How does one decide whether the complex number a = exp(2 π √-1 b) is a root of unity, that is fulfills the equation

aⁿ = 1
for some natural number n? A very elementary answer is to say that it is equivalent to
b ∈ ℚ,
that is b is a rational number. Starting from it, we can ask separately what makes a to be a root of unity, or b to be a rational number.

Kronecker gave an analytic answer: among the complex numbers a which fulfill an equation of the shape aⁿ + c₁a^n-1 + ... + c_n = 0 for some integers c_i ∈ ℤ, the roots of unity are those for which all the complex solutions have absolute value 1. He also gave an arithmetic answer: among the complex numbers b which fulfill an equation of the shape b^m + q₁a^n-1 + ... + q_m = 0 for some rational numbers q_i ∈ ℚ, the rational numbers are those for which for all prime numbers p not dividing the denominators of the q_i, the mod p reduction of b fulfills (b mod p)^p = (b mod p). In the lecture, we present a classical analog of those two theorems for linear differential equations. The analogy relates b with a linear differential equation and a with its monodromy. The characterization of the solutions being algebraic is the content of Grothendieck's famous p-curvature conjecture. We shall give a panorama of the mathematics hidden behind the conjecture.