How does one decide whether the complex number a = exp(2 π √-1 b) is a root of unity, that is fulfills the equation
Kronecker gave an analytic answer: among the complex numbers a which fulfill an equation of the shape a^{n} + c_{1}a^{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} + q_{1}a^{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.