Semi-galois categories II: An arithmetic analogue of Christol's theorem

Abstract

In connection with our previous work on semi-galois categories[1,2], this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series ξ=∑ξntn∈Fq[[t]] over finite field Fq is algebraic over the polynomial ring Fq[t]. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F1-) variant of Christol's theorem in the sense that it replaces the polynomial ring Fq[t] with the ring OK of integers of a number field K and the ring Fq[[t]] of formal power series with the ring of Witt vectors. We also study some related problems.