Field of moduli of generalized Fermat curves of type (k, 3) with an application to non-hyperelliptic dessins d'enfants. (November 2015)

• Record Type: Journal Article
• Authors: Hidalgo, Ruben A.; Johnson, Pilar
• Is Part Of: Journal of symbolic computation. Volume 71(2016)
• Page Start: 60
• Page End: 72
• Publication Date: 2015-11
• Subjects: 30F10 -- 14H37 -- 14H10 -- 14H45; Algebraic curves -- Riemann surfaces -- Field of moduli -- Field of definition; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2014.09.042
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: A generalized Fermat curve of type ( k, 3 ), where k ≥ 2, is a closed Riemann surface admitting a group H ≅ Z k 3 as a group of conformal automorphisms so that the quotient orbifold S / H is the Riemann sphere and it has exactly 4 cone points, each one of order k . Every genus one Riemann surface is a generalized Fermat curve of type ( 2, 3 ) and, if k ≥ 3, then a generalized Fermat curve of type ( k, 3 ) is non-hyperelliptic. For each generalized Fermat curve, we compute its field of moduli and note that it is a field of definition. Moreover, for k = e 2 i π / p, where p ≥ 5 is a prime integer, we produce explicit algebraic models over the corresponding field of moduli. As a byproduct, we observe that the absolute Galois group Gal ( Q ¯ / Q ) acts faithfully at the level of non-hyperelliptic dessins d'enfants. This last fact was already known for dessins of genus 0, 1 and for hyperelliptic ones, but it seems that the non-hyperelliptic situation is not explicitly given in the existent literature.