Field of Moduli and Generalized Fermat Curves

Field of Moduli and Generalized Fermat Curves

A generalized Fermat curve of type (p,n) is a closed Riemann surface S admitting a group H \cong Zp n of conformal automorphisms with S/H being the Riemann sphere with exactly n+1 cone points, each one of order p. If (p-1)(n-1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by \operatornameAutH(S) the normalizer of H in \operatornameAut(S). If p is a prime, and either (i) n=4 or (ii) n is even and \operatornameAutH(S)/H is not a non-trivial cyclic group or (iii) n is odd and \operatornameAutH(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3,4}, then we also compute the field of moduli of S.

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Main Authors: HIDALGO,RUBEN A., REYES-CAROCCA,SEBASTIÁN, VALDÉS,MARÍA ELISA
Format: Digital revista
Language:English
Published: Universidad Nacional de Colombia y Sociedad Colombiana de Matemáticas 2013
Online Access:http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0034-74262013000200007
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spelling oai:scielo:S0034-742620130002000072014-02-14Field of Moduli and Generalized Fermat CurvesHIDALGO,RUBEN A.REYES-CAROCCA,SEBASTIÁNVALDÉS,MARÍA ELISA Algebraic curves Riemann surfaces Field of moduli Field of definition A generalized Fermat curve of type (p,n) is a closed Riemann surface S admitting a group H \cong Zp n of conformal automorphisms with S/H being the Riemann sphere with exactly n+1 cone points, each one of order p. If (p-1)(n-1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by \operatornameAutH(S) the normalizer of H in \operatornameAut(S). If p is a prime, and either (i) n=4 or (ii) n is even and \operatornameAutH(S)/H is not a non-trivial cyclic group or (iii) n is odd and \operatornameAutH(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3,4}, then we also compute the field of moduli of S.info:eu-repo/semantics/openAccessUniversidad Nacional de Colombia y Sociedad Colombiana de MatemáticasRevista Colombiana de Matemáticas v.47 n.2 20132013-12-01info:eu-repo/semantics/articletext/htmlhttp://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0034-74262013000200007en
institution SCIELO
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country Colombia
countrycode CO
component Revista
access En linea
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region America del Sur
libraryname SciELO
language English
format Digital
author HIDALGO,RUBEN A.
REYES-CAROCCA,SEBASTIÁN
VALDÉS,MARÍA ELISA
spellingShingle HIDALGO,RUBEN A.
REYES-CAROCCA,SEBASTIÁN
VALDÉS,MARÍA ELISA
Field of Moduli and Generalized Fermat Curves
author_facet HIDALGO,RUBEN A.
REYES-CAROCCA,SEBASTIÁN
VALDÉS,MARÍA ELISA
author_sort HIDALGO,RUBEN A.
title Field of Moduli and Generalized Fermat Curves
title_short Field of Moduli and Generalized Fermat Curves
title_full Field of Moduli and Generalized Fermat Curves
title_fullStr Field of Moduli and Generalized Fermat Curves
title_full_unstemmed Field of Moduli and Generalized Fermat Curves
title_sort field of moduli and generalized fermat curves
description A generalized Fermat curve of type (p,n) is a closed Riemann surface S admitting a group H \cong Zp n of conformal automorphisms with S/H being the Riemann sphere with exactly n+1 cone points, each one of order p. If (p-1)(n-1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by \operatornameAutH(S) the normalizer of H in \operatornameAut(S). If p is a prime, and either (i) n=4 or (ii) n is even and \operatornameAutH(S)/H is not a non-trivial cyclic group or (iii) n is odd and \operatornameAutH(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3,4}, then we also compute the field of moduli of S.
publisher Universidad Nacional de Colombia y Sociedad Colombiana de Matemáticas
publishDate 2013
url http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0034-74262013000200007
work_keys_str_mv AT hidalgorubena fieldofmoduliandgeneralizedfermatcurves
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AT valdesmariaelisa fieldofmoduliandgeneralizedfermatcurves
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