A new Gal(Q)over-bar/Q)-invariant of dessins d'enfants

作者：Jagadeesan Ravi^*

来源：Proceedings of the London Mathematical Society, 2015, 111(4): 911-935.

DOI：10.1112/plms/pdv039

We study the action of Gal(Q/Q) on the category of Belyi functions (finite etale covers of P 1/Q {0, 1,8}). We describe a new combinatorial Gal(Q/Q)-invariant for Belyi functions whose monodromy cycle types above 0 and 8 are the same. We use a version of our invariant to prove that Gal(Q/Q) acts faithfully on the set of Belyi functions whose monodromy cycle types above 0 and 8 are the same; the proof of this result involves a version of Belyi's Theorem for meromorphic functions of odd degree. Using our invariant, we obtain that for all k < 2(root 2/3) and all positive integers N, there exists a positive integer n <= N such that the set of degree n Belyi functions of a particular rational Nielsen class must split into at least Omega(k(root N)) Galois orbits.

• 出版日期2015-10