Arithmetic and Geometry

doi:10.1017/CBO9781316106877

The 'Arithmetic and Geometry' trimester, held at the Hausdorff Research Institute for Mathematics in Bonn, focussed on recent work on Serre's conjecture and on rational points on algebraic varieties. The resulting proceedings volume provides a modern overview of the subject for graduate students in arithmetic geometry and Diophantine geometry. It is also essential reading for any researcher wishing to keep abreast of the latest developments in the field. Highlights include Tim Browning's survey on applications of the circle method to rational points on algebraic varieties and Per Salberger's chapter on rational points on cubic hypersurfaces.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-viii
- By Luis V. Dieulefait, Universitat de Barcelona, Gerd Faltings, Max-Planck-Institut für Mathematik (Bonn), D. R. Heath-Brown, University of Oxford, Yu. I. Manin, Max-Planck-Institut für Mathematik, Boris Z. Moroz, Max-Planck-Institut für Mathematik (Bonn), Jean-Pierre Wintenberger, Université de Strasbourg
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Introduction
pp ix-ix
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List of participants
pp x-xii
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Trimester Seminar
pp xiii-xxii
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Workshop on Serre's conjecture
pp xxiii-xxvii
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The research conference
pp xxviii-xxxiv
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1 - Galois groups of local fields, Lie algebras and ramification
pp 1-23
- By Victor Abrashkin, University of Durham
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2 - A characterisation of ordinary modular eigenforms with CM
pp 24-35
- By Rajender Adibhatla, Universität Regensburg, Panagiotis Tsaknias, University of Luxembourg
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3 - Selmer complexes and p-adic Hodge theory
pp 36-88
- By Denis Benois, Université de Bordeaux
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4 - A survey of applications of the circle method to rational points
pp 89-113
- By Tim D. Browning, University of Bristol
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5 - Arithmetic differential equations of Painlevé VI type
pp 114-138
- By Alexandru Buium, University of New Mexico, Yuri I. Manin, Max-Planck-Institut für Mathematik
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6 - Differential calculus with integers
pp 139-187
- By Alexandru Buium, University of New Mexico
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7 - Un calcul de groupe de Brauer et une application arithmétique
pp 188-192
- By Jean-Louis Colliot-Thélène, Université Paris-Sud
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8 - Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity
pp 193-216
- By Luis V. Dieulefait, Universitat de Barcelona, Ariel Pacetti, Universidad de Buenos Aires
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9 - Big image of Galois representations and congruence ideals
pp 217-254
- By Haruzo Hida, UCLA, Los Angeles, Jacques Tilouine, Université Paris
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10 - The skew-symmetric pairing on the Lubin–Tate formal module
pp 255-263
- By Mikhail A. Ivanov, Saint-Petersburg State University, Sergei V. Vostokov, University of St. Petersburg
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11 - Equations in matrix groups and algebras over number fields and rings: prolegomena to a lowbrow noncommutative Diophantine geometry
pp 264-282
- By Boris Kunyavskiĭ, Bar-Ilan University
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12 - On the l-adic regulator as an ingredient of Iwasawa theory
pp 283-317
- By Leonid V. Kuz'min, National Research Center Kurchatov Institute
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13 - On a counting problem for G-shtukas
pp 318-350
- By Ngo Dac Tuan, Université de Paris
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14 - Modular forms and Calabi-Yau varieties
pp 351-372
- By Kapil Paranjape, IISER, Mohali, India, Dinakar Ramakrishnan, Caltech
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15 - Derivative of symmetric square p-adic L-functions via pull-back formula
pp 373-400
- By Giovanni Rosso, Université Paris
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16 - Uniform bounds for rational points on cubic hypersurfaces
pp 401-421
- By Per Salberger, Chalmers University of Technology
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17 - Descent on toric fibrations
pp 422-435
- By Alexei N. Skorobogatov, Imperial College London
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18 - On filtrations of vector bundles over P1Z
pp 436-457
- By Aleksander Smirnov, Steklov Institute of Mathematics in St. Petersburg
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19 - On the dihedral Euler characteristics of Selmer groups of Abelian varieties
pp 458-492
- By Jeanine Van Order, The Hebrew University of Jerusalem
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20 - CM values of higher Green's functions and regularized Petersson products
pp 493-503
- By Maryna Viazovska, National University of Kyiv
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Arithmetic and Geometry

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Introduction
pp ix-ix

List of participants
pp x-xii

Trimester Seminar
pp xiii-xxii

Workshop on Serre's conjecture
pp xxiii-xxvii

The research conference
pp xxviii-xxxiv

1 - Galois groups of local fields, Lie algebras and ramification
pp 1-23

2 - A characterisation of ordinary modular eigenforms with CM
pp 24-35

3 - Selmer complexes and p-adic Hodge theory
pp 36-88

4 - A survey of applications of the circle method to rational points
pp 89-113

5 - Arithmetic differential equations of Painlevé VI type
pp 114-138

6 - Differential calculus with integers
pp 139-187

7 - Un calcul de groupe de Brauer et une application arithmétique
pp 188-192

8 - Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity
pp 193-216

9 - Big image of Galois representations and congruence ideals
pp 217-254

10 - The skew-symmetric pairing on the Lubin–Tate formal module
pp 255-263

11 - Equations in matrix groups and algebras over number fields and rings: prolegomena to a lowbrow noncommutative Diophantine geometry
pp 264-282

12 - On the l-adic regulator as an ingredient of Iwasawa theory
pp 283-317

13 - On a counting problem for G-shtukas
pp 318-350

14 - Modular forms and Calabi-Yau varieties
pp 351-372

15 - Derivative of symmetric square p-adic L-functions via pull-back formula
pp 373-400

16 - Uniform bounds for rational points on cubic hypersurfaces
pp 401-421

17 - Descent on toric fibrations
pp 422-435

18 - On filtrations of vector bundles over P1Z
pp 436-457

19 - On the dihedral Euler characteristics of Selmer groups of Abelian varieties
pp 458-492

20 - CM values of higher Green's functions and regularized Petersson products
pp 493-503

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Arithmetic and Geometry

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