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

Abstract

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. 543), Λ-rings (Definition 2, pp. 544), integral models of finite etale Λ-rings over K (Remark 4, pp. 544) and our arithmetic analogue of Christol's theorem (Theorem 3.4, pp. 557) in particular. Also, for the categorical concepts and results on semi-galois categories, we refer the reader to [14].