Book contents
- Frontmatter
- Contents
- Preface
- Index of authors
- List of participants
- Conference photograph, with key
- The trace problem for totally positive algebraic integers
- Mahler's measure: from Number Theory to Geometry
- Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps
- The merit factor problem
- Barker sequences and flat polynomials
- The Hansen-Mullen primitivity conjecture: completion of proof
- An inequality for the multiplicity of the roots of a polynomial
- Newman's inequality for increasing exponential sums
- On primitive divisors of n2 + b
- Irreducibility and greatest common divisor algorithms for sparse polynomials
- Consequences of the continuity of the monic integer transfinite diameter
- Nonlinear recurrence sequences and Laurent polynomials
- Conjugate algebraic numbers on conics: a survey
- On polynomial ergodic averages and square functions
- Polynomial inequalities, Mahler's measure, and multipliers
- Integer transfinite diameter and computation of polynomials
- Smooth divisors of polynomials
- Self-inversive polynomials with all zeros on the unit circle
- The Mahler measure of algebraic numbers: a survey
Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps
Published online by Cambridge University Press: 04 May 2010
- Frontmatter
- Contents
- Preface
- Index of authors
- List of participants
- Conference photograph, with key
- The trace problem for totally positive algebraic integers
- Mahler's measure: from Number Theory to Geometry
- Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps
- The merit factor problem
- Barker sequences and flat polynomials
- The Hansen-Mullen primitivity conjecture: completion of proof
- An inequality for the multiplicity of the roots of a polynomial
- Newman's inequality for increasing exponential sums
- On primitive divisors of n2 + b
- Irreducibility and greatest common divisor algorithms for sparse polynomials
- Consequences of the continuity of the monic integer transfinite diameter
- Nonlinear recurrence sequences and Laurent polynomials
- Conjugate algebraic numbers on conics: a survey
- On polynomial ergodic averages and square functions
- Polynomial inequalities, Mahler's measure, and multipliers
- Integer transfinite diameter and computation of polynomials
- Smooth divisors of polynomials
- Self-inversive polynomials with all zeros on the unit circle
- The Mahler measure of algebraic numbers: a survey
Summary
Abstract. In this paper we give an introduction to Belyi-maps and Grothendieck's dessins d'enfant. In addition we provide an explicit method to compute the Belyi-maps corresponding to all semi-stable families of elliptic curves with six singular fibers.
Introduction
In a paper by Miranda and Persson, the authors study semi-stable elliptic fibrations over ℙ1 of K3-surfaces with 6 singular fibres. In their paper the authors give a list of possible fiber types for such fibrations. It turns out that there are 112 cases. The corresponding J-invariant is a so-called Belyifunction. More particularly, J is a rational function of degree 24, it ramifies of order 3 in every point above 0, it ramifies of order 2 in every point above 1, and the only other ramification occurs above infinity. To every such map we can associate a so-called ‘dessin d'enfant’ (a name coined by Grothendieck) which in its turn uniquely determines the Belyi map. If f: C → ℙ1 is a Belyi map, the dessin is the inverse image under f of the real segment.
Several papers, e.g., have been devoted to the calculation of some of the rational J-invariants for the Miranda-Persson list. It turns out that explicit calculations quickly become too cumbersome (even for a computer) if one is not careful enough. The goal of this paper is to compute all J-invariants corresponding to the Miranda-Persson list.
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- Number Theory and Polynomials , pp. 33 - 51Publisher: Cambridge University PressPrint publication year: 2008
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