 7.Hypothesis 1: structure of global L-packets for GSp(4).- 8.An analytic estimate for the conjugates of Maass forms.- 9.Hypothesis 2: Galois representations attached to Siegel modular forms of higher weight.- 10.The main theorem.- Le Groupe Fondamental De La Droite Projective Moins Trois Points.- 0.Terminologie et notations.- 1.Motifs mixtes.- 2.Exemples.- 3.Torseurs sous Z(n).- 4.Rappels sur les Ind-objets.- 5.Géométrie algébrique dans une catégorie tannakienne.- 6.Le groupe fondamental d'une catégorie tannakienne.- 7.Géométrie algébrique dans la catégorie tannakienne des systèmes de réalisations: interprétations.- 8.Extensions itérées de motifs de Tate.- 9.Rappels sur les groupes unipotents.- 10.Théories du ?1.- Groupoïdes.- Théorie classique.- Théorie profinie.- Théorie algébrique.- 11.Le Frobenius cristallin du ?1 de Rham.- 12.La filtration de Hodge du ?1.- 13.Le ?1 motivique.- 14.Exemple: le ?1 motivique de Gm.- 15.Points bases à l'infini.- Théorie classique.- Théorie profinie.- Théorie algébrique.- Compatibilités.- Théorie motivique.- 16.P1 moins trois points: un quotient de ?1 motivique.- 17.Relations de distribution: voie géométrique.- 18.Le torseur Pd.k + ( ? 1 )k ? Pd.k est de torsion: voie géométrique.- 19.Comparaison des Z(h)-torseurs des paragraphes 3 et 16.- Index des notations.- The Galois Representation Arising from P1 ? {0, 1,?} and Tate Twists of Even Degree.- 1.Preliminaries and statement of the Theorem.- 2.Reducing the proof of the Theorem to two key lemmas.- 3.Proof of Key Lemma A.- 4.Proof of Key Lemma B.- 5.Remarks and discussion.- On the ?-Adic Cohomology of Varieties Over Number Fields and its Galois Cohomology.- 1.The basic conjecture.- 2.Connections with algebraic K-theory.- 3.Connections with Iwasawa theory.- 4.Global results.- 5.The local case.- 6.The case n ? i + 1 ? 2n.- 7.The case i = 1: abelian varieties.- Rationality Criteria for Galois Extensions.- 1.Fundamental groups.- 2.Class numbers of generators.- 3.Topological automorphisms.- 4.Braids.- 5.Braids and topological automorphisms together.- Deforming Galois Representations.- 1Universal deformation of representations.- 1.1Deformations.- 1.2Existence of universal deformation rings.- 1.3Functoriality.- 1.4One-dimensional representations.- 1.5The duality involution.- 1.6Obstructions.- 1.7Ordinary representations.- 1.8Schur-type results.- 1.9A few simple examples.- 1.10 Global Galois representations.- 1.11Remarks on Galois representations to SL2(Fp).- 1.12Neat residual representations.- 1.13Neat S3-extensions of ?.- 2.The internal structure of universal deformation spaces.- 2.1General glossary.- 2.2Special dihedral representations.- 2.3The origin.- 2.4The globally dihedral locus.- 2.5The ordinary locus.- 2.6The inertially reducible locus.- 2.7The inertially metabelian and the inertially dihedral locus.- 2.8Loci of constant p-adic Hodge type.- Galois Groups of Poincaré Type Over Algebraic Number Fields.- 1.The function field case.- 2.Classification of Demuskin groups.- 3.p-adic number fields.- 4.n-local fields.- 5.The absolute Galois group of a p-adic number field.- 6.Global number fields.