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Automorphisms of certain Niemeier lattices and elliptic fibrations. (English) Zbl 1382.14010

Summary: K.-i. Nishiyama [Jpn. J. Math., New Ser. 22, No. 2, 293–347 (1996; Zbl 0889.14015)] introduced a lattice theoretic classification of the elliptic fibrations on a \(K3\) surface. In a previous paper we used his method to exhibit 52 elliptic fibrations, up to isomorphisms, of the singular \(K3\) surface of discriminant \(-12\). We prove here that the list is complete with a 53th fibration, thanks to a remark of N. Elkies [private communication] and M. Schütt and T. Shioda [Adv. Stud. Pure Math. 60, 51–160 (2010; Zbl 1216.14036)]. We characterize the fibration both theoretically and with a Weierstrass model.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
11F23 Relations with algebraic geometry and topology
11G05 Elliptic curves over global fields
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