Grothendieck-Katz conjecture

TL;DR

This paper proves that linear differential equations with algebraic solutions exhibit zero m-curvature modulo p^k for almost all primes p, offering a stronger reformulation of the Grothendieck-Katz conjecture.

Contribution

It provides a new formulation of the Grothendieck-Katz conjecture under stronger conditions, linking algebraic solutions to curvature properties modulo prime powers.

Findings

Linear differential equations with algebraic solutions have zero m-curvature modulo p^k for almost all primes p.

The result holds for all k,m in natural numbers with specific divisibility conditions.

The paper offers a reformulation of the Grothendieck-Katz conjecture with enhanced hypotheses.

Abstract

In this article we prove that linear differential equations with only algebraic solutions have zero $m$-curvature modulo $p^k$ for all except a finite number of primes $p$ and all $k,m\in\mathbb N$ with ${\rm ord}_pm!\geq k$. This provides us with a reformulation of Grothendieck-Katz conjecture with stronger hypothesis.