Nearby slopes and boundedness for $\mathrm{\ell }$-adic sheaves in positive characteristic

The goal of this paper is to motivate a boundedness conjecture on nearby slopes of $\mathrm{\ell }$-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible $\mathrm{\ell }$-adic sheaf, we prove the finiteness of the set of nearby slopes associated to a given morphism. For the constant sheaf, we prove the vanishing of nearby slopes in case of generalized semi-stable reduction.