Galois action on the Neron-Severi group of Dwork surfaces

Abstract

We study the Galois action attached to the Dwrok surfaces $X_{\lambda }:X_0^4+X_1^4+X_2^4+X_3^4-4\lambda X_0{X}_1{X}_2{X}_3=0$ with parameter $\lambda$ in a number field $F$. We show that when $X_{\lambda }$ has geometric Picard number $19$, its Néron-Severi group $NS({\overline{X}}_{\lambda })\otimes \mathbb{Q}$ is a direct sum of quadratic characters. We provide two proofs to this conclusion in our article. In particular, the geometrically proof determines the conductor of each of quadratic characters. Our result matches the one in \cite{Voight2}. With this decomposition, we give another proof to a result of Wan.