Computational Aspects of Modular Forms and Galois Representations : (Record no. 79497)

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fixed length control field 08745nam a22005413i 4500
001 - CONTROL NUMBER
control field EBC670341
003 - CONTROL NUMBER IDENTIFIER
control field MiAaPQ
005 - DATE AND TIME OF LATEST TRANSACTION
control field 20191126110007.0
006 - FIXED-LENGTH DATA ELEMENTS--ADDITIONAL MATERIAL CHARACTERISTICS
fixed length control field m o d |
007 - PHYSICAL DESCRIPTION FIXED FIELD--GENERAL INFORMATION
fixed length control field cr cnu||||||||
008 - FIXED-LENGTH DATA ELEMENTS--GENERAL INFORMATION
fixed length control field 191125s2011 xx o ||||0 eng d
020 ## - INTERNATIONAL STANDARD BOOK NUMBER
International Standard Book Number 9781400839001
Qualifying information (electronic bk.)
020 ## - INTERNATIONAL STANDARD BOOK NUMBER
Canceled/invalid ISBN 9780691142029
035 ## - SYSTEM CONTROL NUMBER
System control number (MiAaPQ)EBC670341
035 ## - SYSTEM CONTROL NUMBER
System control number (Au-PeEL)EBL670341
035 ## - SYSTEM CONTROL NUMBER
System control number (CaPaEBR)ebr10456327
035 ## - SYSTEM CONTROL NUMBER
System control number (CaONFJC)MIL305180
035 ## - SYSTEM CONTROL NUMBER
System control number (OCoLC)729386470
040 ## - CATALOGING SOURCE
Original cataloging agency MiAaPQ
Language of cataloging eng
Description conventions rda
-- pn
Transcribing agency MiAaPQ
Modifying agency MiAaPQ
050 #4 - LIBRARY OF CONGRESS CALL NUMBER
Classification number QA247 -- .C638 2011eb
082 0# - DEWEY DECIMAL CLASSIFICATION NUMBER
Classification number 512.32
100 1# - MAIN ENTRY--PERSONAL NAME
Personal name Edixhoven, Bas.
9 (RLIN) 66055
245 10 - TITLE STATEMENT
Title Computational Aspects of Modular Forms and Galois Representations :
Remainder of title How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176).
264 #1 - PRODUCTION, PUBLICATION, DISTRIBUTION, MANUFACTURE, AND COPYRIGHT NOTICE
Place of production, publication, distribution, manufacture Princeton :
Name of producer, publisher, distributor, manufacturer Princeton University Press,
Date of production, publication, distribution, manufacture, or copyright notice 2011.
264 #4 - PRODUCTION, PUBLICATION, DISTRIBUTION, MANUFACTURE, AND COPYRIGHT NOTICE
Date of production, publication, distribution, manufacture, or copyright notice ©2011.
300 ## - PHYSICAL DESCRIPTION
Extent 1 online resource (438 pages)
336 ## - CONTENT TYPE
Content type term text
Content type code txt
Source rdacontent
337 ## - MEDIA TYPE
Media type term computer
Media type code c
Source rdamedia
338 ## - CARRIER TYPE
Carrier type term online resource
Carrier type code cr
Source rdacarrier
490 1# - SERIES STATEMENT
Series statement Annals of Mathematics Studies ;
Volume/sequential designation v.176
505 0# - FORMATTED CONTENTS NOTE
Formatted contents note Cover -- Title -- Copyright -- Contents -- Preface -- Acknowledgments -- Author information -- Dependencies between the chapters -- Chapter 1. Introduction, main results, context -- 1.1 Statement of the main results -- 1.2 Historical context: Schoof's algorithm -- 1.3 Schoof's algorithm described in terms of étale cohomology -- 1.4 Some natural new directions -- 1.5 More historical context: congruences for Ramanujan's τ-function -- 1.6 Comparison with p-adic methods -- Chapter 2. Modular curves, modular forms, lattices, Galois representations -- 2.1 Modular curves -- 2.2 Modular forms -- 2.3 Lattices and modular forms -- 2.4 Galois representations attached to eigenforms -- 2.5 Galois representations over finite fields, and reduction to torsion in Jacobians -- Chapter 3. First description of the algorithms -- Chapter 4. Short introduction to heights and Arakelov theory -- 4.1 Heights on Q and Q -- 4.2 Heights on projective spaces and on varieties -- 4.3 The Arakelov perspective on height functions -- 4.4 Arithmetic surfaces, intersection theory, and arithmetic Riemann-Roch -- Chapter 5. Computing complex zeros of polynomials and power series -- 5.1 Polynomial time complexity classes -- 5.2 Computing the square root of a positive real number -- 5.3 Computing the complex roots of a polynomial -- 5.4 Computing the zeros of a power series -- Chapter 6. Computations with modular forms and Galois representations -- 6.1 Modular symbols -- 6.2 Intermezzo: Atkin-Lehner operators -- 6.3 Basic numerical evaluations -- 6.4 Numerical calculations and Galois representations -- Chapter 7. Polynomials for projective representations of level one forms -- 7.1 Introduction -- 7.2 Galois representations -- 7.3 Proof of the theorem -- 7.4 Proof of the corollary -- 7.5 The table of polynomials -- Chapter 8. Description of X[sub(1)](5l).
505 8# - FORMATTED CONTENTS NOTE
Formatted contents note 8.1 Construction of a suitable cuspidal divisor on x[sub(1)].5l -- 8.2 The exact setup for the level one case -- Chapter 9. Applying Arakelov theory -- 9.1 Relating heights to intersection numbers -- 9.2 Controlling D[sub(x)]-D[sub(0)] -- Chapter 10. An upper bound for Green functions on Riemann surfaces -- Chapter 11. Bounds for Arakelov invariants of modular curves -- 11.1 Bounding the height of X[sub(1)](pl) -- 11.2 Bounding the theta function on Pic[sup(g-1)](X[sub(1)](pl)) -- 11.3 Upper bounds for Arakelov Green functions on the curves X[sub(1)](pl) -- 11.4 Bounds for intersection numbers on X[sub(1)](pl) -- 11.5 A bound for h(x[sup(′)][sub(l)](Q)) in terms of h(b([sub(l)](Q)) -- 11.6 An integral over X[sub(1)](5l) -- 11.7 Final estimates of the Arakelov contribution -- Chapter 12. Approximating V[sub(f)] over the complex numbers -- 12.1 Points, divisors, and coordinates on X -- 12.2 The lattice of periods -- 12.3 Modular functions -- 12.4 Power series -- 12.5 Jacobian and Wronskian determinants of series -- 12.6 A simple quantitative study of the Jacobi map -- 12.7 Equivalence of various norms -- 12.8 An elementary operation in the Jacobian variety -- 12.9 Arithmetic operations in the Jacobian variety -- 12.10 The inverse Jacobi problem -- 12.11 The algebraic conditioning -- 12.12 Heights -- 12.13 Bounding the error in X[sub(g)] -- 12.14 Final result of this chapter -- Chapter 13. Computing V[sub(f)] modulo p -- 13.1 Basic algorithms for plane curves -- 13.2 A first approach to picking random divisors -- 13.3 Pairings -- 13.4 Divisible groups -- 13.5 The Kummer map -- 13.6 Linearization of torsion classes -- 13.7 Computing V[sub(f)] modulo p -- Chapter 14. Computing the residual Galois representations -- 14.1 Main result -- 14.2 Reduction to irreducible representations -- 14.3 Reduction to torsion in Jacobians -- 14.4 Computing the Q(ζl.
505 8# - FORMATTED CONTENTS NOTE
Formatted contents note l)-algebra corresponding to V -- 14.5 Computing the vector space structure -- 14.6 Descent to Q -- 14.7 Extracting the Galois representation -- 14.8 A probabilistic variant -- Chapter 15. Computing coefficients of modular forms -- 15.1 Computing τ(p) in time polynomial in log p -- 15.2 Computing Tn for large n and large weight -- 15.3 An application to quadratic forms -- Epilogue -- Bibliography -- Index -- Non-alphabetic symbols -- Greek symbols -- Roman symbols -- Words -- A -- C -- D -- E -- F -- G -- H -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- W.
520 ## - SUMMARY, ETC.
Summary, etc. 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
520 8# - SUMMARY, ETC.
Summary, etc. 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.
588 ## - SOURCE OF DESCRIPTION NOTE
Source of description note Description based on publisher supplied metadata and other sources.
590 ## - LOCAL NOTE (RLIN)
Local note Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
650 #0 - SUBJECT ADDED ENTRY--TOPICAL TERM
Topical term or geographic name entry element Galois modules (Algebra);Class field theory.
9 (RLIN) 66056
655 #4 - INDEX TERM--GENRE/FORM
Genre/form data or focus term Electronic books.
9 (RLIN) 66057
700 1# - ADDED ENTRY--PERSONAL NAME
Personal name Couveignes, Jean-Marc.
9 (RLIN) 66058
700 1# - ADDED ENTRY--PERSONAL NAME
Personal name de Jong, Robin de.
9 (RLIN) 66059
700 1# - ADDED ENTRY--PERSONAL NAME
Personal name Bosman, Johan.
9 (RLIN) 66060
700 1# - ADDED ENTRY--PERSONAL NAME
Personal name Merkl, Franz.
9 (RLIN) 66061
776 08 - ADDITIONAL PHYSICAL FORM ENTRY
Relationship information Print version:
Main entry heading Edixhoven, Bas
Title Computational Aspects of Modular Forms and Galois Representations : How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)
Place, publisher, and date of publication Princeton : Princeton University Press,c2011
International Standard Book Number 9780691142029
797 2# - LOCAL ADDED ENTRY--CORPORATE NAME (RLIN)
Corporate name or jurisdiction name as entry element ProQuest (Firm)
830 #0 - SERIES ADDED ENTRY--UNIFORM TITLE
Uniform title Annals of Mathematics Studies
9 (RLIN) 66062
856 40 - ELECTRONIC LOCATION AND ACCESS
Uniform Resource Identifier <a href="https://ebookcentral.proquest.com/lib/thebc/detail.action?docID=670341">https://ebookcentral.proquest.com/lib/thebc/detail.action?docID=670341</a>
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