In a recent paper, Wei Tao and Guo Xuejun calculate the largest power of the uniformizer that guarantees a congruence between traces of powers of an integral matrix over a p‑adic field.

For a finite extension K of ℚ_p with uniformizer π and residue field 𝔽_q, they prove that for every r ≥ 1 the trace of A^{q^r} and A^{q^{r-1}} differ by a multiple of π^{E_r}, where E_r is the maximum exponent that works for all matrices with a fixed residual similarity class. The authors give explicit formulas for both nilpotent and non‑nilpotent residual classes and, importantly, construct integral lifts that achieve these optimal exponents, confirming the sharpness of the bounds.

However, the study shows that having the same ramification index and residue field is not enough to fix the exponent. Even when K and another unramified extension share e and 𝔽_q, the optimal E_r can vary depending on more delicate arithmetic data. This illustrates hidden subtleties in local field behaviour that were previously unnoticed.

The work expands our understanding of matrix trace congruences, sharpening classical Euler results and highlighting how subtle field variations affect trace divisibility. While the results are theoretically complete for the cases studied, they leave open which additional field invariants control these exponents.