p-adic Stieltjes--Schwarzian 方程的环形 Frobenius 分类

中文摘要 AI 辅助

我们给出了由发展微分定义的 $p$-adic Schwarzian 方程的显式环形正规形，并对其 Frobenius 结构进行了分类。对于 Robba 环的一个单位，降阶将相关模表示为由留数确定的单幂等扩张的二次秩一扭转。Frobenius 的存在性由平方类条件刻画。在奇素数处，恰好有四种微分模类型，一个一致的支配单项式假设给出了在零留数轨迹上的解析族中的 Frobenius 公式。对于有理非整数环形幂，我们在奇素数处获得了一个对角正规形和一个同余判据；当单位因子是显式平方时，这两个结果在每个素数处都成立。应用于可约模方程 $\theta^2y-\ell^2E_4y/144=0$（其中 $\gcd(\ell,6)=1$），这在每个素数处恰好给出两种环形类型，并确定了在 $q\mapsto q^{p^s}$ 下 Frobenius 存在的最小 $s\geq1$。子族 $\ell=12n+1$ 具有单一环形类型。Heine--Stieltjes 留数消去和非退化 Jacobi 构型提供了显式例子。我们还证明了 Robba 环发展映射的常射影 Frobenius 等变性的一个障碍。

英文摘要

We give explicit annular normal forms and classify Frobenius structures for $p$-adic Schwarzian equations defined by developing differentials. For a unit of the Robba ring, reduction of order expresses the associated module as a quadratic rank-one twist of a unipotent extension determined by the residue. Frobenius existence is characterized by a square-class condition. At odd primes, there are exactly four differential-module types, and a uniform dominant-monomial hypothesis gives a Frobenius formula in analytic families across the zero-residue locus. For rational nonintegral annular powers, we obtain a diagonal normal form and a congruence criterion at odd primes; both results hold at every prime when the unit factor is an explicit square. Applied to the reducible modular equations $θ^2y-\ell^2E_4y/144=0$, with $\gcd(\ell,6)=1$, this gives exactly two annular types at every prime and determines the least $s\geq1$ for which Frobenius under $q\mapsto q^{p^s}$ exists. The subfamily $\ell=12n+1$ has a single annular type. Heine--Stieltjes residue cancellation and nondegenerate Jacobi configurations provide explicit examples. We also prove an obstruction to constant projective Frobenius equivariance of Robba-ring developing maps.

发表机构

• Cape Breton University（布雷顿角大学）
• University of Ottawa（渥太华大学）

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