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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. 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 fix ... More

*Keywords: *
modular forms,
arithmetic geometry,
computing algorithms,
Fourier coefficients,
computing coefficients,
polynomial time,
Ramanujan's tau,
Galois representations,
Langlands program,
computation

Print publication date: 2011 | Print ISBN-13: 9780691142012 |

Published to Princeton Scholarship Online: October 2017 | DOI:10.23943/princeton/9780691142012.001.0001 |

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## Front Matter

###
Chapter One Introduction, main results, context

### Bas Edixhoven

###
Chapter Three First description of the algorithms

### Jean-Marc Couveignes and Bas Edixhoven

###
Chapter Four Short introduction to heights and Arakelov theory

### Bas Edixhoven and Robin de Jong

###
Chapter Five Computing complex zeros of polynomials and power series

### Jean-Marc Couveignes

###
Chapter Eight Description of X1(5l)

### Bas Edixhoven

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Chapter Nine Applying Arakelov theory

### Bas Edixhoven and Robin de Jong

###
Chapter Eleven Bounds for Arakelov invariants of modular curves

### Bas Edixhoven and Robin de Jong

###
Chapter Twelve Approximating Vf over the complex numbers

### Jean-Marc Couveignes

###
Chapter Thirteen Computing Vf modulo p

### Jean-Marc Couveignes

###
Chapter Fifteen Computing coefficients of modular forms

### Bas Edixhoven

## End Matter

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