Fields of moduli of some special curves
Let \(C\) denote a smooth projective algebraic curve given by the zeros of the polynomials \(P_1,\dots ,P_s \in {\mathbb L}[x_0,\dots ,x_n]\) where \(\mathbb{L}\) is an algebraic closure of a perfect field \(\mathbb{K}\) of characteristic \(q\neq 2\). If the polynomials \(P_1,\dots ,P_s \) are defined over a subfield \(\mathbb{F}\) of \(\mathbb{L}\), then we call \(\mathbb{F}\) a field of definition of \(C\). For \(\sigma \in \Gamma =\text{Gal} (\mathbb{L} /\mathbb{K})\), the polynomials \(P_1^{\sigma} ,\dots ,P_s^{\sigma}\) define a new smooth projective curve \(C^{\sigma}\) which is, in general, not isomorphic to \(C\). We define the moduli field \(\mathbb{M}\) of \(C\) relative to the extension \(\mathbb{L} /\mathbb{K}\) to be the fixed field of the group \[ U=\left\{ \sigma \in \Gamma =\text{Gal} (\mathbb{L} /\mathbb{K}) \bigg| C \text{ is isomorphic to } C^{\sigma} \text{ over } \mathbb{L} \right\}. \] Clearly the moduli field is a subfield of any field of definition of \(C\). The moduli field however need not be a field of definition. Necessary and sufficient conditions for when the field of moduli is a field of definition were given in [\textit{A. Weil}, Am. J. Math. 78, 509--524 (1956; Zbl 0072.16001)]. A consequence of Weil's conditions is that if the automorphism group \(\text{Aut} (C)\) of \(C\) is trivial, then the moduli field is a field of definition. Unfortunately, when \(\text{Aut} (C)\) is non-trivial, Weil's conditions are often difficult to test. This has motivated an interest in developing alternate conditions for when the moduli field is a field of definition, and this is the primary goal of the authors in the current article. In order to present these conditions, we need the following definitions. The signature of a subgroup \(H\) of \(\text{Aut} (C)\) is defined to be the tuple \((g_0;m_1,\dots ,m_r)\) where the curve \(C/H\) has genus \(g_0\) and the quotient map \(C\rightarrow C/H\) is branched over \(r\) points with branching orders \(m_1,\dots ,m_r\). The subgroup \(H\) is said to be unique up to conjugation if for any other subgroup \(K\) of \(\text{Aut} (C)\) with the same signature as \(H\), \(K\) is conjugate to \(H\). In the article under review, the authors prove that if \(\text{Aut} (C)\) contains a subgroup \(H\) that is unique up to conjugation, the quotient surface \(C/H\) has genus \(0\), and the group \(N_{\text{Aut} (C)} (H)/H\) is neither trivial nor cyclic when the characteristic \(q=0\), nor cyclic of order relatively prime to \(q\) for \(q\neq 0\), then \(C\) can be defined over its field of moduli relative to the extension \(\mathbb{L} /\mathbb{K}\). The method of proof is direct -- the authors show that given these specific conditions, Weil's conditions hold and hence the field of moduli is a field of definition. This result is a generalization of a result developed for hyperelliptic curves [\textit{B. Huggins}, Math. Res. Lett. 14, No. 2, 249--262 (2007; Zbl 1126.14036)] and cyclic \(p\)-gonal curves [\textit{A. Kontogeorgis}, J. Théor. Nombres Bordx. 21, No. 3, 679--692 (2009; Zbl 1201.14020)]. It should be noted that the conditions presented are not necessary -- that is, there exist explicit examples of curves which fail the conditions yet are still defined over their field of moduli.
scientific article; zbMATH DE number 6499128
scientific article