On pq -fold regular covers of the projective line
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On pq-fold regular covers of the projective line Sebastián Reyes-Carocca1 Received: 27 June 2020 / Accepted: 7 November 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract Let p and q be odd prime numbers. In this paper we study non-abelian pq-fold regular covers of the projective line, determine algebraic models for some special cases and provide a general isogeny decomposition of the corresponding Jacobian varieties. We also give a classification and description of the one-dimensional families of compact Riemann surfaces as before. Keywords Compact Riemann surfaces · Group actions · Automorphisms · Jacobians Mathematics Subject Classification 30F10 · 14H37 · 30F35 · 14H40
1 Introduction and statement of the results Compact Riemann surfaces (or, equivalently, smooth complex projective algebraic curves) and their automorphism groups have been extensively studied since the nineteenth century. Foundational results concerning that are: (1) if the genus of the compact Riemann surface is greater than one then its automorphism group is finite (see [21] and [41], and also [14]), and (2) each finite group acts as a group of automorphisms of some compact Riemann surface of a suitable genus greater than one (see [17] and also [26]). A general problem that arises naturally with regard to this is to determine necessary and sufficient conditions under which a given group acts as a group of automorphisms of a compact Riemann surface satisfying some prescribed conditions. This problem was successfully studied for cyclic groups by Harvey in [18] and soon after for abelian groups by Maclachlan in [29]. The same problem for dihedral groups, among other aspects, was completely solved by Bujalance, Cirre, Gamboa and Gromadzki in [7]. See also [49]. This article is devoted to study those compact Riemann surfaces that are branched pq-fold regular covers of the projective line, where p and q are prime numbers. Since the abelian
Partially supported by Fondecyt Grants 11180024, 1190991 and Redes Grant 2017-170071.
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Sebastián Reyes-Carocca [email protected] Departamento de Matemática y Estadística, Universidad de La Frontera, Avenida Francisco Salazar, 01145 Temuco, Chile 0123456789().: V,-vol
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and dihedral cases have been already classified, we shall consider compact Riemann surfaces endowed with a non-abelian group of automorphisms isomorphic to the semidirect product of two cyclic groups of odd prime order, in such a way that the corresponding orbit space is isomorphic to the projective line. Let p and q be odd primes such that p divides q − 1 and let r be a primitive p-th root of unity in the field of q elements. Throughout the article the unique non-abelian group of order pq will be denote by G p,q := a, b : a q = b p = 1, bab−1 = a r ∼ = Cq C p . The first result of this paper establishes a simple necessary and sufficient condition for G p,q to act on a compact Riemann surface of genus greater than one. In order to state it, we need to bring in the concept of sig
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