2020/07/27 by Takeo Uramoto, Uramoto, Takeo · 1 citation
Mathematics · #11R37 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2007.13367
openalex publication_date 2020/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Based on our previous work on an arithmetic analogue of Christol's theorem, this paper studies in more detail the structure of the lambda-ring EK = K ⊗ WOKa (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(\widehatℤ) × \mathfrakH → ℂ of modular functions of finite level always define algebraic Witt vectors \widehatf by their special values, and conversely, every algebraic Witt vector ξ∈ EK is realized in this way, that is, ξ= \widehatf for some deformation family f: M2(\widehatℤ) × \mathfrakH → ℂ. This gives a rather explicit description of the lambda-ring EK for imaginary quadratic fields K, which is stated as the identity EK=MK between the lambda-ring EK and the K-algebra MK of modular vectors \widehatf.