Eisenstein Series and Applications pdf epub mobi txt 电子书 下载 2026

简体网页||繁体网页

☆☆☆☆☆

出版者:Springer Verlag

作者:Gan, Wee Teck (EDT)/ Kudla, Stephen S. (EDT)/ Tschinkel, Yuri (EDT)

出品人:

页数:324

译者:

出版时间:2008-1

价格:$ 123.17

装帧:HRD

isbn号码:9780817644963

丛书系列:Progress in Mathematics

图书标签:

• 数学
• Eisenstein series
• Modular forms
• Number theory
• Arithmetic
• L-functions
• Automorphic forms
• Representation theory
• Algebraic number theory
• q-series
• Special functions

Eisenstein Series and Applications A Comprehensive Exploration of Modular Forms, Number Theory, and Spectral Geometry This volume offers an in-depth and rigorous treatment of Eisenstein series, charting their profound connections across number theory, the theory of modular forms, and modern spectral geometry. Far from a mere introduction, this text delves into the intricate machinery that underpins these crucial mathematical objects, providing a unified framework for understanding their behavior and diverse applications. The book begins with a foundational treatment of the modular group $ ext{SL}_2(mathbb{Z})$ and its action on the upper half-plane $mathcal{H}$. We meticulously construct the principal congruence subgroups and detail the geometric structure of the modular surface, emphasizing the fundamental domain and the role of cusps. This sets the stage for the rigorous definition of Eisenstein series. We explore the classical Eisenstein series, $E_k(z)$, derived from the Poincaré series, analyzing their transformation properties under the modular group and establishing their automorphy. Crucially, the text details the meromorphic continuation and the functional equations satisfied by these series, treating the central $L$-functions associated with them—specifically the Riemann zeta function and Dirichlet $L$-functions—as immediate consequences of the constant terms of these series evaluated at the cusps. A significant portion of the work is dedicated to the theory of Fourier expansions, or Fourier coefficients, of Eisenstein series. We introduce the divisor functions $sigma_k(n)$ and establish the precise relationships between the coefficients of $E_k(z)$ and these number-theoretic sums. The discussion extends to Kronecker’s limit formula and its historical and modern significance in the calculation of class numbers and the study of quadratic forms. The narrative then transitions into the modern framework of modular forms theory. The text provides a thorough exposition of Maass wave forms, which generalize classical holomorphic modular forms to functions that satisfy the Laplace-Beltrami equation on the modular surface. Eisenstein series in this context, often referred to as continuous spectrum Eisenstein series, are central to understanding the spectral decomposition of the Laplacian on the quotient space $ ext{SL}_2(mathbb{Z}) setminus mathcal{H}$. We develop the necessary tools from automorphic representation theory to define these non-holomorphic series, analyze their poles and residues, and connect their growth behavior to the analytic properties of the Selberg zeta function. The spectral interpretation forms a powerful lens through which to view the arithmetic properties encoded in the Fourier coefficients. The applications section is rich and multifaceted. We devote substantial attention to the Langlands correspondence in the context of $ ext{GL}_2$. The construction of Eisenstein series provides the fundamental initial step in the construction of general automorphic representations for $ ext{GL}_n$. This involves a detailed study of induced representations, particularly the principal series representations, and the construction of the "standard" Eisenstein series via the notion of Jacquet-Langlands functoriality in the $ ext{GL}_2$ setting over number fields. The functional equations for Artin $L$-series and the deep connections to Galois representations are explored through the lens of the Langlands functoriality principle, positioning Eisenstein series as the arithmetic backbone of this vast program. Further applications delve into the realm of geometry and topology. The connection between the spectrum of the Laplacian on hyperbolic manifolds (which includes the modular surface) and the structure of automorphic forms is elaborated upon. We explore how the eigenvalues of the Laplacian (which appear as parameters in the continuous spectrum Eisenstein series) are intimately related to the arithmetic complexity of the underlying quotient spaces. This section bridges analysis with geometric invariants. Finally, the text concludes with a modern perspective on the arithmetic of special values. We examine the relationship between Eisenstein series and values of $L$-functions at critical points. Advanced topics include the study of Petersson products, the theory of periods, and the role of Eisenstein series in conjectures related to special values, such as those arising from the theory of motivic cohomology and arithmetic intersection theory. The book assumes a solid undergraduate background in complex analysis and abstract algebra, dedicating sufficient preparatory material to ensure readers can navigate the subtleties of $p$-adic methods and representation theory when they are introduced. The goal is to equip the advanced graduate student or researcher with the foundational knowledge and the specialized tools necessary to conduct cutting-edge research utilizing the full breadth of Eisenstein series theory. The text incorporates numerous worked examples and challenging exercises designed to solidify theoretical understanding.

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆

评分☆☆☆☆☆