Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations
Abstract
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 θ2y−ℓ2E4y/144=0, with gcd(ℓ,6)=1, this gives exactly two annular types at every prime and determines the least s≥1 for which Frobenius under q↦ qps exists. The subfamily ℓ=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.
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BibTeX
@misc{mohajer2026annular,
title = {{Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations}},
author = {Mohammadreza Mohajer and Abdellah Sebbar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21213},
eprint = {2609.21213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21213},
}
RIS
TY - UNPB
TI - Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations
AU - Mohammadreza Mohajer
AU - Abdellah Sebbar
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21213
UR - https://arxiv.org/abs/2609.21213
AB - 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−ℓ^2E_4y/144=0, with gcd(ℓ,6)=1, this gives exactly two annular types at every prime and determines the least s≥1 for which Frobenius under q↦ q^p^s exists. The subfamily ℓ=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.
ER -