On the Classification of Elliptic Fibrations modulo Isomorphism on K3 Surfaces with large Picard Number

Motivated by a problem originating in string theory, we study elliptic fibrations on K3 surfaces with large Picard number modulo isomorphism. We give methods to determine upper bounds for the number of inequivalent K3 surfaces sharing the same frame lattice. For any given Neron--Severi lattice $S_X$, such a bound on the `multiplicity' can be derived by investigating the quotient of the isometry group of $S_X$ by the automorphism group. The resulting bounds are strongest for large Picard numbers and multiplicities of unity do indeed occur for a number of K3 surfaces with Picard number 20. Under a few extra conditions, a more refined analysis is also possible by explicitly studying the embedding of $S_X$ into the even unimodular lattice ${\rm II}_{1,25}$ and exploiting the detailed structure of the isometry groups of $S_X$ and ${\rm II}_{1,25}$. We illustrate these methods in examples and derive bounds for the number of elliptic fibrations on Kummer surfaces of Picard numbers 17 and 20. As an intermediate step, we also discuss coarser classification schemes and review known results.

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