nLab Habiro ring

Idea

Roughly, the Habiro ring is a ring of functions defined on formal neighborhoods of all roots of unity simultaneously, satisfying appropriate gluing conditions.

Definition

The Habiro ring over $\mathbb{Z}$ is defined as an inverse limit:

$$\mathcal{H}_{\mathbb{Z}} \;\simeq\; \lim_n \, \mathbb{Z}[q]/\big( (q;q)_n \big) \,,$$

where $(q;q)_n = \prod_{i=1}^n(1-q^i)$, or equivalently, an inverse limit:

$$\mathcal{H}_{\mathbb{Z}} \;\simeq\; \lim_{n,m} \, \mathbb{Z}[q]/\big((1-q^n)^m\big) \,,$$

where

(1)$$(n_0,m_0) \le (n_1, m_1) \;\;\;\colon\Leftrightarrow\;\;\; m_0\le m_1 \;\wedge\; n_0|n_1 \,.$$

Properties

For any root of unity $\omega$ and $f \in \mathcal{H}_{\mathbb{Z}}$, one defines $f_\omega(x) \coloneqq f(\omega + x) \in \mathbb{Z}[\omega][[x]]$. These satisfy compatibility conditions for varying $\omega$.

Related concepts

• Chern-Simons invariant

• Donaldson-Thomas invariant

References

• Stavros Garoufalidis, Peter Scholze, Campbell Wheeler, Don Zagier: The Habiro ring of a number field [arXiv:2412.04241]