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



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Titolo: Computational aspects of modular forms and Galois representations [[electronic resource] ] : how one can compute in polynomial time the value of Ramanujan's tau at a prime / / edited by Jean-Marc Couveignes and Bas Edixhoven Visualizza cluster
Pubblicazione: Princeton, N.J., : Princeton University Press, c2011
Edizione: Course Book
Descrizione fisica: 1 online resource (438 p.)
Disciplina: 512/.32
Soggetto topico: Galois modules (Algebra)
Class field theory
Soggetto non controllato: Arakelov invariants
Arakelov theory
Fourier coefficients
Galois representation
Galois representations
Green functions
Hecke operators
Jacobians
Langlands program
Las Vegas algorithm
Lehmer
Peter Bruin
Ramanujan's tau function
Ramanujan's tau-function
Ramanujan's tau
Riemann surfaces
Schoof's algorithm
Turing machines
algorithms
arithmetic geometry
arithmetic surfaces
bounding heights
bounds
coefficients
complex roots
computation
computing algorithms
computing coefficients
cusp forms
cuspidal divisor
eigenforms
finite fields
height functions
inequality
lattices
minimal polynomial
modular curves
modular forms
modular representation
modular representations
modular symbols
nonvanishing conjecture
p-adic methods
plane curves
polynomial time algorithm
polynomial time algoriths
polynomial time
polynomials
power series
probabilistic polynomial time
random divisors
residual representation
square root
square-free levels
tale cohomology
torsion divisors
torsion
Classificazione: MAT001000MAT012010
Altri autori: EdixhovenB <1962-> (Bas)  
CouveignesJean-Marc  
Note generali: Description based upon print version of record.
Nota di bibliografia: Includes bibliographical references and index.
Nota di contenuto: Front matter -- Contents -- Preface -- Acknowledgments -- Author information -- Dependencies between the chapters -- Chapter 1. Introduction, main results, context / Edixhoven, Bas -- Chapter 2. Modular curves, modular forms, lattices, Galois representations / Edixhoven, Bas -- Chapter 3. First description of the algorithms / Couveignes, Jean-Marc / Edixhoven, Bas -- Chapter 4. Short introduction to heights and Arakelov theory / Edixhoven, Bas / de Jong, Robin -- Chapter 5. Computing complex zeros of polynomials and power series / Couveignes, Jean-Marc -- Chapter 6. Computations with modular forms and Galois representations / Bosman, Johan -- Chapter 7. Polynomials for projective representations of level one forms / Bosman, Johan -- Chapter 8. Description of X1(5l) / Edixhoven, Bas -- Chapter 9. Applying Arakelov theory / Edixhoven, Bas / de Jong, Robin -- Chapter 10. An upper bound for Green functions on Riemann surfaces / Merkl, Franz -- Chapter 11. Bounds for Arakelov invariants of modular curves / Edixhoven, B. / de Jong, R. -- Chapter 12. Approximating Vf over the complex numbers / Couveignes, Jean-Marc -- Chapter 13. Computing Vf modulo p / Couveignes, Jean-Marc -- Chapter 14. Computing the residual Galois representations / Edixhoven, Bas -- Chapter 15. Computing coefficients of modular forms / Edixhoven, Bas -- Epilogue -- Bibliography -- Index
Sommario/riassunto: "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'"--
Titolo autorizzato: Computational aspects of modular forms and Galois representations  Visualizza cluster
ISBN: 1-283-05180-X
9786613051806
1-4008-3900-9
Formato: Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione: Inglese
Record Nr.: 9910789850303321
Lo trovi qui: Univ. Federico II
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Serie: Annals of mathematics studies ; ; no. 176.