The $p$-curvature conjecture and monodromy about simple closed loops

TL;DR

This paper proves that for a generic curve, every simple closed loop has finite monodromy, supporting the $p$-curvature conjecture which links vanishing $p$-curvature to finite monodromy.

Contribution

It establishes that on a generic curve, all simple closed loops exhibit finite monodromy, advancing understanding of the $p$-curvature conjecture in this setting.

Findings

Simple closed loops on generic curves have finite monodromy.

Supports the $p$-curvature conjecture relating $p$-curvature and monodromy.

Advances the understanding of differential equations on algebraic varieties.

Abstract

The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes modulo $p$, for almost all primes $p$. We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy.