Computational Aspects Of Modular Forms And Galois Representations
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Author | : Bas Edixhoven |
Publisher | : Princeton University Press |
Total Pages | : 438 |
Release | : 2011-05-31 |
Genre | : Mathematics |
ISBN | : 1400839009 |
Download Computational Aspects of Modular Forms and Galois Representations Book in PDF, Epub and Kindle
Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.
Author | : Bas Edixhoven |
Publisher | : |
Total Pages | : 442 |
Release | : 1940 |
Genre | : Class field theory |
ISBN | : |
Download Computational Aspects of Modular Forms and Galois Representations Book in PDF, Epub and Kindle
"This book represents a major step forward from explicit class field theory, and it could be described as the start of the 'explicit Langlands program'"--
Author | : Bas Edixhoven |
Publisher | : Princeton University Press |
Total Pages | : 438 |
Release | : 2011-06-20 |
Genre | : Mathematics |
ISBN | : 0691142017 |
Download Computational Aspects of Modular Forms and Galois Representations Book in PDF, Epub and Kindle
Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.
Author | : Dennis Charles |
Publisher | : |
Total Pages | : 134 |
Release | : 2005 |
Genre | : |
ISBN | : |
Download Computational Aspects of Modular Forms and Elliptic Curves Book in PDF, Epub and Kindle
Author | : Gebhard Böckle |
Publisher | : Springer Science & Business Media |
Total Pages | : 377 |
Release | : 2014-01-23 |
Genre | : Mathematics |
ISBN | : 3319038478 |
Download Computations with Modular Forms Book in PDF, Epub and Kindle
This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.
Author | : William A. Stein |
Publisher | : American Mathematical Soc. |
Total Pages | : 290 |
Release | : 2007-02-13 |
Genre | : Mathematics |
ISBN | : 0821839608 |
Download Modular Forms, a Computational Approach Book in PDF, Epub and Kindle
This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments. Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty. The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience. As well as being an invaluable companion to those learning the theory in a more traditional way, this book will be a great help to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations. There is also a useful Appendix by Gunnells on extensions to more general modular forms, which has enough in it to inspire many PhD theses for years to come. While the book's main readership will be graduate students in number theory, it will also be accessible to advanced undergraduates and useful to both specialists and non-specialists in number theory. --John E. Cremona, University of Nottingham William Stein is an associate professor of mathematics at the University of Washington at Seattle. He earned a PhD in mathematics from UC Berkeley and has held positions at Harvard University and UC San Diego. His current research interests lie in modular forms, elliptic curves, and computational mathematics.
Author | : Alejandro Argáez García |
Publisher | : |
Total Pages | : 168 |
Release | : 2016 |
Genre | : Galois modules (Algebra) |
ISBN | : |
Download Computational Aspects of Galois Representations Book in PDF, Epub and Kindle
Author | : Lloyd James Peter Kilford |
Publisher | : World Scientific Publishing Company |
Total Pages | : 252 |
Release | : 2015-03-12 |
Genre | : Mathematics |
ISBN | : 1783265477 |
Download Modular Forms: A Classical And Computational Introduction (2nd Edition) Book in PDF, Epub and Kindle
Modular Forms is a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to various subjects, such as the theory of quadratic forms, the proof of Fermat's Last Theorem and the approximation of π. The text gives a balanced overview of both the theoretical and computational sides of its subject, allowing a variety of courses to be taught from it.This second edition has been revised and updated. New material on the future of modular forms as well as a chapter about longer-form projects for students has also been added.
Author | : Greville G. Corbett |
Publisher | : Cambridge University Press |
Total Pages | : 364 |
Release | : 1993-06-24 |
Genre | : Language Arts & Disciplines |
ISBN | : 9780521402453 |
Download Heads in Grammatical Theory Book in PDF, Epub and Kindle
A study of the idea of the 'head' or dominating element of a phrase.
Author | : H. Kisilevsky |
Publisher | : American Mathematical Soc. |
Total Pages | : 208 |
Release | : 1994-01-01 |
Genre | : Mathematics |
ISBN | : 9780821870358 |
Download Elliptic Curves and Related Topics Book in PDF, Epub and Kindle
This book represents the proceedings of a workshop on elliptic curves held in St. Adele, Quebec, in February 1992. Containing both expository and research articles on the theory of elliptic curves, this collection covers a range of topics, from Langlands's theory to the algebraic geometry of elliptic curves, from Iwasawa theory to computational aspects of elliptic curves. This book is especially significant in that it covers topics comprising the main ingredients in Andrew Wiles's recent result on Fermat's Last Theorem.