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 \(\widehat{f}\) .
