Bounding the degree of Belyi polynomials

JOURNAL OF NUMBER THEORY

卷:133

Article

Rodriguez, Jose

关键词: Belyi;    Dessin d'enfant;    Newton polygon;    p-Adics;    Height functions;    Heights;    Belyi height;    Belyi functions;   

DOI  :  10.1016/j.jnt.2012.12.019

来源: Elsevier

【 摘 要 】

Text. Belyi's theorem states that a Riemann surface X, as an algebraic curve, is defined over (Q) over bar if and only if there exists a holomorphic function B taking X to (PC)-C-1 with at most three critical values {0, 1, infinity}. By restricting to the case where X = (PC)-C-1 and our holomorphic functions are Belyi polynomials, for an algebraic number lambda, we define a Belyi height H(lambda) to be the minimal degree of the set of Belyi polynomials with B(lambda) is an element of {0, 1}. We prove for non-zero lambda with non-zero p-adic valuation, the Belyi height of lambda is greater than or equal to p using the combinatorics of Newton polygons. We also give examples of algebraic numbers with relatively low height and show that our bounds are sharp. (c) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free