On a purely inseparable analogue of the Abhyankar conjecture for affine curves

Let$U$be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck’s étale fundamental group$\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U)$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori’s profinite fundamental group scheme$\unicode[STIX]{x1D70B}^{N}(U)$, and give a partial answer to it.

Congruence kernels around affine curves

Let