Minimal integral models for principal series Weil characters

Abstract

We prove a conjecture of Udo Riese about the minimal ring of definition for principal series Weil characters of ${\text{SL}}_2({\mathbb{F}}_p)$, for p an odd prime. More precisely, we show that the $(p+1)/2$-dimensional Weil characters can be realized over the ring of integers of $\mathbb{Q}\left(\sqrt{\epsilon p}\right)$, where $\epsilon ={(-1)}^{(p-1)/2}$, and we provide explicit integral models over these quadratic rings. We do so by studying the Galois action on the integral models of Weil characters recently discovered by Yilong Wang.

Keywords:

• Conjecture
• integral models
• minimal ring
• Weil characters

2020 Mathematics Subject Classification:

• 11F27
• 20C10
• 57R56

Acknowledgments

We would like to thank Siu-Hung Ng, Yilong Wang and Shaul Zemel for sharing many great ideas and conversations about this problem, as well as the referee for many insightful comments.

Additional information

Funding

The first author is supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award Number DE-SC-SC0022134. The second author is supported by a 2021-2022 Thomas C. Rumble University Graduate Fellowship through the Department of Mathematics at Wayne State University.