Algebraicity of vector fields over ${\mathbb C}$ and over ${\mathbb F}_p$

We will discuss a conjecture that a vector field over $\mathbb Q$ is algebraic (has algebraic leaves) if and only if it is such after the reduction modulo most primes. This conjecture is closely related to the "$p$-curvature conjecture" of Grothendieck. This is joint work in progress with Dima Leshchiner.