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

TL;DR

This paper explores the structure of algebraic Witt vectors over number fields, establishing a modularity theorem that links deformation families of modular functions to algebraic Witt vectors, providing an explicit description for imaginary quadratic fields.

Contribution

It proves a modularity theorem connecting deformation families of modular functions with algebraic Witt vectors, offering an explicit characterization of the lambda-ring for imaginary quadratic fields.

Findings

Algebraic Witt vectors are characterized by deformation families of modular functions.

The lambda-ring $E_K$ is explicitly described as the algebra of modular vectors for imaginary quadratic fields.

The identity $E_K=M_K$ is established, linking Witt vectors and 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 $\xi \in E_K$ 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 $\widehat{f}$.