Automorphic Forms And Applications
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| Author | : Peter Sarnak |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 443 |
| Release | : 2007 |
| Genre | : Mathematics |
| ISBN | : 0821828738 |
The theory of automorphic forms has seen dramatic developments in recent years. In particular, important instances of Langlands functoriality have been established. This volume presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on automorphic forms and their applications. It addresses some of the general aspects of automorphic forms, as well as certain recent advances in the field. The book starts with the lectures of Borel on the basic theory of automorphic forms, which lay the foundation for the lectures by Cogdell and Shahidi on converse theorems and the Langlands-Shahidi method, as well as those by Clozel and Li on the Ramanujan conjectures and graphs. The analytic theory of GL(2)-forms and $L$-functions are the subject of Michel's lectures, while Terras covers arithmetic quantum chaos. The volume also includes a chapter by Vogan on isolated unitary representations, which is related to the lectures by Clozel. This volume is recommended for independent study or an advanced topics course. It is suitable for graduate students and researchers interested in automorphic forms and number theory. the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
| Author | : Kathrin Bringmann |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 409 |
| Release | : 2017-12-15 |
| Genre | : Mathematics |
| ISBN | : 1470419440 |
Modular forms and Jacobi forms play a central role in many areas of mathematics. Over the last 10–15 years, this theory has been extended to certain non-holomorphic functions, the so-called “harmonic Maass forms”. The first glimpses of this theory appeared in Ramanujan's enigmatic last letter to G. H. Hardy written from his deathbed. Ramanujan discovered functions he called “mock theta functions” which over eighty years later were recognized as pieces of harmonic Maass forms. This book contains the essential features of the theory of harmonic Maass forms and mock modular forms, together with a wide variety of applications to algebraic number theory, combinatorics, elliptic curves, mathematical physics, quantum modular forms, and representation theory.
| Author | : D. Bump |
| Publisher | : Springer |
| Total Pages | : 196 |
| Release | : 2006-12-08 |
| Genre | : Mathematics |
| ISBN | : 3540390553 |
| Author | : Henryk Iwaniec |
| Publisher | : American Mathematical Society, Revista Matemática Iberoamericana (RMI), Madrid, Spain |
| Total Pages | : 220 |
| Release | : 2021-11-17 |
| Genre | : Mathematics |
| ISBN | : 1470466228 |
Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style. The first edition of this book was an underground classic, both as a textbook and as a respected source for results, ideas, and references. Iwaniec treats the spectral theory of automorphic forms as the study of the space of $L^2$ functions on the upper half plane modulo a discrete subgroup. Key topics include Eisenstein series, estimates of Fourier coefficients, Kloosterman sums, the Selberg trace formula and the theory of small eigenvalues. Henryk Iwaniec was awarded the 2002 Cole Prize for his fundamental contributions to number theory.
| Author | : Gaëtan Chenevier |
| Publisher | : Springer |
| Total Pages | : 428 |
| Release | : 2019-02-28 |
| Genre | : Mathematics |
| ISBN | : 3319958917 |
This book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur. Its connecting thread is a question about lattices of rank 24: the problem of p-neighborhoods between Niemeier lattices. This question, whose expression is quite elementary, is in fact very natural from the automorphic point of view, and turns out to be surprisingly intriguing. We explain how the new advances in the Langlands program mentioned above pave the way for a solution. This study proves to be very rich, leading us to classical themes such as theta series, Siegel modular forms, the triality principle, L-functions and congruences between Galois representations. This monograph is intended for any mathematician with an interest in Euclidean lattices, automorphic forms or number theory. A large part of it is meant to be accessible to non-specialists.
| Author | : H. Jacquet |
| Publisher | : Springer |
| Total Pages | : 156 |
| Release | : 2006-11-15 |
| Genre | : Mathematics |
| ISBN | : 3540376127 |
| Author | : Peter Sarnak |
| Publisher | : Cambridge University Press |
| Total Pages | : 124 |
| Release | : 1990-11-15 |
| Genre | : Mathematics |
| ISBN | : 1316582442 |
The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications. In order to keep the presentation reasonably self-contained, Professor Sarnak begins by developing the necessary background material in modular forms. He then considers the solution of three problems: the Ruziewicz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs: 'expander graphs' and 'Ramanujan graphs'; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares. These applications are carried out in detail. The book therefore should be accessible to a wide audience of graduate students and researchers in mathematics and computer science.
| Author | : Dorian Goldfeld |
| Publisher | : Cambridge University Press |
| Total Pages | : 65 |
| Release | : 2006-08-03 |
| Genre | : Mathematics |
| ISBN | : 1139456202 |
L-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy-to-read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2,R) and GL(3,R), and then for the general case of GL(n,R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.
| Author | : Toshiyuki Kobayashi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 220 |
| Release | : 2007-10-10 |
| Genre | : Mathematics |
| ISBN | : 0817646469 |
This volume uses a unified approach to representation theory and automorphic forms. It collects papers, written by leading mathematicians, that track recent progress in the expanding fields of representation theory and automorphic forms and their association with number theory and differential geometry. Topics include: Automorphic forms and distributions, modular forms, visible-actions, Dirac cohomology, holomorphic forms, harmonic analysis, self-dual representations, and Langlands Functoriality Conjecture, Both graduate students and researchers will find inspiration in this volume.
| Author | : Jan Hendrik Bruinier |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 273 |
| Release | : 2008-02-10 |
| Genre | : Mathematics |
| ISBN | : 3540741194 |
This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.
