Elliptic Fibrations on the Modular Surface Associated to Γ 1(8)

Abstract

We give all the elliptic fibrations of the K3 surface associated to the modular group Γ 1 (8).

Women in Numbers Europe

“…In particular, we will see that in our case (as in the case described in Bertin and Lecacheux 2013) the classification that we obtain for the elliptic fibrations on a certain K3 surface using the Nishiyama's method is in fact a J 1 -classification (and not only a J 2classification).…”

“…Some years later, Nishiyama (1996) proposed a lattice-theoretic technique to produce such classifications, recovering Oguiso's results and classifying other Kummer and K3 surfaces. Since then, results of the same type have been obtained by various authors (Kumar 2014;Elkies and Schütt 2014;Bertin and Lecacheux 2013).…”

“…We denote by r the rank.MW.E//, and we use Bourbaki notations for A n ; D n ; E k as in Bertin and Lecacheux (2013).…”

“…Nishiyama applied this method in order to obtain a J 2 -classification of the elliptic fibrations, both on the K3 surface already considered in Oguiso (1989) and on other K3 surfaces (cyclic quotients of the product of two special elliptic curves) to which the method by Oguiso cannot be applied. Later, in Bertin and Lecacheux (2013), the method is used to give a J 2 -classification of elliptic fibrations on a K3 surface whose transcendental lattice is h4i˚h2i.…”

“…Since the method is very well described both in the original paper (Nishiyama 1996) and in some other papers where it is applied, e.g. Bertin and Lecacheux (2013) and Braun et al (2013), we summarize it in an algorithm which allows us to compute all the results given in Table 1. In the next section we will describe in detail some peculiar cases, in order to show how this algorithm can be applied.…”