Galois representations in arithmetic algebraic geometry / edited by A.J. Scholl, R.L. Taylor.

Includes bibliographical references.

Print version record.

This book contains conference proceedings from the 1996 Durham Symposium on 'Galois representations in arithmetic algebraic geometry'. The title was interpreted loosely and the symposium covered recent developments on the interface between algebraic number theory and arithmetic algebraic geometry. The book reflects this and contains a mixture of articles. Some are expositions of subjects which have received substantial attention, e.g. Erez on geometric trends in Galois module theory; Mazur on rational points on curves and varieties; Moonen on Shimura varieties in mixed characteristics; Rubin and Scholl on the work of Kato on the Birch-Swinnerton-Dyer conjecture; and Schneider on rigid geometry. Others are research papers by authors such as Coleman and Mazur, Goncharov, Gross and Serre.

Cover; Title; Copyright; Contents; Preface; List of Participants; Lecture programme; The Eigencurve; Chapter 1. Rigid analytic varieties.; 1.1 Rigid analytic spaces attached to complete local noetherianrings.; 1.2. Irreducible components and component parts.; 1.3. Fredholm varieties.; 1.4 Weight space.; 1.5. The eigencurve as the Fredholm closure of the classical modularlocus. (Statement of the main theorems); Chapter 2. Modular Forms.; 2.1 Affinoid sub-domains in modular curves.; 2.2 Eisenstein series.; 2.3. Katz p-adic Modular Functions.

2.4 Convergent modular forms and Katz modular functions. Chapter 3. Hecke Algebras; 3.1 Hecke eigenvectors and generalized eigenvectors.; 3.2. Action on Mk; 3.3 Action on Katz Modular Functions.; 3.4. Action on Mt(N).; 3.5. Action on weight K forms.; 3.6. Remarks about cusp forms and Eisenstein series.; Chapter 4. Fredholm determinants.; 4.1. Completely continuous operators and Fredholm determinants; 4.2. Factoring Characteristic Series.; 4.3. Analytic variation of the Fredholm determinant.; 4.4 The Spectral Curves.; Chapter 5. Galois representations and pseudo-representations.

5.1. Deforming representations and pseudo-representations. 5.2. Pseudo-representations attached to Katz modular functions.; Chapter 6. The Eigencurve.; 6.1. The definition of the eigencurve.; 6.2. The points of the eigencurve are overconvergent eigenforms.; 6.3. The projections of the eigencurve to the spectral curves.; 6.4. The Eisenstein curve.; Chapter 7. The eigencurve as a finite cover of a spectral curve.; 7.1. Local pieces.; 7.2. Gluing.; 7.3. The relationships among the curves; 7.4. D is reduced.; 7.5. Equality of D and C; 7.6. Consequences of the relationship between D and C.

7 Equivariant motivesReferences; Mixed elliptic motives; 1 Introduction; 2 Two basic examples: a survey; 3 Mixed motives and motivic Lie algebras; 4 Conjectures on the motivic Galois group; 5 Towards the Lie coalgebra; 6 Reflections on elliptic motivic complexes; 7 The complexes; 8 The regulator integrals, Eisenstein-Kroneckerseries and a conjecture on; 9 The complexes B(E; n).and motivic elliptic polylogarithms; 10 Elliptic Chow polylogarithms and generalized Eisenstein-Kronecker series; REFERENCES; On the Satake isomorphism; 1. The algebraic group; 2. The Gelfand pair (G, K)

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