The Conference On L Functions
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| Author | : Lin Weng |
| Publisher | : World Scientific |
| Total Pages | : 383 |
| Release | : 2007 |
| Genre | : Mathematics |
| ISBN | : 9812772391 |
This invaluable volume collects papers written by many of the world''s top experts on L -functions. It not only covers a wide range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole. The contributions reflect the latest, most advanced and most important aspects of L- functions. In particular, it contains Hida''s lecture notes at the conference and at the Eigenvariety semester in Harvard University and Weng''s detailed account of his works on high rank zeta functions and non-abelian L -functions. Sample Chapter(s). Chapter 1: Quantum Maass Forms (435 KB). Contents: Quantum Maass Forms (R Bruggeman); o-invariant of p -Adic L -Functions (H Hida); Siegel Modular Forms of Weight Three and Conjectural Correspondence of Shimura Type and Langlands Type (T Ibukiyama); Convolutions of Fourier Coefficients of Cusp Forms and the Circle Method (M Jutila); On an Extension of the Derivation Relation for Multiple Zeta Values (M Kaneko); On Symmetric Powers of Cusp Forms on GL 2 (H H Kim); Zeta Functions of Root Systems (Y Komori et al.); Sums of Kloosterman Sums Revisted (Y Motohashi); The LindelAf Class of L -Functions (K Murty); A Proof of the Riemann Hypothesis for the Weng Zeta Function of Rank 3 for the Rationals (M Suzuki); Elliptic Curves Arising from the Spectral Zeta Function for Non-Commutative Harmonic Oscillators and o 0 (4)-Modular Forms (K Kimoto & M Wakayama); A Geometric Approach to L -Functions (L Weng). Readership: Graduate students, lecturers, and active researchers in various branches of mathematics, such as algebra, analysis, geometry and mathematical physics."
| Author | : Wen-Ching Winnie Li |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 106 |
| Release | : 2019-03-01 |
| Genre | : Mathematics |
| ISBN | : 1470449005 |
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.
| Author | : M. Rapoport |
| Publisher | : Academic Press |
| Total Pages | : 399 |
| Release | : 2014-07-14 |
| Genre | : Mathematics |
| ISBN | : 1483263304 |
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.
| Author | : Carlos J. Moreno |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 313 |
| Release | : 2005 |
| Genre | : Mathematics |
| ISBN | : 0821842668 |
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.
| Author | : Armand Borel |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 394 |
| Release | : 1979-06-30 |
| Genre | : Mathematics |
| ISBN | : 0821814370 |
Part 2 contains sections on Automorphic representations and $L$-functions, Arithmetical algebraic geometry and $L$-functions
| Author | : Freydoon Shahidi |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 218 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821849891 |
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.
| Author | : Jan Hendrik Bruinier |
| Publisher | : Springer |
| Total Pages | : 367 |
| Release | : 2018-02-22 |
| Genre | : Mathematics |
| ISBN | : 3319697129 |
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.
| Author | : Hugh L. Montgomery |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 242 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821807374 |
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.
| Author | : Jörn Steuding |
| Publisher | : Springer |
| Total Pages | : 320 |
| Release | : 2007-05-26 |
| Genre | : Mathematics |
| ISBN | : 3540448225 |
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.
| Author | : Nicholas M. Katz |
| Publisher | : Princeton University Press |
| Total Pages | : 262 |
| Release | : 2002-02-24 |
| Genre | : Mathematics |
| ISBN | : 9780691091518 |
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.
