Zeta and $L$-functions in Number Theory and Combinatorics

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Publisher : American Mathematical Soc.
ISBN 13 : 1470449005
Total Pages : 106 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Zeta and $L$-functions in Number Theory and Combinatorics by : Wen-Ching Winnie Li

Download or read book Zeta and $L$-functions in Number Theory and Combinatorics written by Wen-Ching Winnie Li and published by American Mathematical Soc.. This book was released on 2019-03-01 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: Zeta and L-functions play a central role in number theory. They provide important information of arithmetic nature. This book, which grew out of the author's teaching over several years, explores the interaction between number theory and combinatorics using zeta and L-functions as a central theme. It provides a systematic and comprehensive account of these functions in a combinatorial setting and establishes, among other things, the combinatorial counterparts of celebrated results in number theory, such as the prime number theorem and the Chebotarev density theorem. The spectral theory for finite graphs and higher dimensional complexes is studied. Of special interest in theory and applications are the spectrally extremal objects, called Ramanujan graphs and Ramanujan complexes, which can be characterized by their associated zeta functions satisfying the Riemann Hypothesis. Explicit constructions of these extremal combinatorial objects, using number-theoretic and combinatorial means, are presented. Research on zeta and L-functions for complexes other than graphs emerged only in recent years. This is the first book for graduate students and researchers offering deep insight into this fascinating and fast developing area.

Beilinson's Conjectures on Special Values of L-Functions

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Publisher : Academic Press
ISBN 13 : 1483263304
Total Pages : 399 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Beilinson's Conjectures on Special Values of L-Functions by : M. Rapoport

Download or read book Beilinson's Conjectures on Special Values of L-Functions written by M. Rapoport and published by Academic Press. This book was released on 2014-07-14 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: Beilinson's Conjectures on Special Values of L-Functions deals with Alexander Beilinson's conjectures on special values of L-functions. Topics covered range from Pierre Deligne's conjecture on critical values of L-functions to the Deligne-Beilinson cohomology, along with the Beilinson conjecture for algebraic number fields and Riemann-Roch theorem. Beilinson's regulators are also compared with those of Émile Borel. Comprised of 10 chapters, this volume begins with an introduction to the Beilinson conjectures and the theory of Chern classes from higher k-theory. The "simplest" example of an L-function is presented, the Riemann zeta function. The discussion then turns to Deligne's conjecture on critical values of L-functions and its connection to Beilinson's version. Subsequent chapters focus on the Deligne-Beilinson cohomology; ?-rings and Adams operations in algebraic k-theory; Beilinson conjectures for elliptic curves with complex multiplication; and Beilinson's theorem on modular curves. The book concludes by reviewing the definition and properties of Deligne homology, as well as Hodge-D-conjecture. This monograph should be of considerable interest to researchers and graduate students who want to gain a better understanding of Beilinson's conjectures on special values of L-functions.

Advanced Analytic Number Theory: L-Functions

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Publisher : American Mathematical Soc.
ISBN 13 : 0821842668
Total Pages : 313 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Advanced Analytic Number Theory: L-Functions by : Carlos J. Moreno

Download or read book Advanced Analytic Number Theory: L-Functions written by Carlos J. Moreno and published by American Mathematical Soc.. This book was released on 2005 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the pioneering work of Euler, Dirichlet, and Riemann, the analytic properties of L-functions have been used to study the distribution of prime numbers. With the advent of the Langlands Program, L-functions have assumed a greater role in the study of the interplay between Diophantine questions about primes and representation theoretic properties of Galois representations. This book provides a complete introduction to the most significant class of L-functions: the Artin-Hecke L-functions associated to finite-dimensional representations of Weil groups and to automorphic L-functions of principal type on the general linear group. In addition to establishing functional equations, growth estimates, and non-vanishing theorems, a thorough presentation of the explicit formulas of Riemann type in the context of Artin-Hecke and automorphic L-functions is also given. The survey is aimed at mathematicians and graduate students who want to learn about the modern analytic theory of L-functions and their applications in number theory and in the theory of automorphic representations. The requirements for a profitable study of this monograph are a knowledge of basic number theory and the rudiments of abstract harmonic analysis on locally compact abelian groups.

Automorphic Forms, Representations and $L$-Functions

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Publisher : American Mathematical Soc.
ISBN 13 : 0821814370
Total Pages : 394 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Automorphic Forms, Representations and $L$-Functions by : Armand Borel

Download or read book Automorphic Forms, Representations and $L$-Functions written by Armand Borel and published by American Mathematical Soc.. This book was released on 1979-06-30 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: Part 2 contains sections on Automorphic representations and $L$-functions, Arithmetical algebraic geometry and $L$-functions

Eisenstein Series and Automorphic $L$-Functions

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Publisher : American Mathematical Soc.
ISBN 13 : 0821849891
Total Pages : 218 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Eisenstein Series and Automorphic $L$-Functions by : Freydoon Shahidi

Download or read book Eisenstein Series and Automorphic $L$-Functions written by Freydoon Shahidi and published by American Mathematical Soc.. This book was released on 2010 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a treatment of the theory of $L$-functions developed by means of the theory of Eisenstein series and their Fourier coefficients, a theory which is usually referred to as the Langlands-Shahidi method. The information gathered from this method, when combined with the converse theorems of Cogdell and Piatetski-Shapiro, has been quite sufficient in establishing a number of new cases of Langlands functoriality conjecture; at present, some of these cases cannot be obtained by any other method. These results have led to far-reaching new estimates for Hecke eigenvalues of Maass forms, as well as definitive solutions to certain problems in analytic and algebraic number theory. This book gives a detailed treatment of important parts of this theory, including a rather complete proof of Casselman-Shalika's formula for unramified Whittaker functions as well as a general treatment of the theory of intertwining operators. It also covers in some detail the global aspects of the method as well as some of its applications to group representations and harmonic analysis. This book is addressed to graduate students and researchers who are interested in the Langlands program in automorphic forms and its connections with number theory.

Value-Distribution of L-Functions

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Publisher : Springer
ISBN 13 : 3540448225
Total Pages : 320 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Value-Distribution of L-Functions by : Jörn Steuding

Download or read book Value-Distribution of L-Functions written by Jörn Steuding and published by Springer. This book was released on 2007-05-26 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.

L-Functions and Automorphic Forms

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Publisher : Springer
ISBN 13 : 3319697129
Total Pages : 367 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis L-Functions and Automorphic Forms by : Jan Hendrik Bruinier

Download or read book L-Functions and Automorphic Forms written by Jan Hendrik Bruinier and published by Springer. This book was released on 2018-02-22 with total page 367 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a collection of carefully refereed research articles and lecture notes stemming from the Conference "Automorphic Forms and L-Functions", held at the University of Heidelberg in 2016. The theory of automorphic forms and their associated L-functions is one of the central research areas in modern number theory, linking number theory, arithmetic geometry, representation theory, and complex analysis in many profound ways. The 19 papers cover a wide range of topics within the scope of the conference, including automorphic L-functions and their special values, p-adic modular forms, Eisenstein series, Borcherds products, automorphic periods, and many more.

Twisted L-Functions and Monodromy. (AM-150)

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Publisher : Princeton University Press
ISBN 13 : 9780691091518
Total Pages : 262 pages
Book Rating : 4.0/5 (915 download)

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Book Synopsis Twisted L-Functions and Monodromy. (AM-150) by : Nicholas M. Katz

Download or read book Twisted L-Functions and Monodromy. (AM-150) written by Nicholas M. Katz and published by Princeton University Press. This book was released on 2002-02-24 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves. The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is essential reading for anyone interested in number theory and algebraic geometry.

Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis

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Publisher : American Mathematical Soc.
ISBN 13 : 0821807374
Total Pages : 242 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis by : Hugh L. Montgomery

Download or read book Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis written by Hugh L. Montgomery and published by American Mathematical Soc.. This book was released on 1994 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains lectures presented by Hugh L. Montgomery at the NSF-CBMS Regional Conference held at Kansas State University in May 1990. The book focuses on important topics in analytic number theory that involve ideas from harmonic analysis. One particularly valuable aspect of the book is that it collects material that was either unpublished or that had appeared only in the research literature. The book should be a useful resource for harmonic analysts interested in moving into research in analytic number theory. In addition, it is suitable as a textbook in an advanced graduate topics course in number theory.

Multiple Dirichlet Series, L-functions and Automorphic Forms

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Publisher : Springer
ISBN 13 : 0817683348
Total Pages : 367 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis Multiple Dirichlet Series, L-functions and Automorphic Forms by : Daniel Bump

Download or read book Multiple Dirichlet Series, L-functions and Automorphic Forms written by Daniel Bump and published by Springer. This book was released on 2012-07-09 with total page 367 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiple Dirichlet Series, L-functions and Automorphic Forms gives the latest advances in the rapidly developing subject of Multiple Dirichlet Series, an area with origins in the theory of automorphic forms that exhibits surprising and deep connections to crystal graphs and mathematical physics. As such, it represents a new way in which areas including number theory, combinatorics, statistical mechanics, and quantum groups are seen to fit together. The volume also includes papers on automorphic forms and L-functions and related number-theoretic topics. This volume will be a valuable resource for graduate students and researchers in number theory, combinatorics, representation theory, mathematical physics, and special functions. Contributors: J. Beineke, B. Brubaker, D. Bump, G. Chinta, G. Cornelissen, C.A. Diaconu, S. Frechette, S. Friedberg, P. Garrett, D. Goldfeld, P.E. Gunnells, B. Heim, J. Hundley, D. Ivanov, Y. Komori, A.V. Kontorovich, O. Lorscheid, K. Matsumoto, P.J. McNamara, S.J. Patterson, M. Suzuki, H. Tsumura.

Modular Forms and Fermat’s Last Theorem

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Publisher : Springer Science & Business Media
ISBN 13 : 1461219744
Total Pages : 592 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Modular Forms and Fermat’s Last Theorem by : Gary Cornell

Download or read book Modular Forms and Fermat’s Last Theorem written by Gary Cornell and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.

Visions in Mathematics

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Publisher : Birkhäuser
ISBN 13 : 9783034604246
Total Pages : 528 pages
Book Rating : 4.6/5 (42 download)

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Book Synopsis Visions in Mathematics by : Noga Alon

Download or read book Visions in Mathematics written by Noga Alon and published by Birkhäuser. This book was released on 2010-04-14 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas. The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest. This is the second part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

Littlewood-Paley Theory and the Study of Function Spaces

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Publisher : American Mathematical Soc.
ISBN 13 : 0821807315
Total Pages : 142 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Littlewood-Paley Theory and the Study of Function Spaces by : Michael Frazier

Download or read book Littlewood-Paley Theory and the Study of Function Spaces written by Michael Frazier and published by American Mathematical Soc.. This book was released on 1991 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: Littlewood-Paley theory was developed to study function spaces in harmonic analysis and partial differential equations. Recently, it has contributed to the development of the *q-transform and wavelet decompositions. Based on lectures presented at the NSF-CBMS Regional Research Conference on Harmonic Analysis and Function Spaces, held at Auburn University in July 1989, this book is aimed at mathematicians, as well as mathematically literate scientists and engineers interested in harmonic analysis or wavelets. The authors provide not only a general understanding of the area of harmonic analysis relating to Littlewood-Paley theory and atomic and wavelet decompositions, but also some motivation and background helpful in understanding the recent theory of wavelets. The book begins with some simple examples which provide an overview of the classical Littlewood-Paley theory. The *q-transform, wavelet, and smooth atomic expansions are presented as natural extensions of the classical theory. Finally, applications to harmonic analysis (Calderon-Zygmund operators), signal processing (compression), and mathematical physics (potential theory) are discussed.

p-Adic Automorphic Forms on Shimura Varieties

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Publisher : Springer Science & Business Media
ISBN 13 : 9780387207117
Total Pages : 414 pages
Book Rating : 4.2/5 (71 download)

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Book Synopsis p-Adic Automorphic Forms on Shimura Varieties by : Haruzo Hida

Download or read book p-Adic Automorphic Forms on Shimura Varieties written by Haruzo Hida and published by Springer Science & Business Media. This book was released on 2004-05-10 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers the following three topics in a manner accessible to graduate students who have an understanding of algebraic number theory and scheme theoretic algebraic geometry: 1. An elementary construction of Shimura varieties as moduli of abelian schemes. 2. p-adic deformation theory of automorphic forms on Shimura varieties. 3. A simple proof of irreducibility of the generalized Igusa tower over the Shimura variety. The book starts with a detailed study of elliptic and Hilbert modular forms and reaches to the forefront of research of Shimura varieties associated with general classical groups. The method of constructing p-adic analytic families and the proof of irreducibility was recently discovered by the author. The area covered in this book is now a focal point of research worldwide with many far-reaching applications that have led to solutions of longstanding problems and conjectures. Specifically, the use of p-adic elliptic and Hilbert modular forms have proven essential in recent breakthroughs in number theory (for example, the proof of Fermat's Last Theorem and the Shimura-Taniyama conjecture by A. Wiles and others). Haruzo Hida is Professor of Mathematics at University of California, Los Angeles. His previous books include Modular Forms and Galois Cohomology (Cambridge University Press 2000) and Geometric Modular Forms and Elliptic Curves (World Scientific Publishing Company 2000).

Motives

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Publisher : American Mathematical Soc.
ISBN 13 : 0821827987
Total Pages : 694 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Motives by :

Download or read book Motives written by and published by American Mathematical Soc.. This book was released on 1994-02-28 with total page 694 pages. Available in PDF, EPUB and Kindle. Book excerpt: 'Motives' were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, and to play the role of the missing rational cohomology. This work contains the texts of the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991.

Handbook of Mathematical Functions

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Publisher : Courier Corporation
ISBN 13 : 9780486612720
Total Pages : 1068 pages
Book Rating : 4.6/5 (127 download)

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Book Synopsis Handbook of Mathematical Functions by : Milton Abramowitz

Download or read book Handbook of Mathematical Functions written by Milton Abramowitz and published by Courier Corporation. This book was released on 1965-01-01 with total page 1068 pages. Available in PDF, EPUB and Kindle. Book excerpt: An extensive summary of mathematical functions that occur in physical and engineering problems

The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series

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Publisher : American Mathematical Soc.
ISBN 13 : 0821833685
Total Pages : 226 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series by : Ken Ono

Download or read book The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series written by Ken Ono and published by American Mathematical Soc.. This book was released on 2004 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: Chapter 1.