Computational Aspects of Modular Forms and Galois Representations (eBook)

How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Bas Edixhoven, Jean-Marc Couveignes (Herausgeber)

eBook Download: EPUB

2011
440 Seiten
Princeton University Press (Verlag)
978-1-4008-3900-1 (ISBN)

Infos

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.

Autor

Bas Edixhoven is professor of mathematics at the University of Leiden. Jean-Marc Couveignes is professor of mathematics at the University of Toulouse le Mirail. Robin de Jong is assistant professor at the University of Leiden. Franz Merkl is professor of applied mathematics at the University of Munich. Johan Bosman is a postdoctoral researcher at the Institut für Experimentelle Mathematik in Essen, Germany.

Produkt-Details

• Erscheint lt. Verlag: 31.5.2011
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Co-Autor: Robin De Jong, Franz Merkl, Johan Bosman
• Zusatzinfo: 6 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: absolute value • Accuracy and precision • Addition • Agence nationale de la recherche • algorithm • algorithms • Approximation • Arakelov invariants • Arakelov Theory • Arithmetic Geometry • arithmetic surfaces • automorphism • Bosman • bounding heights • bounds • Calculation • Characteristic Polynomial • coefficient • Coefficients • cohomology • combination • Complex conjugate • complex number • complex roots • Computation • Computing Algorithms • computing coefficients • conjugacy class • cusp form • Cusp forms • cuspidal divisor • Determinant • Diamond operator • differential form • Dimension (vector space) • discriminant • Division by zero • Divisor • Divisor (algebraic geometry) • dot product • Eigenform • eigenforms • Eigenvalues and Eigenvectors • Eisenstein series • Elliptic Curve • Embedding • Equation • estimation • existential quantification • Explicit formulae (L-function) • exponential function • exponentiation • Factorization • finite field • Finite Fields • Formal power series • Fourier coefficients • Frobenius endomorphism • functional equation • Galois extension • Galois representation • Galois representations • Green functions • GRH • Hecke algebra • Hecke Operator • Hecke operators • height functions • holomorphic function • Inequality • Integer • Irreducible component • Jacobian matrix and determinant • Jacobians • Jacobian variety • langlands program • Las Vegas algorithm • lattices • Lehmer • Linear combination • line bundle • logarithm • Maximal Ideal • minimal polynomial • Minimal polynomial (field theory) • modular curve • modular curves • modular form • Modular Forms • Modular representation • modular representations • Modular symbols • monic polynomial • Morphism • Natural number • nonvanishing conjecture • Numerical analysis • p-adic methods • Pairing • Parity (mathematics) • Permutation • Peter Bruin • plane curves • polynomial • polynomials • polynomial time • polynomial time algorithm • polynomial time algoriths • power series • prime factor • Prime number • probabilistic polynomial time • quadratic form • Ramanujan's tau • Ramanujan's tau function • Ramanujan's tau-function • random divisors • randomized algorithm • Rational number • real number • residual representation • residue field • Riemann hypothesis • Riemann–Roch theorem • Riemann surface • Riemann Surfaces • Ring of integers • Root of unity • Schoof's algorithm • scientific notation • Series expansion • Special case • square-free levels • square root • SUBGROUP • Summation • Surjective function • System of polynomial equations • tale cohomology • Tangent Space • Theorem • time complexity • Torsion • torsion divisors • Turing Machine • Turing Machines • Unit disk • Upper and lower bounds • Upper half-plane • Variable (mathematics) • Vector Space • Weil pairing
• ISBN-10: 1-4008-3900-9 / 1400839009
• ISBN-13: 978-1-4008-3900-1 / 9781400839001