ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P3.02
https://doi.org/10.26493/1855-3974.2570.5e8
(Also available at http://amc-journal.eu)
On p-gonal fields of definition*
Ruben A. Hidalgo †
Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile
Received 24 February 2021, accepted 28 November 2023, published online 11 June 2024
Abstract
Let S be a closed Riemann surface of genus g ≥ 2 and φ be a conformal automorphism
of S of prime order p such that S/⟨φ⟩ has genus zero. Let K ≤ C be a field of definition of
S. We prove the existence of a field extension F of K, of degree at most 2(p−1), for which
S is definable by a curve of the form yp = F (x) ∈ F[x], in which case φ corresponds to
(x, y) 7→ (x, e2πi/py). If, moreover, φ is also definable over K, then F can be chosen to
be at most a quadratic extension of K. For p = 2, that is when S is hyperelliptic and φ
is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and
Lercier-Ritzenthaler-Sijslingit in the case that Aut(S)/⟨φ⟩ is non-trivial.
Keywords: Riemann surfaces, p-gonal curves, automorphisms.
Math. Subj. Class. (2020): 30F10, 30F20, 14H37, 14H55
*The author would like to express their gratitude to both referees for their valuable feedback, suggestions, and
corrections that have significantly improved the paper.
†Partially supported by projects Fondecyt 1230001 and 1220261.
E-mail address: ruben.hidalgo@ufrontera.cl (Ruben A. Hidalgo)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P3.02
https://doi.org/10.26493/1855-3974.2570.5e8
(Dostopno tudi na http://amc-journal.eu)
O p-gonalnih definicijskih obsegih*
Ruben A. Hidalgo †
Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile
Prejeto 24. februarja 2021, sprejeto 28. novembra 2023, objavljeno na spletu 11. junija 2024
Povzetek
Naj bo S sklenjena Riemannova ploskev roda g ≥ 2 in naj bo φ tak konformni avto-
morfizem ploskve S praštevilskega reda p, da ima S/⟨φ⟩ rod nič. Naj bo K ≤ C defini-
cijski obseg ploskve S. Dokažemo obstoj razširitve obsega F obsega K, stopnje največ
2(p − 1), za katero je S definiran s krivuljo oblike yp = F (x) ∈ F[x], kjer φ ustreza
(x, y) 7→ (x, e2πi/py). Če je, poleg tega, φ definiran tudi nad K, potem se da F izbrati
tako, da je kvečjemu kvadratna razširitev definicijskega obsega K. Za p = 2, ko je ploskev
S hipereliptična in je φ njena hipereliptična involucija, je za ta rezultat zaslužen Mestre (za
sodi rod), ter Huggins in Lercier-Ritzenthaler-Sijslingit, ki so to pokazali za primer, ko je
Aut(S)/⟨φ⟩ netrivialen.
Ključne besede: Riemannove ploskve, p-gonske krivulje, avtomorfizmi.
Math. Subj. Class. (2020): 30F10, 30F20, 14H37, 14H55
*Avtor bi rad izrazil svojo hvaležnost obema recenzentoma za njune dragocene povratne informacije, predloge
in popravke, ki so znano izboljšali članek.
†Delno podprt s projektoma Fondecyt 1230001 in 1220261.
E-poštni naslov: ruben.hidalgo@ufrontera.cl (Ruben A. Hidalgo)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/