Slope stability conjectures for Z_p-towers of curves

Let $X_\infty=(X_n)_{n\ge 0}$ be a geometric $\mathbf Z_p$ -tower of smooth projective curves over a finite field of characteristic $p$ , with base $X_0=\mathbf P^1$ and ramification only at infinity. Does genus stability of the tower, or equivalently the relevant eventual stability of its ramification-break data, imply slope stability of the Newton polygons of the zeta functions $Z(X_n,T)$ ? In particular, under the hypothesis of eventual minimal ramification-break ratios, does slope stability follow without imposing an additional degree-gap condition?

References

Primary source

arXiv

Additional references

Slope Stability for Z_p-Towers of Curves — arXiv — Daqing Wan

Posted 2 days ago · edited 2 days ago

Progress summary

Refreshed 2 days ago

Claimed progress

A recent paper reports counterexamples to two proposed implications, but the broader stability problem remains open.

The conjectures concern when geometric $\mathbb{Z}_p$ -towers of curves exhibit stable Newton-slope patterns. Earlier work established important special cases and conditional implications, but did not settle the general questions.

Known results

Wan’s notes: strong genus stability gives approximate slope stability, and full slope stability under an explicit degree-gap condition.
Kosters–Zhu (2018): for $K=\mathbb{F}_{q}(x)$ and $G_{\infty}=\mathbb{Z}_{p}$ , genus stability implies slope uniformity.
Newton-polygon results give exact or stable slopes under additional ramification and congruence hypotheses.

October 2026 counterexamples and partial stability

Daqing Wan’s arXiv paper reports negative answers to removing the degree-gap hypothesis and to genus stability implying slope stability, while proving exact stability on growing intervals and sharper bounds. This is a claimed advance, not an independently verified resolution of the broad conjectures.

Current status (as of October 2026): Special-case and conditional results are established, while the general slope-stability questions remain open and Wan’s reported counterexamples and partial results are unverified.

Sources

No solutions have been posted yet.