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 $\mathbb{Q}(\sqrt{\epsilon p})$, 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.