2021/05/03 by Rubén A. Hidalgo, Hidalgo, Ruben A.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2105.01182
openalex publication_date 2021/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p ≥ 3 be a prime integer and, for l ≥ 1, let G ≅ \mathbb Zpl be a group of conformal automorphisms of some closed Riemann surface S of genus g ≥ 2. By the Riemann-Hurwitz formula, either p ≤ g+1 or p=2g+1. If l=1 and p=2g+1, then S/G is the sphere with exactly three cone points and, if moreover p ≥ 7, then G is the unique p-Sylow subgroup of \rm Aut(S). If l=1 and p=g+1, then S/G is the sphere with exactly four cone points and, if moreover p ≥ 13, then G is again the unique p-Sylow subgroup. The above unique facts permited many authors to obtain algebraic models and the corresponding groups \rm Aut(S) in these situations. Now, let us assume l ≥ 2. If p ≥ 5, then either (i) pl ≤ g-1 or (ii) S/G has genus zero, pl-1(p-3) ≤ 2(g-1) and 2 ≤ l ≤ r-1, where r ≥ 3 is the number of cone points of S/G. Let us assume we are in case (ii). If r=3, then l=2 and S happens to be the classical Fermat curve of degree p, whose group of automorphisms is well known. The next case, r=4, is studied in this paper. We provide an algebraic curve representation for S, a description of its group of conformal automorphisms, a discussion of its field of moduli and an isogenous decomposition of its jacobian variety.