The separable-extension conjecture for characteristic-p L-series

Let $L(s)$, with $s=(x,y)\in S_\infty$, be a characteristic-$p$ $L$-series arising from arithmetic, and let $\mathbf K_L$ be the finite extension of $\mathbf K_{\mathbf V}$ generated by its coefficients. Let $\mathbf K_L(y)$ be obtained from $\mathbf K_L$ by adjoining the roots of $L(x,y)$ for each $y$. The separable-extension conjecture. The maximal sub-extension of $\mathbf K_L(y)$ separable over $\mathbf K_L$ is finite over $\mathbf K$ for all $y\in\mathbb Z_p$. The analogous maximal separable subfields obtained by adjoining the $v$-adic zeroes should also be finite for each $s_v\in S_v$.

This is presented as a function-field analogue of the classical generalized Riemann hypothesis, translating a statement about zeroes into finiteness of the separable field generated by them. The paper states that the implications of the function-field conjectures were not yet known.