Computational Aspects of Modular Forms and Galois Representations

Computational Aspects of Modular Forms and Galois Representations

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

In: Annals of Mathematics Studies, 176

With contributions by: Robin de Jong, Franz Merkl and Johan Bosman
Edited by: Bas Edixhoven and Jean-Marc Couveignes

Princeton University Press | 2011

https://doi.org/10.1515/9781400839001

Overview
Contents

About this book

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.

Author information

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.

Reviews

The book is well written and provides sufficient detail and reminders about the big picture. It gives a nice exposition of the material involved and should be accessible to graduate students or researchers with a sufficient background in number theory and algebraic geometry.---Jeremy A. Rouse, Mathematical Reviews Clippings

Frontmatter Free access PDF PDF	i
Contents Free access PDF PDF	v
Preface Access restricted Content is available PDF PDF	ix
Acknowledgments Access restricted Content is available PDF PDF	x
Author information Access restricted Content is available PDF PDF	xi
Dependencies between the chapters Access restricted Content is available PDF PDF	xii
Chapter 1. Introduction, main results, context Bas Edixhoven Access restricted Content is available PDF PDF	1
Chapter 2. Modular curves, modular forms, lattices, Galois representations Bas Edixhoven Access restricted Content is available PDF PDF	29
Chapter 3. First description of the algorithms Jean-Marc Couveignes and Bas Edixhoven Access restricted Content is available PDF PDF	69
Chapter 4. Short introduction to heights and Arakelov theory Bas Edixhoven and Robin de Jong Access restricted Content is available PDF PDF	79
Chapter 5. Computing complex zeros of polynomials and power series Jean-Marc Couveignes Access restricted Content is available PDF PDF	95
Chapter 6. Computations with modular forms and Galois representations Johan Bosman Access restricted Content is available PDF PDF	129
Chapter 7. Polynomials for projective representations of level one forms Johan Bosman Access restricted Content is available PDF PDF	159
Chapter 8. Description of X1(5l) Bas Edixhoven Access restricted Content is available PDF PDF	173
Chapter 9. Applying Arakelov theory Bas Edixhoven and Robin de Jong Access restricted Content is available PDF PDF	187
Chapter 10. An upper bound for Green functions on Riemann surfaces Franz Merkl Access restricted Content is available PDF PDF	203
Chapter 11. Bounds for Arakelov invariants of modular curves B. Edixhoven and R. de Jong Access restricted Content is available PDF PDF	217
Chapter 12. Approximating Vf over the complex numbers Jean-Marc Couveignes Access restricted Content is available PDF PDF	257
Chapter 13. Computing Vf modulo p Jean-Marc Couveignes Access restricted Content is available PDF PDF	337
Chapter 14. Computing the residual Galois representations Bas Edixhoven Access restricted Content is available PDF PDF	371
Chapter 15. Computing coefficients of modular forms Bas Edixhoven Access restricted Content is available PDF PDF	383
Epilogue Access restricted Content is available PDF PDF	399
Bibliography Access restricted Content is available PDF PDF	403
Index Access restricted Content is available PDF PDF	423

(Deutsch)

Computational Aspects of Modular Forms and Galois Representations

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

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