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We determine the N\'eron-Severi lattices of $K3$ hypersurfaces with largePicard number in toric three-folds derived from Fano polytopes. Such $K3$surfaces are important objects of research in mirror symmetry. On each $K3$surface, we introduce a particular elliptic fibration which induces aMordell-Weil group of rank $1$. In the proof of the main theorem, we show thatthe N\'eron-Severi lattice of each $K3$ surface is generated by a generalfibre, sections and appropriately selected components of the singular fibres ofour elliptic fibration. Our argument gives a down-to-earth proof of theDolgachev conjecture for Fano polytopes, which is a conjecture on mirrorsymmetry for $K3$ surfaces.

Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes