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
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 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 book begins with a concise and concrete introduction that makes it accessible to readers without an extensive background in arithmetic geometry, and it includes a chapter that describes actual computations.Less
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 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 book begins with a concise and concrete introduction that makes it accessible to readers without an extensive background in arithmetic geometry, and it includes a chapter that describes actual computations.
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
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, Schoof's algorithm, Schoof's algorithm described in terms of ètale cohomology, other cases where ètale cohomology can be used to construct polynomial time algorithms for counting rational points of varieties over finite fields, congruences for Ramanujan's tau-function, and comparison with p-adic methods.Less
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, Schoof's algorithm, Schoof's algorithm described in terms of ètale cohomology, other cases where ètale cohomology can be used to construct polynomial time algorithms for counting rational points of varieties over finite fields, congruences for Ramanujan's tau-function, and comparison with p-adic methods.
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
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 and weight up to 22. The chapter presents the result in terms of polynomials associated with the projectivized representations. As an application, it will improve a known result on Lehmer's nonvanishing conjecture for Ramanujan's tau function.Less
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 and weight up to 22. The chapter presents the result in terms of polynomials associated with the projectivized representations. As an application, it will improve a known result on Lehmer's nonvanishing conjecture for Ramanujan's tau function.
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
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 discriminant modular form, that is, the computation of Ramanujan's tau-function at primes, and then deals with the more general case of forms of level one and arbitrary weight k, reformulated as the computation of Hecke operators Tⁿ as ℤ-linear combinations of the Tᵢ with i < k = 12. The chapter gives an application to theta functions of even, unimodular positive definite quadratic forms over ℤ.Less
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 discriminant modular form, that is, the computation of Ramanujan's tau-function at primes, and then deals with the more general case of forms of level one and arbitrary weight k, reformulated as the computation of Hecke operators Tⁿ as ℤ-linear combinations of the Tᵢ with i < k = 12. The chapter gives an application to theta functions of even, unimodular positive definite quadratic forms over ℤ.
