  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 $\xi \in E_K$ is realized in this way, that is,
$\xi = \widehat{f}$ 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 $\widehat{f}$.

Number Theory;Category Theory