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 = ξnt n ∈ 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.1 Some background on Eilenberg theory and its relationship to semi-galois categories will be briefly summarized in §5.2 for the neccessity of our discussion. For more comprehensive texts on Eilenberg theory and its historical aspect, the reader is referred to e.g. [19] and [17]; also, for the axiomatization of Eilenberg theory based on semi-galois categories, see §1, §5 and §7 in [2].

Semi-galois Categories III: Witt vectors by deformations of modular functions

“…This paper is a continuation of our previous work on arithmetic analogue of Christol's theorem [15]. This theorem claims that a (generalized) Witt vector ξ ∈ W OK (O K ) ( §2 [15]) is integral over the ring O K of integers in a number field K if and only if the orbit of ξ under the action of the Frobenius lifts ψ p : W OK (O K ) → W OK (O K ) is finite (cf.…”

“…This section summarizes necessary terminology and results from [15,9,10,12,16]. But in order to avoid duplications, we refer the reader to §2 - §4 [15] for the detailed definitions of basic concepts and results in [15], say, Witt vectors (Definition 1, pp. 543), Λ-rings (Definition 2, pp.…”

“…Theorem 3.4, [15]); we then deduced that, with the aid of the work of Borger and de Smit [3], which heavily relies on class field theory, this is also precisely when the ghost components ξ a of ξ are periodic with respect to some modulus f of K (cf. Corollary 2, [15]). With this background, the major goal of the current paper is then to study in more detail the structure of the Λ-ring E K := K ⊗ W a OK (O K ) (where W a OK (O K ) is the Λ-ring of integral Witt vectors) in the case where K is an imaginary quadratic field; in particular, we prove that the Λ-ring E K coincides, as K-subalgebras of (K ab ) IK , with the K-algebra M K of modular vectors-i.e.…”

“…This theorem claims that a (generalized) Witt vector ξ ∈ W OK (O K ) ( §2 [15]) is integral over the ring O K of integers in a number field K if and only if the orbit of ξ under the action of the Frobenius lifts ψ p : W OK (O K ) → W OK (O K ) is finite (cf. Theorem 3.4, [15]); we then deduced that, with the aid of the work of Borger and de Smit [3], which heavily relies on class field theory, this is also precisely when the ghost components ξ a of ξ are periodic with respect to some modulus f of K (cf. Corollary 2, [15]).…”

“…

Based on our previous work on an arithmetic analogue of Christol's theorem [15], this paper studies in more detail the structure of the Λ-ring EK = K ⊗W a O K (O K ) of algebraic Witt vectors for number fields K. First developing general results concerning EK, we apply them to the case when K is an imaginary quadratic field. The main results include the "modularity theorem" for algebraic Witt vectors, which claims that certain deformation families f : M2( Z) × H → C of modular functions of finite level always define algebraic Witt vectors f by their special values, and conversely, every algebraic Witt vector ξ ∈ EK is realized in this way, that is, ξ = f for some deformation family f : M2( Z) × H → C. This gives a rather explicit description of the Λ-ring EK for imaginary quadratic fields K, which is stated as the identity EK = MK between the Λ-ring EK and the K-algebra MK of modular vectors f .

PreliminariesThis section summarizes necessary terminology and results from [15,9,10,12,16]. But in order to avoid duplications, we refer the reader to §2 - §4 [15] for the detailed definitions of basic concepts and results in [15], say, Witt vectors (Definition 1, pp.

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