Minimal Integral Models for Principal Series Weil Characters

TL;DR

This paper proves a conjecture about the minimal ring of definition for principal series Weil characters of SL_2(F_p), showing they can be realized over specific quadratic integer rings and providing explicit models.

Contribution

It establishes the minimal ring of definition for these Weil characters and constructs explicit integral models over quadratic rings, confirming a conjecture of Udo Riese.

Findings

Weil characters are realizable over rings of integers in quadratic fields.

Explicit integral models for Weil characters are constructed.

The Galois action on these models is analyzed.

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 $\mathbb{Q}(\sqrt{\varepsilon p})$, where $\varepsilon =(-1)^{(p-1)/2}$, 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.