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Fields of moduli of some special curves
Article
Full-text available
  • May 2015
In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic p-gonal curves by A. Kontogeorgis.
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... G is isomorphic to one of the following groups Z n , D n , A 4 , S 4 , A 5 : (if q = 0 or if |G| is prime to q), Z t q , Z t q Z m , PGL 2 (F q r ), PSL 2 (F q r ) : (if |G| is divisible by q), where (n, q) = 1, r > 0, t ≤ r, and m is a divisor of q t − 1. Moreover, the signature of the quotient orbifold P 1 L /G is given in the Table 1, where α = q r (q r −1) ...
Context 2
... K = Z t q , we may proceed as in the previous case by considering a regular branched cover Q : P 1 → P 1 with K as its deck group and with B = {b} where b is the unique branch value of Q with total order, see Table 1. ...
Context 3
... K is either isomorphic to either Z t q Z m , PSL 2 (F q r ) or PGL 2 (F q r ), then the regular branched cover map Q : P 1 → P 1 with deck(Q) = K has branch values set equal to B = {b 1 , b 2 } and the divisor D = [R(b 1 )] + [R(b 2 )] is K-rational. Since the two branch points b 1 and b 2 have different ramification index, see Table 1, we may proceed as in the previous case by considering a K-rational divisor D = [R(b i )] of degree one, where i = 1, 2. ...

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Citations

... (2) In complex arithmetic geometry, the problem of studying fields of definition versus fields of moduli for a Riemann surface S has attracted a lot of recent research. For example, we refer to [1,2,7,9,14,15,16,18,21]. ...
On pseudo-real finite subgroups of PGL3(C)
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Let G be a finite subgroup of PGL⁡3(C)\operatorname{PGL}_3(\mathbb{C}), and let σ\sigma be the generator of Gal⁡(C/R)\operatorname{Gal}(\mathbb{C}/\mathbb{R}). We say that G has a \emph{real field of moduli} if σG^{\sigma}G and G are PGL⁡3(C)\operatorname{PGL}_3(\mathbb{C})-conjugates, that is, if ∃ ϕ∈PGL⁡3(C)\exists\,\phi\in\operatorname{PGL}_3(\mathbb{C}) such that \phi^{-1}\,G\,\phi=\,^{\sigma}G. Furthermore, we say that R\mathbb{R} is \emph{a field of definition for G} or that \emph{G is definable over R\mathbb{R}} if G is PGL⁡3(C)\operatorname{PGL}_3(\mathbb{C})-conjugate to some G′⊂PGL⁡3(R)G'\subset\operatorname{PGL}_3(\mathbb{R}). In this situation, we call G′G' \emph{a model for G over R\mathbb{R}}. If G has R\mathbb{R} as a field of definition but is not definable over R\mathbb{R}, then we call G \emph{pseudo-real}. In this paper, we first show that any finite cyclic subgroup G=Z/nZG=\mathbb{Z}/n\mathbb{Z} in PGL⁡3(C)\operatorname{PGL}_3(\mathbb{C}) has {a real field of moduli} and we provide a necessary and sufficient condition for G=Z/nZG=\mathbb{Z}/n\mathbb{Z} to be definable over R\mathbb{R}; see Theorems 2.1, 2.2, and 2.3. We also prove that any dihedral group D⁡2n\operatorname{D}_{2n} with n≥3n\geq3 in PGL⁡3(C)\operatorname{PGL}_3(\mathbb{C}) is definable over R\mathbb{R}; see Theorem 2.4. Furthermore, we study all six classes of finite primitive subgroups of PGL⁡3(C)\operatorname{PGL}_3(\mathbb{C}), and show that all of them except the icosahedral group A⁡5\operatorname{A}_5 are pseudo-real; see Theorem 2.5, whereas A⁡5\operatorname{A}_5 is definable over R\mathbb{R}. Finally, we explore the connection of these notions in group theory with their analogues in arithmetic geometry; see Theorem 2.6 and Example 2.7.
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... If H is unique in Aut(C), then it is a normal subgroup; so we may consider the reduced group Aut(C) = Aut(C)/H, which is a group of automorphisms of the quotient orbifold C/H. In [53] the following sufficient condition for a curve to definable over its field of moduli was obtained; ...
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