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 JeanMarc 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 JeanMarc Couveignes and Bas Edixhoven
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The item 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 JeanMarc Couveignes and Bas Edixhoven represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of San Diego Libraries.This item is available to borrow from 1 library branch.
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The item 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 JeanMarc Couveignes and Bas Edixhoven represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of San Diego Libraries.
This item is available to borrow from 1 library branch.
 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 taufunction 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 precisionin other words, bounds for the height of the rational numbers that describe the Galois representation to be computedare 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'"
 Language
 eng
 Extent
 1 online resource (xi, 425 pages)
 Contents

 Introduction, main results, context / Bas Edixhoven
 Modular curves, modular forms, lattices, Galois representations / Bas Edixhoven
 First description of the algorithms / JeanMarc 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 / JeanMarc 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 / JeanMarc Couveignes
 Computing V[subscript f] modulo p / JeanMarc Couveignes
 Computing the residual Galois representations / Bas Edixhoven
 Computing coefficients of modular forms / Bas Edixhoven
 Isbn
 9781400839001
 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 JeanMarc Couveignes and Bas Edixhoven
 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 taufunction 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 precisionin other words, bounds for the height of the rational numbers that describe the Galois representation to be computedare 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'"
 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, JeanMarc
 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 JeanMarc Couveignes and Bas Edixhoven
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references (pages 403421) 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 / JeanMarc 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 / JeanMarc 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 / JeanMarc Couveignes  Computing V[subscript f] modulo p / JeanMarc 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 JeanMarc Couveignes and Bas Edixhoven
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references (pages 403421) 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 / JeanMarc 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 / JeanMarc 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 / JeanMarc Couveignes  Computing V[subscript f] modulo p / JeanMarc 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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