Artin–Schreier varieties with actions of finite orthogonal groups and Weil representations

Abstract

We study Artin–Schreier varieties with natural actions of finite orthogonal groups. We compute the characters of the cohomology groups of these varieties as representations of the groups. Our main ingredient is Shinoda’s character formulae on Weil representations for symplectic groups. We give trace formulae for projective hypersurfaces with actions of finite orthogonal groups generalizing Fermat hypersurfaces.

KEYWORDS:

• Artin–Schreier variety
• finite orthogonal group
• trace formula
• Weil representation

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14F20
• 11F27
• Secondary: 20C33
• 14J50

Acknowledgments

The author would like to thank the reviewer sincerely for reading the previous version of this paper very carefully and giving him many helpful comments and suggestions.

Additional information

Funding

This work was supported by JSPS KAKENHI Grant Numbers 20K03529/21H00973.