Cover image for Arithmetic Geometry and Number Theory.
Arithmetic Geometry and Number Theory.
Title:
Arithmetic Geometry and Number Theory.
Author:
Weng, Lin.
ISBN:
9789812773531
Personal Author:
Physical Description:
1 online resource (411 pages)
Series:
Series on Number Theory & Its Applications ; v.1

Series on Number Theory & Its Applications
Contents:
Contents -- Foreword -- Preface -- On Local y-Factors -- 1 Introduction -- 2 Basic Properties of Local y-Factors -- 2.1 Multiplicativity -- 2.2 Stability -- 2.3 Remarks -- 3 Local Converse Theorems -- 3.1 The case of GLn(F) -- 3.2 A conjectural LCT -- 3.3 The case of SO2n+1(F) -- 4 Poles of Local y-Factors -- 4.1 The case of G = SO2n+1 -- 4.2 Other classical groups -- Deligne Pairings over Moduli Spaces of Punctured Riemann Surfaces -- 1 WP Metrics and TZ Metrics -- 2 Line Bundles over Moduli Spaces -- 3 Fundamental Relations on MgN' Algebraic Story -- 4 Fundamental Relation on MgN- Arithmetic Story -- 5 Deligne Tuple in General -- 6 Degeneration of TZ Metrics: Analytic Story -- References -- Vector Bundles on Curves over Cp -- 1 Introduction -- 2 Complex Vector Bundles -- 3 Fundamental Groups of p-Adic Curves -- 4 Finite Vector Bundles -- 5 A Bigger Category of Vector Bundles -- 6 Parallel Transport on Bundles in Bxcp -- 7 Working Outside a Divisor on Xcp -- 8 Properties of Parallel Transport -- 9 Semistable Bundles -- 10 A Simpler Description of Bxcp D -- 11 Strongly Semistable Reduction -- 12 How Big are our Categories of Bundles? -- 13 Representations of the Fundamental Group -- 14 Mumford Curves -- References -- Absolute CM-periods -- Complex and p-Adic -- 1 Introduction -- 2 Notation -- 2.1 Complex Theory -- 2.2 p-Adic Theory -- References -- Special Zeta Values in Positive Characteristic -- 1 Introduction -- 2 Carlitz Theory -- 3 Anderson-Thakur Theory -- 4 t-Motives -- 5 Algebraic Independence of the Special Zeta Values -- References -- Automorphic Forms & Eisenstein Series and Spectral Decompositions -- Day One: Basics of Automorphic Forms -- 1 Basic Decompositions -- 1.1 Langlands Decomposition -- 1.2 Reduction Theory: Siegel Sets -- 1.3 Moderate Growth and Rapidly Decreasing -- 1.4 Automorphic Forms.

2 Structural Results -- 2.1 Moderate Growth and Rapid Decreasing -- 2.2 Semi-Simpleness -- 2.3 3-Finiteness -- 2.4 Philosophy of Cusp Forms -- 2.5 L2-Automorphic Forms -- Day Two: Eisenstein Series -- 3 Definition -- 3.1 Equivalence Classes of Automorphic Representations -- 3.2 Eisenstein Series and Intertwining Operators -- 3.3 Convergence -- 4 Constant Terms of Eisenstein Series -- 5 Fundamental Properties of Eisenstein Series -- Day Three: Pseudo-Eisenstein Series -- 6 Paley-Wiener Functions -- 6.1 Paley-Wiener Functions -- 6.2 Fourier Transforms -- 6.3 Paley-Wiener on p -- 7 Pseudo-Eisenstein Series -- 8 First Decomposition of L2(G(F)\G(A))\ -- 8.1 Inner Product Formula for P-ESes -- 8.2 Decomposition of L2-Spaces According to Cuspidal Data -- 8.3 Constant Terms of P-SEes -- 9 Decomposition of Automorphic Forms According to Cuspidal Data -- 9.1 Main Result -- 9.2 Langlands Operators -- 9.3 Key Bridge -- Day Four: Spectrum Decomposition: Residual Process -- 10 Why Residue? -- 10.1 Pseudo-Eisenstein Series and Residual Process -- 10.2 What do we have? -- 10.3 Difficulties -- 11 Main Results -- 11.1 Functional Analysis -- 11.2 Main Theorem: Rough Version -- 11.3 Main Theorem: Refined Version -- 11.4 How to Prove? -- Day Five: Eisenstein Systems and Spectral Decomposition (II) -- 12 Eisenstein Systems -- 12.1 Relative Theory -- 12.2 Discrete Spectrum -- 12.3 Eisenstein Systems -- 13 Why or Better How to Get Theorem B?! -- 13.1 Bridge -- 13.2 Basic Facts -- 14 Spectrum Decomposition: Levi Interpretation -- 15 Spectral Decomposition: Arthur's Version of Langlands -- 15.1 In Terms of Levi -- 15.2 Relation with Residual Approach: The Proof -- Day Six: Arthur's Truncation and Meromorphic Continuation -- 16 Arthur's Analytic Truncation -- 16.1 Positive Chamber and Positive Cone -- 16.2 Arthur's Analytic Truncation.

17 Meromorphic Continuation of Eisenstein Series: Deduction -- 17.1 Preparation -- 17.2 Deduction to Relative Rank 1 -- 18 The Situation of Relative Rank 1 -- 18.1 Working Site -- 18.2 Constant Terms -- 18.3 Basic Properties -- 18.4 Functional Equation for Truncated Eisenstein Series -- 18.5 Application of Resolvent Theory -- 18.6 Injectivity -- 18.7 Meromorphic Continuation -- 18.8 Functional Equation -- References -- Geometric Arithmetic: A Program -- A Representation of Galois Group Stability and Tannakian Category -- A.l. Summary -- A.2. Non-Abelian CFT for Function Fields over C -- A.3. Towards Non-Abelian CFT for Global Fields -- B Non-Abelian L-Functions -- B.l. Non-Abelian Zeta Functions for Curves -- Appendix to B.l: Weierstrass Groups -- B.2. New Non-Abelian Zeta Functions for Number Fields -- B.3. Non-Abelian L-Functions for Number Fields -- B.4. Geometric and Analytic Truncations: A Bridge -- B.5. Rankin-Selberg Method -- C Explicit Formula & Functional Equation and Geo-Ari Intersection -- C.l. The Riemann Hypothesis for Curves -- C.2. Geo-Ari Intersection: A Mathematics Model -- C.3. Towards a Geo-Ari Cohomology in Lower Dimensions -- C.4. Riemann Hypothesis in Rank Two -- References -- Appendix: Non-Abelian CFT for Function Fields over C -- 1. Refined Structures for Tannakian Categories -- 2. An Example -- 3. Reciprocity Map -- 4. Main Theorem -- 5. Proof -- 6. An Application to Inverse Galois Problem -- References.
Abstract:
Mathematics is very much a part of our culture; and this invaluable collection serves the purpose of developing the branches involved, popularizing the existing theories and guiding our future explorations.More precisely, the goal is to bring the reader to the frontier of current developments in arithmetic geometry and number theory through the works of Deninger-Werner in vector bundles on curves over p-adic fields; of Jiang on local gamma factors in automorphic representations; of Weng on Deligne pairings and Takhtajan-Zograf metrics; of Yoshida on CM-periods; of Yu on transcendence of special values of zetas over finite fields. In addition, the lecture notes presented by Weng at the University of Toronto from October to November 2005 explain basic ideas and the reasons (not just the language and conclusions) behind Langlands' fundamental, yet notably difficult, works on the Eisenstein series and spectral decompositions.And finally, a brand new concept by Weng called the Geometric Arithmetic program that uses algebraic and/or analytic methods, based on geometric considerations, to develop the promising and yet to be cultivated land of global arithmetic that includes non-abelian Class Field Theory, Riemann Hypothesis and non-abelian Zeta and L Functions, etc.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Subject Term:

Added Author:
Electronic Access:
Click to View
Holds: Copies: