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)

Bas Edixhoven
Jean-Marc Couveignes
Robin de Jong
Franz Merkl
Johan Bosman
Copyright Date: 2011
Pages: 438
https://www.jstor.org/stable/j.ctt7rzmb
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    Computational Aspects of Modular Forms and Galois Representations
    Book Description:

    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.

    eISBN: 978-1-4008-3900-1
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Preface
    (pp. ix-x)
  4. ACKNOWLEDGMENTS
    (pp. x-x)
  5. AUTHOR INFORMATION
    (pp. xi-xi)
  6. DEPENDENCIES BETWEEN THE CHAPTERS
    (pp. xii-xii)
  7. Chapter One Introduction, main results, context
    (pp. 1-28)
    Bas Edixhoven

    As the final results in this book are about fast computation of coefficients of modular forms, we start by describing the state of the art in this subject.

    A convenient way to view modular forms and their coefficients in this context is as follows, in terms of Hecke algebras. ForNandkpositive integers, letSk(Γ₁(N)) be the finite dimensional complex vector space of cusp forms of weightkon the congruence subgroup Γ₁(N) of SL₂(ℤ). EachfinSk(Γ₁(N)) has a power series expansion$f = \sum\nolimits_{n \geq 1} {{a_n}(f){q^n}} $, a complex power series converging on the open unit disk. Thesean(f)...

  8. Chapter Two Modular curves, modular forms, lattices, Galois representations
    (pp. 29-68)
    Bas Edixhoven

    As a good reference for getting an overview of the theory of modular curves and modular forms we recommend the article [Di-Im] by Fred Diamond and John Im. This reference is quite complete as results are concerned, and gives good references for the proofs of those results. Moreover, it is one of the few references that treats the various approaches to the theory of modular forms, from the classical analytic theory on the upper half-plane to the more modern representation theory of adelic groups. Another good first introduction is the book [Di-Sh]. Let us also mention the book [Conr] by...

  9. Chapter Three First description of the algorithms
    (pp. 69-78)
    Jean-Marc Couveignes and Bas Edixhoven

    We put ourselves in the situation of Theorem 2.5.7, and we ask how we can compute the Galois representation. More explicitly, letnandkbe positive integers, 𝔽 a finite field andlits characteristic, andf:𝕋(n,k) → 𝔽 a surjective ring morphism. Assume that 2 <k≤ l + 1, and that the associated Galois representation$\rho:{\text{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow {\text{GL}}_2(\mathbb{F})$is absolutely irreducible. Letf₂:𝕋(nl, 2) → 𝔽 be the weight two eigenform as in Theorem 2.5.7 and letm= ker(f₂) Assume that the multiplicity ofρin$V: = {J_1}(nl)(\overline{\mathbb{Q}})[m]$is one, that is, thatρis realized byV.

    We...

  10. Chapter Four Short introduction to heights and Arakelov theory
    (pp. 79-94)
    Bas Edixhoven and Robin de Jong

    Chapter 3 explained how the computation of the Galois representationsVattached to modular forms over finite fields should proceed. The essential step is to approximate the minimal polynomialPof (3.1) with sufficient precision so thatPitself can be obtained. The topic to be addressed now is to bound from above the precision that is needed for this. This means that we must bound the heights of the coefficients ofP. As was hinted at in Chapter 3, we get such bounds using Arakelov theory, a tool we discuss in this section. It is not at all excluded...

  11. Chapter Five Computing complex zeros of polynomials and power series
    (pp. 95-128)
    Jean-Marc Couveignes

    The purpose of this chapter is twofold. We first want to prove the two Theorems 5.3.1 and 5.4.2 below about the complexity of computing complex roots of polynomials and zeros of power series. The existence of a deterministic polynomial time algorithm for these purposes plays an important role in this book. More important, we want to explain what it means to compute with real or complex data in polynomial time. All the necessary concepts and algorithms already exist, and are provided partly by numerical analysis and partly by algorithmic complexity theory. However, the computational model of numerical analysis is not...

  12. Chapter Six Computations with modular forms and Galois representations
    (pp. 129-158)
    Johan Bosman

    In this chapter we will discuss several aspects of the practical side of computating with modular forms and Galois representations. We start by discussing computations with modular forms and from there work toward the computation of polynomials that give the Galois representations associated with modular forms. Throughout this chapter, we will denote the space of cusp forms of weightk, group Γ₁(N), and character ε bySk(N,ε).

    Modular symbols provide a way of doing symbolic calculations with modular forms, as well as the homology of modular curves. In this section our intention is to give the reader an idea of...

  13. Chapter Seven Polynomials for projective representations of level one forms
    (pp. 159-172)
    Johan Bosman

    In this chapter we explicitly compute mod-ℓ Galois representations attached to modular forms. To be precise, we look at cases with ℓ ≤ 23, and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the result in terms of polynomials associated with the projectivized representations. As an application, we will improve a known result on Lehmer’s nonvanishing conjecture for Ramanujan’s tau function (see [Leh], p. 429).

    To fix a notation, for anyk∈ ℤ satisfying dimSk(SL₂(ℤ)) = 1 we will denote the unique normalized cusp form inSk(SL₂(ℤ)) by...

  14. Chapter Eight Description of X₁(5l)
    (pp. 173-186)
    Bas Edixhoven

    In this section we put ourselves in the situation of Theorem 2.5.13:lis a prime number,kis an integer such that 2 <kl+ 1, andfis a surjective ring morphism 𝕋(1,k) → 𝔽 a with 𝔽 a finite field of characteristicl, such that the associated Galois representation$\rho :{\text{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\text{GL}}_2 (\mathbb{F})$is absolutely irreducible. We letVdenote the two-dimensional 𝔽-vector space in$J_1 (l)(\overline{\mathbb{Q}})[l] $that realizesρ.

    As explained in Chapter 3, we would like to have an effective divisorD₀ onX₁(l)of degree the genus ofX₁(l) such that for all nonzeroxin...

  15. Chapter Nine Applying Arakelov theory
    (pp. 187-202)
    Bas Edixhoven and Robin de Jong

    In this chapter we start applying Arakelov theory in order to derive a bound for the height of the coefficients of the polynomials$P_{D_0 ,f_l ,m} $as in (8.2.10). We proceed in a few steps. The first step, taken in Section 9.1, is to relate the height of the$b_l (Q_{x,i} )$as in Section 8.2 to intersection numbers onXl. The second step, taken in Section 9.2, is to get some control on the difference of the divisorsD₀ andDxas in (3.4). Certain intersection numbers concerning this difference are bounded in Theorem 9.2.5, in terms of a number of invariants in...

  16. Chapter Ten An upper bound for Green functions on Riemann surfaces
    (pp. 203-216)
    Franz Merkl

    We begin with explaining the setup and the results of this chapter. LetXbe a compact Riemann surface, endowed with a 2-form μ ≥ 0 that fulfills ∫Xμ = 1. Let * denote rotation by 90° in the cotangential spaces (with respect to the holomorphic structure); in a coordinatez=x+iythis means *dx=dy, *dy= -dxand, equivalently, *dz= -idz, *dz̄=idz̄. In particular, the Laplace operator on realCfunctions onXcan be written as$d*d = 2i\partial \bar \partial $.

    Fora,bX, letga,b:X- {a,b} → ℝ...

  17. Chapter Eleven Bounds for Arakelov invariants of modular curves
    (pp. 217-256)
    Bas Edixhoven and Robin de Jong

    In this chapter, we give bounds for all quantities on the right-hand side in the inequality in Theorems 9.1.1 and 9.2.5, in the context of the modular curvesX₁(5l) withl> 5 prime, using the upper bounds for Green functions from the previous chapter. The final estimates are given in the last section.

    As before, forl> 5 prime, we letXlbe the modular curveX₁(5l), over a suitable base that will be clear from the notation. We letgldenote the genus ofXl; we havegl> 1. A modelXl,ℤis given by [Ka-Ma], as well as...

  18. Chapter Twelve Approximating Vf over the complex numbers
    (pp. 257-336)
    Jean-Marc Couveignes

    In this chapter, we address the problem of computing torsion divisors on modular curves with an application to the explicit calculation of modular representations. We assume that we are given an even integerk> 2, a prime integerl> 6(k- 1), a finite field 𝔽 with characteristicl, and a ring epimorphismf: 𝕋(1,k) → 𝔽. We want to compute the associated Galois representation$\rho_{f}:{\text{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\text{GL}}_2 (\mathbb{F})$. This representation lies in the Jacobian variety of the modular curveX₁(l). Indeed, letf₂ : 𝕋(l,2) → 𝔽 be the unique ring homomorphism such thatf₂(Tm) =f(Tmfor every positive...

  19. Chapter Thirteen Computing Vf modulo p
    (pp. 337-370)
    Jean-Marc Couveignes

    In this chapter we address the problem of computing in the group oflk-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing modular representations.

    Letpbe a prime and let$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$be the field withpelements. Let$\overline{\mathbb{F}}_p $be an algebraic closure of$\mathbb{F}_p $. For any powerqofpwe call$\mathbb{F}_q \subset \overline{\mathbb{F}}_p $the field withqelements. Let$\mathbb{A}^2 \subset \mathbb{P}^2 $be the affine and projective planes over$\mathbb{F}_q $. Let$C \subset \mathbb{P}^2 $be a plane projective geometrically integral curve over$\mathbb{F}_q $. LetXbe its smooth projective model...

  20. Chapter Fourteen Computing the residual Galois representations
    (pp. 371-382)
    Bas Edixhoven

    In this chapter we first combine the results of Chapters 11 and 12 in order to work out the strategy of Chapter 3 in the setup of Section 8.2. This gives the main result, Theorem 14.1.1: a deterministic polynomial time algorithm, based on computations with complex numbers. The crucial transition from approximations to exact values is done in Section 14.4, and the proof of Theorem 14.1.1 is finished in Section 14.7. In Section 14.8 we replace the complex computations with the computations over finite fields from Chapter 13, and give a probabilistic (Las Vegas type) polynomial time variant of the...

  21. Chapter Fifteen Computing coefficients of modular forms
    (pp. 383-398)
    Bas Edixhoven

    In this chapter we apply our main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. In Section 15.1 we treat the case of the discriminant modular form, that is, the computation of Ramanujan’s τ-function at primes. In Section 15.2 we deal with the more general case of forms of level one and arbitrary weightk, reformulated as the computation of Hecke operatorsTnas ℤ-linear combinations of theTiwithi<k/12. In Section 15.3 we give an application to theta functions of even, unimodular positive definite...

  22. Epilogue
    (pp. 399-402)

    Theorems 14.1.1 and 15.2.1 will certainly be generalized to spaces of cusp forms of arbitrarily varying level and weight. This has already been done for the probabilistic variant of Theorem 14.1.1, in the case of square-free levels (and of level two times a square-free number for reasons that will become clear below) by Peter Bruin in his PhD thesis (Leiden, September 2010), see [Bru]. We describe some details and some applications of Bruin’s work, and a perspective on point counting outside the context of modular forms. Deterministic generalizations of the two theorems mentioned above will lead to deterministic applications.

    Peter...

  23. Bibliography
    (pp. 403-422)
  24. Index
    (pp. 423-425)