Understanding Galois representations is one of the central goals of number theory. Around 1990, Fontaine devised a strategy to compare such p-adic Galois representations to seemingly much simpler objects of (semi)linear algebra, the so-called etale (phi, gamma)-modules. This book is the first to provide a detailed and self-contained introduction to this theory. The close connection between the absolute Galois groups of local number fields and local function fields in positive characteristic is established using the recent theory of perfectoid fields and the tilting correspondence. The author works in the general framework of LubinTate extensions of local number fields, and provides an introduction to LubinTate formal groups and to the formalism of ramified Witt vectors. This book will allow graduate students to acquire the necessary basis for solving a research problem in this area, while also offering researchers many of the basic results in one convenient location.
Recenzijos
'Much of this material is available in the literature, but it has never been presented so cleanly and concisely in a single place before In this book, Schneider has done a remarkable job of displaying the beauty and power of perfectoid theoretic techniques. His text is sure to occupy and satisfy the attention of students and researchers working on Galois representations, or those who suspect that perfectoid-style techniques might be relevant for their work. We recommend Schneider's text to anyone with even a passing interest in the perfectoid revolution initiated by Scholze.' Cameron Franc, Mathematical Reviews
Daugiau informacijos
A detailed and self-contained introduction to a key part of local number theory, ideal for graduate students and researchers.
| Preface |
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vii | |
| Overview |
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1 | (4) |
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5 | (78) |
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1.1 Ramified Witt Vectors |
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7 | (17) |
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1.2 Unramified Extensions |
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24 | (4) |
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1.3 Lubin--Tate Formal Group Laws |
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28 | (13) |
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1.4 Tilts and the Field of Norms |
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41 | (23) |
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1.5 The Weak Topology on Witt Vectors |
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64 | (4) |
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1.6 The Isomorphism between HL and HEL |
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68 | (7) |
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1.7 A Two-Dimensional Local Field |
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75 | (8) |
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83 | (27) |
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84 | (14) |
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98 | (9) |
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107 | (3) |
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3 An Equivalence of Categories |
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110 | (26) |
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111 | (12) |
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3.2 The Case of Characteristic p Coefficients |
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123 | (6) |
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129 | (7) |
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136 | (8) |
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136 | (1) |
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137 | (1) |
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4.3 (φL, ΓL)-Modules over the Robba Ring |
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138 | (2) |
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4.4 (φL, ΓL)-Modules and Character Varieties |
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140 | (1) |
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4.5 Multivariate (φL, ΓL)-Modules |
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141 | (1) |
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4.6 Variation of (φL, ΓL)-Modules |
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141 | (1) |
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4.7 (φL, ΓL)-Modules and p-adic Local Langlands |
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142 | (2) |
| References |
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144 | (2) |
| Notation |
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146 | (2) |
| Subject Index |
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148 | |
Peter Schneider is a professor in the Mathematical Institute at the University of Münster. His research interests lie within the Langlands program, which relates Galois representations to representations of p-adic reductive groups, as well as in number theory and in representation theory. He is the author of Nonarchimedean Functional Analysis (2001), p-Adic Lie Groups (2011) and Modular Representation Theory of Finite Groups (2012), and he is a member of the National German Academy of Science Leopoldina and of the Academia Europaea.
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