Galois action on the Neron-Severi group of Dwork surfaces

TL;DR

This paper investigates the Galois action on the Néron-Severi group of Dwork surfaces, revealing a decomposition into quadratic characters when the Picard number is 19, and provides two proofs including conductor determination.

Contribution

It offers two proofs of the Galois action decomposition on the Néron-Severi group of Dwork surfaces with Picard number 19, including a geometrically grounded proof that determines conductors.

Findings

Néron-Severi group decomposes into quadratic characters for Picard number 19

Two independent proofs of the Galois action decomposition are provided

The geometric proof determines conductors of the quadratic characters

Abstract

We study the Galois action attached to the Dwrok surfaces $X_{\lambda}:X_0^4+X_1^4+X_2^4+X_3^4-4\lambda X_0X_1X_2X_3=0$ with parameter $\lambda$ in a number field $F$. We show that when $X_{\lambda}$ has geometric Picard number $19$, its N\'eron-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.