Arithmetic of the [19, 1, 1, 1, 1, 1] fibration (original) (raw)

This paper investigates the arithmetic properties of a specific elliptic K3-surface with a distinct fibration characterized by six singular fibers and a maximal Néron-Severi group. It establishes results regarding the ranks and structures of corresponding lattices over various primes, particularly focusing on the behavior of the surface under reduction mod primes including 3 and 19. Key findings include verifying Shioda's conjecture on the similarity of lattice structures and detailing the nature of the Mordell-Weil group.