L-Functions and Galois Representations

doi:10.1017/CBO9780511721267

This collection of survey and research articles brings together topics at the forefront of the theory of L-functions and Galois representations. Highlighting important progress in areas such as the local Langlands programme, automorphic forms and Selmer groups, this timely volume treats some of the most exciting recent developments in the field. Included are survey articles from Khare on Serre's conjecture, Yafaev on the André-Oort conjecture, Emerton on his theory of Jacquet functors, Venjakob on non-commutative Iwasawa theory and Vigneras on mod p representations of GL(2) over p-adic fields. There are also research articles by: Böckle, Buzzard, Cornut and Vatsal, Diamond, Hida, Kurihara and R. Pollack, Kisin, Nekovář, and Bertolini, Darmon and Dasgupta. Presenting the very latest research on L-functions and Galois representations, this volume is indispensable for researchers in algebraic number theory.

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Contents

Frontmatter
pp i-vi
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Contents
pp vii-viii
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Preface
pp ix-x
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List of participants
pp xi-xii
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Stark–Heegner points and special values of L-series
pp 1-23
- By Massimo Bertolini, Dipartimento di Matematica, Universita' degli Studi di Milano, Via Saldini 50 20133, Milano, Italy, Massimo.Bertolini@mat.unimi.it, Henri Darmon, McGill University Mathematics Department, 805 Sherbrooke Street West Montreal, QC H3A-2K6 CANADA, darmon@math.mcgill.ca, Samit Dasgupta, Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, MA 02138, U.S.A., dasgupta@math.harvard.edu
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Presentations of universal deformation rings
pp 24-58
- By Gebhard Böckle, Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany, boeckle@iem.uni-due.de
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Eigenvarieties
pp 59-120
- By Kevin Buzzard, Department of Mathematics Imperial College London U.K. buzzard@ic.ac.uk
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Nontriviality of Rankin-Selberg L-functions and CM points
pp 121-186
- By Christophe Cornut, Institut de Mathématiques de Jussieu 4 place Jussieu F-75005 Paris France cornut@math.jussieu.fr, Vinayak Vatsal, University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 Canada vatsal@math.ubc.ca
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A correspondence between representations of local Galois groups and Lie-type groups
pp 187-206
- By Fred Diamond, Department of Mathematics, Brandeis University, Waltham, MA 02454, USA. Current address: Department of Mathematics, King's College London, WC2R 2LS, UK. fred.diamond@kcl.ac.uka
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Non-vanishing modulo p of Hecke L–values and application
pp 207-269
- By Haruzo Hida, Department of Mathematics UCLA Los Angeles, Ca 90095-1555 U.S.A. hida@math.ucla.edu
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Serre's modularity conjecture: a survey of the level one case
pp 270-299
- By Chandrashekhar Khare, Department of Mathematics 155 South 1400 East, Room 233 Salt Lake City, UT 84112-0090 U.S.A. shekhar@math.utah.edu
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Two p-adic L-functions and rational points on elliptic curves with supersingular reduction
pp 300-332
- By Masato Kurihara, Department of Mathematics Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama, 223-8522, Japan kurihara@math.keio.ac.jp, Robert Pollack, Department of Mathematics and Statistics Boston University Boston, MA 02215, USA rpollack@math.bu.edu
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From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey
pp 333-380
- By Otmar Venjakob, Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg, Germany. otmar@mathi.uni-heidelberg.de
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The André-Oort conjecture - a survey
pp 381-406
- By Andrei Yafaev, University College London, Dept. of Mathematics, 25 Gordon Street, WC1H 0AH, U.K. yafaev@math.ucl.ac.uk
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Locally analytic representation theory of p-adic reductive groups: a summary of some recent developments
pp 407-437
- By Matthew Emerton, Northwestern University Department of Mathematics 2033 Sheridan Rd. Evanston, IL 60208-2730 USA emerton@math.northwestern.edu
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Modularity for some geometric Galois representations - with an appendix by Ofer Gabber
pp 438-470
- By Mark Kisin, Department of Mathematics University of Chicago USA kisin@math.uchicago.edu
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The Euler system method for CM points on Shimura curves
pp 471-547
- By Jan Nekovář, Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu, Théorie des Nombres, Case 247, 4, place Jussieu, F-75252 Paris cedex 05, FRANCE, nekovar@math.jussieu.fr
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Représentations irréductibles de GL(2, F) modulo p
pp 548-563
- By Marie-France Vignéras, Institut de Mathematiques de Jussieu 175 rue du Chevaleret Paris 75013 France vigneras@math.jussieu.fr
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L-Functions and Galois Representations

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

List of participants
pp xi-xii

Stark–Heegner points and special values of L-series
pp 1-23

Presentations of universal deformation rings
pp 24-58

Eigenvarieties
pp 59-120

Nontriviality of Rankin-Selberg L-functions and CM points
pp 121-186

A correspondence between representations of local Galois groups and Lie-type groups
pp 187-206

Non-vanishing modulo p of Hecke L–values and application
pp 207-269

Serre's modularity conjecture: a survey of the level one case
pp 270-299

Two p-adic L-functions and rational points on elliptic curves with supersingular reduction
pp 300-332

From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey
pp 333-380

The André-Oort conjecture - a survey
pp 381-406

Locally analytic representation theory of p-adic reductive groups: a summary of some recent developments
pp 407-437

Modularity for some geometric Galois representations - with an appendix by Ofer Gabber
pp 438-470

The Euler system method for CM points on Shimura curves
pp 471-547

Représentations irréductibles de GL(2, F) modulo p
pp 548-563

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