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The Resource Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime, edited by Jean-Marc Couveignes and Bas Edixhoven

Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime, edited by Jean-Marc Couveignes and Bas Edixhoven

Label
Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime
Title
Computational aspects of modular forms and Galois representations
Title remainder
how one can compute in polynomial time the value of Ramanujan's tau at a prime
Statement of responsibility
edited by Jean-Marc Couveignes and Bas Edixhoven
Contributor
Subject
Genre
Language
eng
Summary
  • "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"--
  • "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'"--
Member of
Assigning source
  • Provided by publisher
  • Provided by publisher
Cataloging source
N$T
Illustrations
illustrations
Index
index present
Language note
In English
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
http://library.link/vocab/relatedWorkOrContributorDate
1962-
http://library.link/vocab/relatedWorkOrContributorName
  • Couveignes, Jean-Marc
  • Edixhoven, B.
Series statement
Annals of mathematics studies
Series volume
no. 176
http://library.link/vocab/subjectName
  • Galois modules (Algebra)
  • Class field theory
  • MATHEMATICS
  • MATHEMATICS
  • MATHEMATICS
  • MATHEMATICS
  • Class field theory
  • Galois modules (Algebra)
Label
Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime, edited by Jean-Marc Couveignes and Bas Edixhoven
Instantiates
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references (pages 403-421) and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
mixed
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
Introduction, main results, context / Bas Edixhoven -- Modular curves, modular forms, lattices, Galois representations / Bas Edixhoven -- First description of the algorithms / Jean-Marc Couveignes and Bas Edixhoven -- Short introduction to heights and Arakelov theory / Bas Edixhoven and Robin de Jong -- Computing complex zeros of polynomials and power series / Jean-Marc Couveignes -- Computations with modular forms and Galois representations / Johan Bosman -- Polynomials for projective representations of level one forms / Johan Bosman -- Description of X1(5l) / Bas Edixhoven -- Applying Arakelov theory / Bas Edixhoven and Robin de Jong -- An upper bound for Green functions on Riemann surfaces / Franz Merkl -- Bounds for Arakelov invariants of modular curves / Bas Edixhoven and Robin de Jong -- Approximating V[subscript f] over the complex numbers / Jean-Marc Couveignes -- Computing V[subscript f] modulo p / Jean-Marc Couveignes -- Computing the residual Galois representations / Bas Edixhoven -- Computing coefficients of modular forms / Bas Edixhoven
Control code
ocn729386470
Dimensions
unknown
Extent
1 online resource (xi, 425 pages)
File format
unknown
Form of item
online
Isbn
9781400839001
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Note
JSTOR
Other control number
10.1515/9781400839001
Other physical details
illustrations
http://library.link/vocab/ext/overdrive/overdriveId
  • cl0500000122
  • 22573/cttt3bg
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
(OCoLC)729386470
Label
Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime, edited by Jean-Marc Couveignes and Bas Edixhoven
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references (pages 403-421) and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
mixed
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
Introduction, main results, context / Bas Edixhoven -- Modular curves, modular forms, lattices, Galois representations / Bas Edixhoven -- First description of the algorithms / Jean-Marc Couveignes and Bas Edixhoven -- Short introduction to heights and Arakelov theory / Bas Edixhoven and Robin de Jong -- Computing complex zeros of polynomials and power series / Jean-Marc Couveignes -- Computations with modular forms and Galois representations / Johan Bosman -- Polynomials for projective representations of level one forms / Johan Bosman -- Description of X1(5l) / Bas Edixhoven -- Applying Arakelov theory / Bas Edixhoven and Robin de Jong -- An upper bound for Green functions on Riemann surfaces / Franz Merkl -- Bounds for Arakelov invariants of modular curves / Bas Edixhoven and Robin de Jong -- Approximating V[subscript f] over the complex numbers / Jean-Marc Couveignes -- Computing V[subscript f] modulo p / Jean-Marc Couveignes -- Computing the residual Galois representations / Bas Edixhoven -- Computing coefficients of modular forms / Bas Edixhoven
Control code
ocn729386470
Dimensions
unknown
Extent
1 online resource (xi, 425 pages)
File format
unknown
Form of item
online
Isbn
9781400839001
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Note
JSTOR
Other control number
10.1515/9781400839001
Other physical details
illustrations
http://library.link/vocab/ext/overdrive/overdriveId
  • cl0500000122
  • 22573/cttt3bg
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
(OCoLC)729386470

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