Gaudin model modulo $p$, Tango structures, and dormant Miura opers

In the present paper, we study the Bethe ansatz equations for Gaudin model and Miura opers in characteristic $p>0$. Our study is based on a work by E. Frenkel, in which solutions to the Bethe ansatz equations are described in terms of Miura opers on the complex projective line. The main result of the present paper provides a positive characteristic analogue of this description. We pay particular attention to the case of Miura $\mathrm{PGL}_2$-opers because dormant generic Miura $\mathrm{PGL}_2$-opers correspond bijectively to Tango structures, which bring various sorts of exotic phenomena in positive characteristic, e.g., counter-examples to the Kodaira vanishing theorem. As a consequence, we construct new examples of Tango structures by means of solutions to the Bethe ansatz equations modulo $p$.