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## Modular curves, modular forms, lattices, Galois representations

*Bas Edixhoven*

### in 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)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0002
- Subject:
- Mathematics, Number Theory

This chapter provides the necessary background concerning modular curves and modular forms. It covers modular curves, modular forms, lattices and modular forms, Galois representations attached to ... More

##
Approximating *V*_{f} over the complex numbers

_{f}

*Jean-Marc Couveignes*

### in 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)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0012
- Subject:
- Mathematics, Number Theory

This chapter addresses the problem of computing torsion divisors on modular curves with an application to the explicit calculation of modular representations. The final result of the chapter is ... More

## 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 and Jean-Marc Couveignes (eds)*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.001.0001
- Subject:
- Mathematics, Number Theory

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 ... More

## Computations with modular forms and Galois representations

*Johan Bosman*

### in 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)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0006
- Subject:
- Mathematics, Number Theory

This chapter discusses several aspects of the practical side of computing with modular forms and Galois representations. It starts by discussing computations with modular forms, and from there work ... More

## Epilogue

*Bas Edixhoven and Jean-Marc Couveignes (eds)*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0016
- Subject:
- Mathematics, Number Theory

This epilogue describes some work on generalizations and applications, as well as a direction of further research outside the context of modular forms. Theorems 14.1.1 and 15.2.1 will certainly be ... More

## Introduction, main results, context

*Bas Edixhoven*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0001
- Subject:
- Mathematics, Number Theory

This chapter provides an introduction to the subject, precise statements of the main results, and places these in a somewhat wider context. Topics discussed include statement of the main results, ... More

## Computing coefficients of modular forms

*Bas Edixhoven*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0015
- Subject:
- Mathematics, Number Theory

This chapter applies the main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. It treats the case of the ... More

## Bounds for Arakelov invariants of modular curves

*Bas Edixhoven and Robin de Jong*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0011
- Subject:
- Mathematics, Number Theory

This chapter gives 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 curves X₁(5l) with l > 5 prime, using the upper bounds ... More

## Polynomials for projective representations of level one forms

*Johan Bosman*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0007
- Subject:
- Mathematics, Number Theory

This chapter explicitly computes mod-ℓ Galois representations attached to modular forms. To be precise, it looks at cases with l ≤ 23, and the modular forms considered will be cusp forms of level 1 ... More

##
Computing *V*_{f} modulo *p*

_{f}

*Jean-Marc Couveignes*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0013
- Subject:
- Mathematics, Number Theory

This chapter addresses the problem of computing in the group of lsuperscript k-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing ... More

## First description of the algorithms

*Jean-Marc Couveignes and Bas Edixhoven*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0003
- Subject:
- Mathematics, Number Theory

This chapter provides the first, informal description of the algorithms. It explains how the computation of the Galois representations V attached to modular forms over finite fields should proceed. ... More

## Applying Arakelov theory

*Bas Edixhoven and Robin de Jong*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0009
- Subject:
- Mathematics, Number Theory

This chapter starts applying Arakelov theory in order to derive a bound for the height of the coefficients of the polynomials Pno hexa conversion found as in (8.2.10). It proceeds in a few steps. The ... More

##
Description of *X*_{1}(5*l*)

*Bas Edixhoven*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0008
- Subject:
- Mathematics, Number Theory

This chapter first discusses the construction of a suitable cuspidal divisor on X₁(5l). It then describes the strategy for computing the residual representations V in the situation of Theorem 2.5.13.

## An upper bound for Green functions on Riemann surfaces

*Franz Merkl*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0010
- Subject:
- Mathematics, Number Theory

This chapter deals with an upper bound for Green functions on Riemann surfaces.

## Short introduction to heights and Arakelov theory

*Bas Edixhoven and Robin de Jong*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0004
- Subject:
- Mathematics, Number Theory

This chapter discusses bounding the heights of the coefficients of minimal polynomial P. As was hinted at in Chapter 3, such bounds are obtained using Arakelov theory, a tool that is discussed in ... More

## Computing complex zeros of polynomials and power series

*Jean-Marc Couveignes*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0005
- Subject:
- Mathematics, Number Theory

The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of ... More

## Computing the residual Galois representations

*Bas Edixhoven*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0014
- Subject:
- Mathematics, Number Theory

This chapter proves the main result on the computation of Galois representations. It provides a detailed description of the algorithm and a rigorous proof of the complexity. It first combines the ... More

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