Publications
Publications
- October 2015
- Proceedings of the London Mathematical Society
A New Gal(ℚ^⎯/ℚ)-invariant of Dessins d'enfants
By: Ravi Jagadeesan
Abstract
We study the action of Gal(ℚ^⎯/ℚ) on the category of Belyĭ functions (finite étale covers of ℙ^1_(ℚ^⎯) \ {0,1,∞}. We describe a new combinatorial Gal(ℚ^⎯/ℚ)‐invariant for Belyĭ functions whose monodromy cycle types above 0 and ∞ are the same. We use a version of our invariant to prove that Gal(ℚ^⎯/ℚ) acts faithfully on the set of Belyĭ functions whose monodromy cycle types above 0 and ∞ are the same; the proof of this result involves a version of Belyĭ's Theorem for meromorphic functions of odd degree. Using our invariant, we obtain that for all k<2^(√2/√3) and all positive integers N, there exists a positive integer n⩽N such that the set of degree n Belyĭ functions of a particular rational Nielsen class must split into at least Ω(k^√N) Galois orbits.
Keywords
Citation
Jagadeesan, Ravi. "A New Gal(ℚ^⎯/ℚ)-invariant of Dessins d'enfants." Proceedings of the London Mathematical Society 111, no. 4 (October 2015): 911–935.