The Degree of the Dormant Operatic Locus

Abstract

Let $X$ be a smooth, projective curve of genus $g\geq 2$ over an algebraically closed field of characteristic $p>0$. I provide a conjectural formula for the degree of the scheme of dormant ${\rm PGL}(r)$-opers on $X$ where $r\geq 2$ (I assume that $p$ is greater than an explicit constant depending on $g,r$). For $r=2$ a dormant ${\rm PGL}(2)$-oper is a dormant indigenous bundle on $X$ in the sense of Shinichi Mochuzki (and his work provides a formula only for $g=2,r=2,p\geq 5$, from a different point of view). Recently Yasuhiro Wakabayashi has shown that my conjectural formula holds for $X$ generic, $r\geq 2$ and $p$ large.