AI 中文总结
本文系统研究有限域上射影直线 $\mathbf Z_p$-塔的斜率稳定性，证明最小分歧断点假设下的部分斜率稳定性，并构造反例否定亏格稳定性蕴含斜率稳定性，给出不依赖常数域次数的改进界。
AI 中文摘要
本文致力于系统研究特征为 $p$ 的有限域上射影直线在无穷远处仅分歧的 $\mathbf Z_p$-塔的斜率稳定性。我们的目标是理解亏格序列或分歧断点序列的几何稳定性在多大程度上蕴含塔中曲线 zeta 函数的 Newton 斜率的算术稳定性。我们证明了正面的结果，并构造了反例以阐明斜率稳定性与其失效之间的边界，在若干情形下给出了精确的界。对于具有最终最小分歧断点比的塔（这是比亏格稳定性更强的条件），Kosters 和 Zhu 在额外的度差条件下证明了斜率稳定性。我们构造了具有最小断点比但不斜率稳定的塔，从而对其关于该差距是否可去除的问题以及更广泛的亏格稳定性是否蕴含斜率稳定性的问题均给出了否定回答。仅在最小断点假设下，我们证明了在趋近第一个块的递增区间上的精确部分斜率稳定性，并在显式边界环带和有限平坦谱片段上给出了一致比较。进一步的反例区分了部分传播与完全第一块传播、第一块传播与两块传播，以及完全经典稳定性与小于一的均匀环形斜率律。我们改进了经典度差常数、部分稳定性损失和充分起始水平的界，所有这些界均不依赖于常数域的次数。证明使用了有限接触块上半线性 Frobenius 的独立精度定理。
英文摘要
This work is devoted to a systematic study of slope stability for $\mathbf Z_p$-towers of the projective line over finite fields of characteristic $p$, ramified only at infinity. Our aim is to understand to what extent geometric stability of the genus sequence, or of the ramification break sequence, implies arithmetic stability of the Newton slopes of the zeta functions of the curves in the tower. We prove positive results and construct counterexamples that clarify the boundary between slope stability and its failure, with sharp bounds in several cases. For towers with eventual minimal ramification break ratios, a condition stronger than genus stability, Kosters and Zhu proved slope stability under an additional degree-gap condition. We construct towers with minimal break ratios that are not slope stable, answering negatively both their question of whether the gap can be removed and their broader question of whether genus stability implies slope stability. Under the minimal-break hypothesis alone, we prove exact partial slope stability on growing intervals approaching the first block, with uniform comparisons over explicit boundary annuli and finite flat spectral pieces. Further counterexamples distinguish partial from complete first-block propagation, first-block from two-block propagation, and full classical stability from a uniform annular slope law below one. We obtain improved bounds for the classical degree-gap constant, the partial-stability loss, and the sufficient starting levels, all independent of the constant-field degree. The proof uses a separate precision theorem for the semilinear Frobenius on finite contact blocks.
Comments150 pages
