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Monodromy representations of $p$-adic differential equations in families

Kiran S. Kedlaya

TL;DR

The paper develops a relative extension of the $p$-adic local monodromy theorem for ordinary differential equations on annuli over mixed-characteristic nonarchimedean fields and then for relative annuli over adic bases. It proves a relative monodromy theorem that does not require $K$ to be discretely valued or to carry a Frobenius structure, enabling a formal relativization via quasicompactness and base change. Three key applications follow: a simpler semistable reduction for overconvergent $F$-isocrystals, a relative version of Berger’s de Rham implies potentially semistable theorem for de Rham local systems, and a multivariate Drinfeld-lemma–type result for fiber products of annuli. These results offer new, robust tools for p-adic cohomology and p-adic Hodge theory, allowing monodromy information to be propagated and analyzed across families. The framework also provides a conceptual bridge between local p-adic differential equations and global geometric phenomena in rigid and nonarchimedean geometry.

Abstract

We derive a relative version of the local monodromy theorem for ordinary differential equations on an annulus over a mixed-characteristic nonarchimedean field, and give several applications in $p$-adic cohomology and $p$-adic Hodge theory. These include a simplified proof of the semistable reduction theorem for overconvergent $F$-isocrystals, a relative version of Berger's theorem that de Rham representations are potentially semistable, and a multivariate version of the local monodromy theorem in the style of Drinfeld's lemma on fundamental groups.

Monodromy representations of $p$-adic differential equations in families

TL;DR

The paper develops a relative extension of the -adic local monodromy theorem for ordinary differential equations on annuli over mixed-characteristic nonarchimedean fields and then for relative annuli over adic bases. It proves a relative monodromy theorem that does not require to be discretely valued or to carry a Frobenius structure, enabling a formal relativization via quasicompactness and base change. Three key applications follow: a simpler semistable reduction for overconvergent -isocrystals, a relative version of Berger’s de Rham implies potentially semistable theorem for de Rham local systems, and a multivariate Drinfeld-lemma–type result for fiber products of annuli. These results offer new, robust tools for p-adic cohomology and p-adic Hodge theory, allowing monodromy information to be propagated and analyzed across families. The framework also provides a conceptual bridge between local p-adic differential equations and global geometric phenomena in rigid and nonarchimedean geometry.

Abstract

We derive a relative version of the local monodromy theorem for ordinary differential equations on an annulus over a mixed-characteristic nonarchimedean field, and give several applications in -adic cohomology and -adic Hodge theory. These include a simplified proof of the semistable reduction theorem for overconvergent -isocrystals, a relative version of Berger's theorem that de Rham representations are potentially semistable, and a multivariate version of the local monodromy theorem in the style of Drinfeld's lemma on fundamental groups.
Paper Structure (17 sections, 49 theorems, 32 equations)

This paper contains 17 sections, 49 theorems, 32 equations.

Table of Contents

  1. Connections on annuli
  2. Preliminaries
  3. Regular connections
  4. Solvable connections
  5. Quasiconstant connections
  6. The p-adic local monodromy theorem
  7. Wild monodromy representations
  8. Relative connections on relative annuli
  9. Relative connections
  10. Decompositions of relative connections
  11. Regular relative connections
  12. Solvable relative connections
  13. The relative monodromy theorem
  14. Applications
  15. Semistable reduction for isocrystals
  16. ...and 2 more sections

Key Result

Theorem 1.1.3

For $i=1,\dots,n$, the function $r \mapsto -\log s_1(\mathcal{E}, e^{-r})- \cdots - \log s_i(\mathcal{E}, e^{-r})$ is continuous, convex, and piecewise affine with slopes in $\frac{1}{n!} \mathbb{Z}$.

Theorems & Definitions (146)

  • Definition 1.1.1
  • Definition 1.1.2
  • Theorem 1.1.3
  • proof
  • Lemma 1.1.4
  • proof
  • Definition 1.2.1
  • Definition 1.2.2
  • Lemma 1.2.3
  • proof
  • ...and 136 more