On unramified normal coverings of hyperelliptic curves
It is well known that if \(C\) is a compact Riemann surface, then the number of unramified coverings of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2} \times {\mathbb Z}_{2}\) is equal to \((2^{4g}-3\cdot 2^{2g}+2)/6\) (\({\mathbb Z}_{2}\) denotes the cyclic group of order \(2\)). In the paper under review, under the additional assumption that \(C\) is a hyperelliptic surface, the authors determine defining equations for each of these covers and give complete conditions for when any two such surfaces are isomorphic. This extends the results of \textit{R. Horiuchi} [J. Math. Kyoto Univ. 19, 497--523 (1979; Zbl 0427.30038)] where defining equations for those covers which are themselves hyperelliptic are determined. Suppose that \(C\) is a hyperelliptic surface of genus \(g\) with defining equation \[ y^2=\prod_{d=1}^{2g+2} (x-\mu_{d} ) \] and let \(X=\{ 1,2,\dots ,2g+2\}\). Suppose that \(\widetilde{C}\) is some unramified cover of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2} \times {\mathbb Z}_{2}\) generated by \(\alpha\) and \(\beta\) and let \(\pi_{1} \colon \widetilde{C} \rightarrow C_{1}=\widetilde{C} /\langle \alpha \rangle\) and \(\pi_{2} \colon \widetilde{C} \rightarrow C_{2}=\widetilde{C} /\langle \beta \rangle\) be the quotient maps and surfaces. Note that \(C_{1}\) and \(C_{2}\) are both unramified covers of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2}\). Let \(f_{1} \colon C_{1} \rightarrow C\) and \(f_{2} \colon C_{2} \rightarrow C\) denote the corresponding quotient maps. The proof follows from two important results: First, the authors show that \(\widetilde{C}\) is isomorphic to the fibre product \(C_{1} \times C_{2}\) defined by the surfaces \(C_{1}\), \(C_{2}\), and \(C\) and the maps between them, \(\pi_{1}\), \(\pi_{2}\), \(f_{1}\) and \(f_{2}\). Secondly, in previous work by the authors [in: The geometry of Riemann surfaces and abelian varieties. III Iberoamerican congress on geometry in honor of Professor Sevin Recillas-Pishmish's 60th birthday, Salamanca, Spain, June 8--12, 2004. Contemp. Math. 397, 73--77 (2006; Zbl 1101.14044)], it is shown that any unramified cover of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2}\) is isomorphic to a space curve \(C_{X,A}\) determined by the two equations \[ z^2=\prod_{k\in X} (x-\mu_{k}) \] and \[ w^2=\prod_{j\in A} (x-\mu_{j} ) \] where \(A\) runs over all non-empty proper subsets of \(X\). Moreover, any two such curves \(C_{X,A}\) and \(C_{X,B}\) are isomorphic if and only if \(A=B\) or \(A=B^{C}\) (the complement). Thus any such cover \(\widetilde{C}\) is the fibre product of two curves \(C_{X,A}\) and \(C_{X,B}\) with defining equations given above. To derive explicit equations from this information, the authors examine the corresponding maps between these surfaces, each map of which can be described explicitly through the action of \({\mathbb Z}_{2} \times {\mathbb Z}_{2}\) on \(\widetilde{C}\). Determination of the isomorphism classes of these covers is performed via a counting argument. Specifically, the authors first group the covers into families of isomorphic curves by explicitly showing each curve in a given family is isomorphic to every other curve in that family. Following this, they show that the number of such families is equal to \((2^{4g}-3\cdot 2^{2g}+2)/6\), the total number of isomorphism classes of covers of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2} \times {\mathbb Z}_{2}\). It follows that two curves from different families cannot be isomorphic. Two interesting corollaries to these results provide conditions for when an unramified cover of a hyperelliptic surface with group of covering transformations isomorphic to \({\mathbb Z}_{2} \times {\mathbb Z}_{2}\) has a defining equation over \(\bar{\mathbb Q}\). Specifically, if \(C\) has defining equation \(y^2=f(x)\) and \(f(x)\) splits into distinct linear factors in \({\mathbb Q}\), then every such cover \(\widetilde{C}\) has a defining equation over \(\bar{\mathbb Q}\). Alternatively, if \(f(x)\) splits into the product of two polynomials of even degree with coefficients in \({\mathbb Q}\), then there exists some such cover \(\widetilde{C}\) with a defining equation over \(\bar{\mathbb Q}\).
scientific article; zbMATH DE number 5083181
scientific article