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.

Format:

Book

Contributor:

Edixhoven, B. (Bas), 1962-

Couveignes, Jean-Marc.

Series:

Annals of mathematics studies ; no. 176.

Annals of mathematics studies ; 176

Language:

English

Subjects (All):

Galois modules (Algebra).

Class field theory.

Physical Description:

1 online resource (438 p.)

Edition:

Course Book

Place of Publication:

Princeton, N.J. : Princeton University Press, c2011.

Language Note:

English

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"-- Provided by publisher.

"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'"-- Provided by publisher.

Contents:

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

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786613051806

9781283051804

128305180X

9781400839001

1400839009

OCLC:

729386470