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出版者:Cambridge University Press

作者:Scholl, A. J.; Taylor, Richard Lawrence; Taylor, R. L.

出品人:

页数:504

译者:

出版时间:1998-11-26

价格:GBP 63.00

装帧:Paperback

isbn号码:9780521644198

丛书系列:London Mathematical Society Lecture Note Series

图书标签:

Galois Representations
Arithmetic Geometry
Algebraic Geometry
Number Theory
Representation Theory
Moduli Spaces
L-functions
Étale Cohomology
Langlands Program
Automorphic Forms

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具体描述

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.

Galois Representations in Arithmetic Algebraic Geometry: A Deeper Dive into Geometric Structures and Number Theoretic Connections This book delves into the intricate relationship between Galois representations and arithmetic algebraic geometry, exploring how profound symmetries in algebraic varieties over number fields manifest as representations of Galois groups. It offers a comprehensive examination of these fundamental objects, their construction, properties, and their pivotal role in unlocking deep arithmetic invariants of geometric objects. The narrative begins by laying a solid foundation in the theory of Galois representations. It systematically introduces the concept of a Galois representation, focusing on those arising from the action of the absolute Galois group of a number field on various moduli spaces and cohomology groups associated with algebraic varieties. Key examples such as Tate modules of elliptic curves and abelian varieties, as well as étale cohomology of smooth projective varieties, are thoroughly analyzed. The book meticulously details the definition and significance of these representations, emphasizing their connection to the arithmetic of the underlying geometric structures. A significant portion of the exposition is dedicated to the study of the deformation theory of Galois representations. This area, crucial for understanding the landscape of possible Galois representations and their moduli, is treated with considerable rigor. The book explores the infinitesimal and higher-order deformations of a given Galois representation, introducing the essential tools and concepts like the deformation functor and its associated universal object. The interplay between geometric moduli spaces of certain objects (e.g., families of elliptic curves) and the deformation spaces of their associated Galois representations is a recurring theme, highlighting the power of this correspondence. Furthermore, the text provides an in-depth treatment of the properties and classifications of Galois representations. It delves into the notions of irreducibility, semisimplicity, and the concept of "crystalline" or "de Rham" representations in the context of p-adic fields, linking them to the arithmetic of varieties over finite fields and p-adic fields respectively. The book also explores the profound implications of the inverse Galois problem and its reformulations in the context of algebraic geometry, examining how Galois representations can encode information about the structure of Galois groups themselves. The authors skillfully weave together concepts from algebraic geometry, number theory, and representation theory. For instance, the construction of Galois representations from étale cohomology is meticulously explained, demonstrating how the geometric structure of a variety over a number field dictates the representation afforded by its étale fundamental group or higher étale cohomology. The book explores the rich properties of these representations, including their behavior under functoriality, base change, and their connections to modular forms and automorphic representations. The latter is a particularly important area, where the book discusses how Galois representations arising from geometry are conjectured to be automorphic, a cornerstone of the Langlands program. The book also addresses advanced topics, such as the theory of motives and their role in unifying different cohomology theories and Galois representations. It explores how the concept of a Chow ring and its relation to cohomology provides further insight into the structure of Galois actions on geometric objects. The p-adic aspect of Galois representations is given due attention, with discussions on their properties over p-adic fields and their connections to p-adic Hodge theory. Throughout the text, numerous examples and detailed calculations are presented to illustrate the abstract concepts. These concrete instances range from classical examples like the Galois representations attached to elliptic curves and modular forms to more recent developments in the study of Galois representations arising from higher-dimensional varieties and their associated zeta functions. The book aims to equip the reader with the necessary theoretical framework and computational tools to understand and contribute to this vibrant field. This volume is designed for graduate students and researchers in mathematics with a strong background in algebraic geometry and number theory. It serves as an invaluable resource for those seeking a comprehensive understanding of the sophisticated interplay between geometric structures and arithmetic symmetries, paving the way for further exploration into cutting-edge research questions at the intersection of these disciplines.

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这本书的装帧和排版给我留下了极其专业的印象，这对于阅读涉及大量符号和复杂结构的书籍至关重要。纸张的质感和字体的选择都体现了出版方对学术质量的尊重，这在阅读体验上有着潜移默化的影响。然而，更重要的是其内容的组织方式。我非常关注它如何处理那些源自古典数论的动机如何被现代的几何语言所重塑和提升的过程。期望看到的是一种清晰的叙事线索，如何从费马大定理的深层结构，逐步过渡到更广阔的算术世界图景。如果作者能巧妙地在严格证明的严密性和概念引入的启发性之间找到一个完美的平衡点，那么这本书的价值将无可估量。毕竟，对于许多研究生乃至年轻研究者来说，这样的专著是他们建立起自己研究框架的基石。我特别希望看到作者在引入新的范式时，能提供足够的历史背景和前因后果的铺垫，避免让初次接触这些概念的读者迷失在纯粹的符号运算之中。好的教科书或专著，不仅要告诉我们“是什么”，更要解释“为什么是这样”。

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这部书的整体论调，从我初步的感知来看，是极其审慎和要求极高的。它似乎并非旨在普及某一概念，而是力求成为该领域内专业人士和顶尖学子的标准参考手册。在阅读这类作品时，我特别关注作者在面对“领域间鸿沟”时的处理策略。代数几何的语言和伽罗瓦理论的结构有时会显得格格不入，如何通过精妙的结构选择来弥合这些差异，是衡量一部跨学科著作水平的关键。我希望能看到对经典对象（比如椭圆曲线或更一般的代数簇）的伽罗瓦表示，如何被现代工具重新审视和深化。如果作者能够提供一些“小费”，关于如何将这些高级理论应用于计算或构造性的例子，那将是雪中送炭。总而言之，这本书散发出一种严肃的、不妥协的学术氛围，它承诺的，是一次对数学结构最深层奥秘的近距离接触，无疑将是书架上最重要的珍藏之一，尽管阅读过程注定是充满挑战的。

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这部著作的标题本身就预示着它将是一场深刻而艰深的学术旅程，它将引导读者深入到现代数学的诸多前沿交叉领域。从封面上就能感受到那种严谨而厚重的学术气息，它不是那种轻松的读物，更像是为那些已经对代数几何和数论有相当基础的探险家准备的地图集。我期待它能在伽罗瓦表示与算术代数几何的交汇点上，提供一种整合性的、全新的视角。特别吸引人的是，它承诺将阐释那些看似孤立的数学结构是如何通过这种表示法紧密联系起来的，那种不同领域概念的融会贯通，是真正令人兴奋的学术体验。我希望作者能够清晰地勾勒出这条脉络，尤其是在处理那些高度抽象的构造时，能提供足够的直觉辅助，而不是仅仅罗列定理和证明。这本书的深度必然要求读者付出巨大的智力投入，但这正是追求数学真理的魅力所在——揭示隐藏在复杂表象之下的根本统一性。这本书无疑是这个特定研究方向上一个里程碑式的贡献，它设定了一个极高的标准，要求读者具备驾驭高维抽象空间的导航能力。

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拿到手后，我立刻翻阅了目录和前言，感受到了作者试图构建一个宏大理论框架的雄心。目录的结构显示出对逻辑递进的精心设计，章节之间的过渡似乎经过了细致的打磨，力图实现从基础工具到核心应用平稳的攀升。我最感兴趣的部分是如何处理动机化（motivic）理论与伽罗瓦表示的实际联系，这通常是连接代数几何直观与分析数论精确性的关键环节。在这样的高深领域，理论的“可操作性”往往是一个挑战，即如何将抽象的结构转化为可以实际计算或推导的工具。如果本书能提供一些关于这些工具如何应用于解决具体算术问题的实例分析，哪怕只是概念性的概述，都将极大地提升其实用价值。我十分期待它能够揭示出，在最深层次上，几何的“形”是如何编码了数论的“量”的，这种深刻的映照关系，正是吸引我深入研读这部作品的核心动力。这种跨越学科藩篱的洞察力，是真正大师级著作的标志。

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从一位资深数学爱好者的角度来看，这类作品的价值往往体现在其对“领域前沿”的定义能力上。它不仅是现有知识的汇编，更应该是一张指向未来研究方向的导航图。我希望这本书能够大胆地提出一些尚未完全解决或仍存在争议的问题，并从作者的视角提供一些前瞻性的见解。在探讨伽罗瓦表示时，对局部-整体原理的深入讨论是不可或缺的，特别是如何在非阿基米德域上的表现能够为全局的黎曼猜想类问题提供深刻的启示。这本书的篇幅和深度暗示了它可能对某些技术细节进行了非常详尽的阐述，这既是优点也是挑战。对于那些需要频繁查阅技术细节的读者来说，清晰的索引和详尽的参考文献列表至关重要。我个人更倾向于那些能够在复杂理论中提炼出核心哲学思想的论述，让读者在掌握工具的同时，也能理解这些工具背后的数学哲学。

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