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, this paper studies in more detail the structure of the lambda-ring $E_K = K \otimes W_{O_K}^a (O_{\bar{K}})$ of algebraic Witt vectors for number fields $K$. First developing general results concerning $E_K$, 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 $f: M_2(\widehat{\mathbb{Z}}) \times \mathfrak{H} \rightarrow \mathbb{C}$ of modular functions of finite level always define algebraic Witt vectors $\widehat{f}$ by their special values, and conversely, every algebraic Witt vector $ξ\in E_K$ is realized in this way, that is, $ξ= \widehat{f}$ for some deformation family $f: M_2(\widehat{\mathbb{Z}}) \times \mathfrak{H} \rightarrow \mathbb{C}$. This gives a rather explicit description of the lambda-ring $E_K$ for imaginary quadratic fields $K$, which is stated as the identity $E_K=M_K$ between the lambda-ring $E_K$ and the $K$-algebra $M_K$ of modular vectors $\widehat{f}$.