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Author: Goro Shimura
Publisher: Springer
ISBN: 3540358323
Category : Mathematics
Languages : en
Pages : 75
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Book Description
Author: Goro Shimura
Publisher: Springer
ISBN: 3540358323
Category : Mathematics
Languages : en
Pages : 75
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Book Description
Author: Gorō Shimura
Publisher: Princeton University Press
ISBN: 9780691080925
Category : Mathematics
Languages : en
Pages : 292
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Book Description
The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.
Author: Dorian Goldfeld
Publisher: Cambridge University Press
ISBN: 1139456202
Category : Mathematics
Languages : en
Pages : 65
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Book Description
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: Anton Deitmar
Publisher: Springer Science & Business Media
ISBN: 144714435X
Category : Mathematics
Languages : en
Pages : 255
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Book Description
Automorphic forms are an important complex analytic tool in number theory and modern arithmetic geometry. They played for example a vital role in Andrew Wiles's proof of Fermat's Last Theorem. This text provides a concise introduction to the world of automorphic forms using two approaches: the classic elementary theory and the modern point of view of adeles and representation theory. The reader will learn the important aims and results of the theory by focussing on its essential aspects and restricting it to the 'base field' of rational numbers. Students interested for example in arithmetic geometry or number theory will find that this book provides an optimal and easily accessible introduction into this topic.
Author: D. Bump
Publisher: Springer
ISBN: 3540390553
Category : Mathematics
Languages : en
Pages : 196
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Book Description
Author: Jacques Hadamard
Publisher: American Mathematical Soc.
ISBN: 9780821890479
Category : Mathematics
Languages : en
Pages : 116
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Book Description
This is the English translation of a volume originally published only in Russian and now out of print. The book was written by Jacques Hadamard on the work of Poincare. Poincare's creation of a theory of automorphic functions in the early 1880s was one of the most significant mathematical achievements of the nineteenth century. It directly inspired the uniformization theorem, led to a class of functions adequate to solve all linear ordinary differential equations, and focused attention on a large new class of discrete groups. It was the first significant application of non-Euclidean geometry. This unique exposition by Hadamard offers a fascinating and intuitive introduction to the subject of automorphic functions and illuminates its connection to differential equations, a connection not often found in other texts.
Author: Toshiyuki Kobayashi
Publisher: Springer Science & Business Media
ISBN: 0817646469
Category : Mathematics
Languages : en
Pages : 220
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Book Description
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: Marvin Isadore Knopp
Publisher: American Mathematical Soc.
ISBN: 0821844881
Category : Mathematics
Languages : en
Pages : 169
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Book Description
Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions, $\eta(\tau)$ and $\vartheta(\tau)$, and their applications to two number-theoretic functions, $p(n)$ and $r_s(n)$. They are well chosen, as at the heart of these particular applications to the treatment of these specific number-theoretic functions lies the general theory of automorphic functions, a theory of far-reaching significance with important connections to a great many fields of mathematics. The book is essentially self-contained, assuming only a good first-year course in analysis. The excellent exposition presents the beautiful interplay between modular forms and number theory, making the book an excellent introduction to analytic number theory for a beginning graduate student. Table of Contents: The Modular Group and Certain Subgroups: 1. The modular group; 2. A fundamental region for $\Gamma(1)$; 3. Some subgroups of $\Gamma(1)$; 4. Fundamental regions of subgroups. Modular Functions and Forms: 1. Multiplier systems; 2. Parabolic points; 3 Fourier expansions; 4. Definitions of modular function and modular form; 5. Several important theorems.The Modular Forms $\eta(\tau)$ and $\vartheta(\tau)$: 1. The function $\eta(\tau)$; 2. Several famous identities; 3. Transformation formulas for $\eta(\tau)$; 4. The function $\vartheta(\tau)$. The Multiplier Systems $\upsilon_{\eta}$ and $\upsilon_{\vartheta}$: 1. Preliminaries; 2. Proof of theorem 2; 3. Proof of theorem 3. Sums of Squares: 1. Statement of results; 2. Lipschitz summation formula; 3. The function $\psi_s(\tau)$; 4. The expansion of $\psi_s(\tau)$ at $-1$; 5. Proofs of theorems 2 and 3; 6. Related results. The Order of Magnitude of $p(n)$: 1. A simple inequality for $p(n)$; 2. The asymptotic formula for $p(n)$; 3. Proof of theorem 2. The Ramanujan Congruences for $p(n)$: 1. Statement of the congruences; 2. The functions $\Phi_{p, r}(\tau)$ and $h_p(\tau)$; 3. The function $s_{p, r}(\tau)$; 4. The congruence for $p(n)$ Modulo 11; 5. Newton's formula; 6. The modular equation for the prime 5; 7. The modular equation for the prime 7. Proof of the Ramanujan Congruences for Powers of 5 and 7: 1. Preliminaries; 2. Application of the modular equation; 3. A digression: The Ramanujan identities for powers of the prime 5; 4. Completion of the proof for powers of 5; 5.Start of the proof for powers of 7; 6. A second digression: The Ramanujan identities for powers of the prime 7; 7. Completion of the proof for powers of 7. Index. (CHEL/337.H
Author: Haruzo Hida
Publisher: JHU Press
ISBN: 9780801878602
Category : Mathematics
Languages : en
Pages : 946
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Book Description
In Contributions to Automorphic Forms, Geometry, and Number Theory, Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi bring together a distinguished group of experts to explore automorphic forms, principally via the associated L-functions, representation theory, and geometry. Because these themes are at the cutting edge of a central area of modern mathematics, and are related to the philosophical base of Wiles' proof of Fermat's last theorem, this book will be of interest to working mathematicians and students alike. Never previously published, the contributions to this volume expose the reader to a host of difficult and thought-provoking problems. Each of the extraordinary and noteworthy mathematicians in this volume makes a unique contribution to a field that is currently seeing explosive growth. New and powerful results are being proved, radically and continually changing the field's make up. Contributions to Automorphic Forms, Geometry, and Number Theory will likely lead to vital interaction among researchers and also help prepare students and other young mathematicians to enter this exciting area of pure mathematics. Contributors: Jeffrey Adams, Jeffrey D. Adler, James Arthur, Don Blasius, Siegfried Boecherer, Daniel Bump, William Casselmann, Laurent Clozel, James Cogdell, Laurence Corwin, Solomon Friedberg, Masaaki Furusawa, Benedict Gross, Thomas Hales, Joseph Harris, Michael Harris, Jeffrey Hoffstein, Hervé Jacquet, Dihua Jiang, Nicholas Katz, Henry Kim, Victor Kreiman, Stephen Kudla, Philip Kutzko, V. Lakshmibai, Robert Langlands, Erez Lapid, Ilya Piatetski-Shapiro, Dipendra Prasad, Stephen Rallis, Dinakar Ramakrishnan, Paul Sally, Freydoon Shahidi, Peter Sarnak, Rainer Schulze-Pillot, Joseph Shalika, David Soudry, Ramin Takloo-Bigash, Yuri Tschinkel, Emmanuel Ullmo, Marie-France Vignéras, Jean-Loup Waldspurger.
Author: Henryk Iwaniec
Publisher: American Mathematical Society, Revista Matemática Iberoamericana (RMI), Madrid, Spain
ISBN: 1470466228
Category : Mathematics
Languages : en
Pages : 220
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Book Description
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.
