From Frobenius Structures to Differential Equations

Frobenius structures are omnipresent in arithmetic geometry. In this note we show that over suitable rings, Frobenius endomorphisms define differential structures and vice versa. This includes, for example, differential rings in positive characteristic and complete non-archimedean differential rings in characteristic zero. Further, in the global case, the existence of sufficiently many Frobenius rings is related to algebraicity properties. These results apply, for example, to t-motives as well as to p-adic and arithmetic differential equations