The elliptic modular surface of level 4 and its reduction modulo 3. (English) Zbl 1448.14036
Elliptic modular surfaces are a class of surfaces displaying an intriguing behaviour both from the geometrical and the arithmetical point of view [T. Shioda, J. Math. Soc. Japan 24, 20–59 (1972; Zbl 0226.14013)].
The present paper is concerned with the automorphism group of the elliptic modular surface of level \(4\), which is a K3 surface \(X_0\), and its reduction modulo \(3\) \(X_3\), which is known to be isomorphic to the Fermat quartic surface by work of T. Shioda [Manifolds, Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 357–364 (1975, Zbl 0311.14007)].
Both \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) have already been calculated by J. Keum and S. Kondō [Trans. Am. Math. Soc. 353, No. 4, 1469–1487 (2001; Zbl 0968.14022); S. Kondō and I. Shimada, Int. Math. Res. Not. 2014, No. 7, 1885–1924 (2014; Zbl 1343.14034)], but in this paper the results are reinterpreted in order to show the relations between them. The faithful representation of \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) on the Néron-Severi lattices \(S_0\) and \(S_3\) allows to identify \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) with subgroups of \(\mathrm{O}^+(S_0)\) and \(\mathrm{O}^+(S_3)\), respectively. Moreover, there exists a primitive embedding \(\rho\colon S_0 \rightarrow S_3\) induced by the specialization of \(X_0\) to \(X_3\). The author provides two descriptions of \(\rho\): a combinatorial description in terms of the Petersen graph and a geometrical description starting from a certain Weierstraß equation.
The positive cone of the even unimodular hyperbolic lattice \(L_{26}\) of rank \(26\) has a well-known decomposition into Conway chambers. Borcherds’ method consists in embedding the Néron-Severi lattice \(S_0\) or \(S_3\) into the even unimodular hyperbolic lattice \(L_{26}\) of rank \(26\) in order to obtain an induced chamber decomposition on the positive cone of the respective surface. The main result of the paper is the identification of the subgroup \(O^+(S_3,S_0) \cap \mathrm{Aut}(X_3)\) with an explicit subgroup of \(\mathrm{Aut}(X_0)\) using Borcherds’ method.
The author is then interested in particular automorphisms of \(X_0\) and \(X_3\), namely Enriques involutions (that is, without fixed points). Up to conjugation there are 9 different ones, but the author concentrates on Enriques involutions inducing an étale double cover \(X \rightarrow Y\) onto Enriques surfaces \(Y\) with \(|{\mathrm{Aut}(Y)}| = 320\) (type \(\mathrm{IV}\) in Nikulin-Kondō-Martin’s classification of Enriques surfaces with finite automorphism group). The explicit description of \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) allows the author to identify Enriques involutions of type \(\mathrm{IV}\) on \(X_0\) with Enriques involutions of type \(\mathrm{IV}\) on \(X_3\) and to interpret the pullbacks of the 20 rational curves on the Enriques quotient as lines on the Fermat quartic surface.
Both \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) have already been calculated by J. Keum and S. Kondō [Trans. Am. Math. Soc. 353, No. 4, 1469–1487 (2001; Zbl 0968.14022); S. Kondō and I. Shimada, Int. Math. Res. Not. 2014, No. 7, 1885–1924 (2014; Zbl 1343.14034)], but in this paper the results are reinterpreted in order to show the relations between them. The faithful representation of \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) on the Néron-Severi lattices \(S_0\) and \(S_3\) allows to identify \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) with subgroups of \(\mathrm{O}^+(S_0)\) and \(\mathrm{O}^+(S_3)\), respectively. Moreover, there exists a primitive embedding \(\rho\colon S_0 \rightarrow S_3\) induced by the specialization of \(X_0\) to \(X_3\). The author provides two descriptions of \(\rho\): a combinatorial description in terms of the Petersen graph and a geometrical description starting from a certain Weierstraß equation.
The positive cone of the even unimodular hyperbolic lattice \(L_{26}\) of rank \(26\) has a well-known decomposition into Conway chambers. Borcherds’ method consists in embedding the Néron-Severi lattice \(S_0\) or \(S_3\) into the even unimodular hyperbolic lattice \(L_{26}\) of rank \(26\) in order to obtain an induced chamber decomposition on the positive cone of the respective surface. The main result of the paper is the identification of the subgroup \(O^+(S_3,S_0) \cap \mathrm{Aut}(X_3)\) with an explicit subgroup of \(\mathrm{Aut}(X_0)\) using Borcherds’ method.
The author is then interested in particular automorphisms of \(X_0\) and \(X_3\), namely Enriques involutions (that is, without fixed points). Up to conjugation there are 9 different ones, but the author concentrates on Enriques involutions inducing an étale double cover \(X \rightarrow Y\) onto Enriques surfaces \(Y\) with \(|{\mathrm{Aut}(Y)}| = 320\) (type \(\mathrm{IV}\) in Nikulin-Kondō-Martin’s classification of Enriques surfaces with finite automorphism group). The explicit description of \(\mathrm{Aut}(X_0)\) and \(\mathrm{Aut}(X_3)\) allows the author to identify Enriques involutions of type \(\mathrm{IV}\) on \(X_0\) with Enriques involutions of type \(\mathrm{IV}\) on \(X_3\) and to interpret the pullbacks of the 20 rational curves on the Enriques quotient as lines on the Fermat quartic surface.
Reviewer: Davide Cesare Veniani (Stuttgart)
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GAPReferences:
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