Modular Zhu algebra theory and Virasoro vertex algebras

We develop the representation theory of vertex algebras over arbitrary rings using higher Zhu algebras and mode transition algebras. Among our results, we give several equivalent conditions for rationality of M\"obius vertex algebras over a field of positive characteristic, generalizing the work of Damiolini, Gibney, and Krashen. As an application of these results, we prove that, for any field $\mathbb{F}$ of characteristic 0 and coprime integers $r,s>1$, the discrete series Virasoro vertex operator algebra $L_{\operatorname{Vir}}(c_{r,s},0)_{\mathbb{F}}$ is rational. The proof involves using a certain integral form for $L_{\operatorname{Vir}}(c_{r,s},0)_{\mathbb{F}}$ to calculate the Zhu algebra over $\mathbb{F}$. If $\operatorname{char}\mathbb{F}=p>2$, then we show that for $p=3,5$ the simple quotient $L_{\operatorname{Vir}}(c,0)_{\mathbb{F}}$ is holomorphic for all $c\in \mathbb{F}_p$. We show the same result for $p=7$ and $c\in \mathbb{F}_p\setminus \{3,6\}$.