Exact Order-Lifting Depth and Ramification in Global Function Fields: A Coefficient-Field Rank Formula and Counterexamples to Kernel-Dimension Conjectures

Abstract

This preprint studies order lifting for rational functions on curves over finite fields. It identifies order-lifting depth with the local multiplicity of the associated morphism and relates this invariant to Fermat quotients, Hasse derivatives, and the lifting of multiplicative orders modulo prime powers. For the higher Fermat quotient operators introduced by Jeong and Li, the paper gives an exact description of the relevant kernels using canonical coefficient fields in finite local rings. This leads to an exact rank formula for their dimensions. Explicit infinite families show that the proposed kernel-dimension formulas fail in every prime characteristic. The same local invariant also admits a global interpretation through ramification and logarithmic differentials. For separable maps, the resulting depth divisors satisfy a logarithmic Riemann–Hurwitz formula and corresponding degree bounds, while the inseparable case reduces naturally to the separable core.