Arithmetic of the [19,1,1,1,1,1] fibration

TL;DR

This paper investigates the arithmetic properties of a specific extremal elliptic K3 surface with a unique configuration of singular fibers, providing explicit models, zeta-function descriptions, and verifying key conjectures.

Contribution

It constructs a model over Q with a Neron-Severi group generated over Q and describes the local zeta-functions via modular forms, advancing understanding of this K3 surface's arithmetic.

Findings

Neron-Severi group generated over Q

Zeta-functions expressed via weight 3 modular form

Verified Tate conjecture at reduction 3

Abstract

This paper studies the arithmetic of the extremal elliptic K3 surface with configuration of singular fibres [19,1,1,1,1,1]. We give a model over Q such that the Neron Severi group is generated by divisors over Q, and we describe the local Hasse-Weil zeta-functions in terms of a modular form of weight 3. Furthermore we verify the Tate conjecture for the reduction at 3 and comment on a conjecture of T. Shioda concerning the similarity of the lattice of transcendental cycles and a lattice resulting from supersingular reduction.