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INTRODUCTION

Haruzo Hida

in Hilbert Modular Forms and Iwasawa Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198571025
eISBN:
9780191718946
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571025.003.0001
Subject:
Mathematics, Algebra

This introductory chapter describes the deformation theoretic techniques of Wiles-Taylor for elliptic modular forms. The chapter concentrates mostly on the Galois side of the picture, recalling the ... More


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


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


Computing the residual 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/9780691142012.003.0014
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


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


Polynomials for projective representations of level one forms

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/9780691142012.003.0007
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 coefficients of modular forms

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/9780691142012.003.0015
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


First description of the algorithms

Jean-Marc Couveignes and 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/9780691142012.003.0003
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


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