The cyclosyntomic regulator of a number field

1. Problem Statement and Motivation

The paper addresses the need for a $q$-deformation of the $p$-adic regulator of a number field that allows for interpolation between different prime numbers $p$.

• Context: The classical $p$-adic regulator (Leopoldt) maps units of a number field to a product of $p$-adic logarithms. Its $p$-adic analogue in the context of Beilinson conjectures is realized via syntomic cohomology (Fontaine-Messing, Bhatt-Morrow-Scholze).
• Limitation: Standard syntomic cohomology is defined relative to a specific prime $p$ (using $p$-adic completions and the $p$-prism). It does not naturally allow for a unified framework where one can interpolate between different primes or view the regulator as a global object independent of a specific $p$-completion.
• Goal: Construct a "decompleted" refinement of syntomic cohomology, termed cyclosyntomic cohomology, which lives over the integers $\mathbb{Z}$ (or $\mathbb{Z}[q]$) rather than $\mathbb{Z}_p$. This new theory should recover the classical $p$-adic regulator upon $p$-adic realization and provide explicit formulas for units, particularly cyclotomic units.

2. Methodology and Key Ingredients

The authors build their construction on several advanced tools from arithmetic geometry and algebraic topology:

A. The Habiro Ring and Cyclotomic Rings

• Habiro Rings ($H_R$): Utilizing the recent work of Garoufalidis–Scholze–Wheeler–Zagier, the authors use the Habiro ring, which can be viewed as a ring of "analytic functions in one variable $q$ at roots of unity." It is defined as a limit of completions of $\mathbb{Z}[q]$ at cyclotomic polynomials $\Phi_e(q)$.
• Cyclotomic Rings ($C_R$): The authors introduce cyclotomic rings as a reduced version of Habiro rings, defined via the theory of big $q$-Witt vectors (developed by Wagner). Specifically, $C_R \cong \lim_{m} q\text{-}W_m(R)$ along Frobenius maps. These rings serve as the coefficient rings for the new cohomology theory.

B. Refinement of Norm Maps

• Cyclotomic Norms: A central technical innovation is the construction of cyclotomic norm maps $\Pi_{m'/m}$ on big $q$-Witt vectors. These maps refine Sulyma's norm maps in prismatic cohomology.
• Function: These norms interpolate between classical power maps ($x \mapsto x^d$) and Frobenius maps at various primes. They are constructed using the universal property of $q$-Witt vectors and $q$-ghost coordinates.

C. The Cyclotomic Logarithm

• The authors define a cyclotomic logarithm ($d\log_{cyc}$), which is a global analogue of the prismatic logarithm ($d\log_\Delta$) introduced by Bhatt–Lurie and refined by Mao.
• Construction: It is defined as a map from the multiplicative group $G_m(R)$ to a "Nygaard twist" of the cyclotomic ring. The construction relies on lifting the cyclotomic norms to the Habiro ring level (using "lifted cyclotomic norms") to define a homotopy that makes the logarithm compatible across different truncation levels $m$.

D. Cyclosyntomic Cohomology

• Definition: The cyclosyntomic complex $R\Gamma_{CycSyn}(R, \mathbb{Z}(1)(d))$ is defined as the fiber of a map between two complexes:
  $$\text{can} - \text{Frob}^{cyc}_d: N_{\geq 1}C_R\{1\} \to C_R^{(d)}\{1\}$$
  Here, $N_{\geq 1}C_R\{1\}$ is the Nygaard twist (analogous to the Nygaard filtration in prismatic cohomology), and $C_R^{(d)}\{1\}$ is a cyclotomic twist. The map involves the canonical map and the $d$-th cyclotomic Frobenius.
• Relation to Prismatic Cohomology: For a prime $p$, the $p$-adic realization of this complex recovers the syntomic cohomology relative to the $q$-prism $(\mathbb{Z}_p[[q-1]], [p]_q)$.

3. Key Contributions and Results

A. The Cyclosyntomic First Chern Class (Theorem A)

The authors construct a natural cyclosyntomic first Chern class:
$$c_1^{CycSyn}: G_m(R)[-1] \to R\Gamma_{CycSyn}(R, \mathbb{Z}(1)(d))$$

• This map recovers the standard syntomic first Chern class upon $p$-adic realization when $d=p$.
• It provides a global, $q$-deformed version of the regulator map that is not $p$-complete.

B. Explicit Formula for Units (Theorem B)

For any unit $u \in R^\times$, the authors provide an explicit formula for the image of $u$ under the cyclosyntomic Chern class:
$$c_1^{CycSyn}(u) = \frac{1}{d} \log \left( \frac{\tilde{\Pi}_d(\tilde{u})}{\text{Frob}^{Hab}_d(\tilde{u})} \right)$$

• Here, $\tilde{\Pi}_d$ is the lifted cyclotomic norm (a $q$-deformation of the $d$-th power map).
• $\text{Frob}^{Hab}_d$ is the Habiro Frobenius.
• $\tilde{u}$ represents a sequence of lifts of $u$ into the Habiro ring.
• This formula generalizes Kato's fundamental expression for the $p$-adic Chern class ($u \mapsto \frac{1}{p} \log(u^p / \text{Frob}_p(u))$).

C. Connection to $q$-Polylogarithms (Theorem C)

The most significant computational result concerns cyclotomic units (elements of the form $1-\zeta$ where $\zeta$ is a root of unity).

• The authors prove that the cyclosyntomic Chern class of $1-\zeta$ is given by the first $q$-polylogarithm:
  $$c_1^{CycSyn}(1-\zeta) = -Li_1^{(d)}([\zeta])_q$$
• The first $q$-polylogarithm is defined as the series:
  $$Li_1^{(d)}(z)_q = \sum_{k \geq 1, d \nmid k} \frac{z^k}{[k]_q}$$
  where $[k]_q = 1 + q + \dots + q^{k-1}$.
• This result establishes a direct link between the cohomological regulator and $q$-deformed special functions, showing that cyclotomic units naturally map to $q$-polylogarithm classes in the cyclosyntomic cohomology.

4. Significance and Implications

1. Interpolation of Primes: The theory provides a framework to study regulators across different primes simultaneously, moving beyond the rigid $p$-adic setting. This is a step toward understanding the "global" nature of $p$-adic L-functions.
2. New Cohomological Theory: The introduction of cyclosyntomic cohomology extends the scope of prismatic cohomology. It suggests that the "decompleted" world of Habiro rings and cyclotomic rings is the natural home for certain arithmetic invariants that are usually only visible after $p$-completion.
3. Polylogarithms and K-Theory: The explicit computation for cyclotomic units connects the regulator map to $q$-polylogarithms. This supports the broader conjecture (related to the Zagier conjecture) that higher K-groups of number fields are related to higher polylogarithms. The authors suggest this work is a precursor to a theory of higher Chern classes and higher $q$-polylogarithms.
4. Technical Innovation: The refinement of norm maps in $q$-Witt vectors and the construction of the cyclotomic logarithm via Habiro lifts represent significant technical advances in the theory of prismatic and syntomic cohomology.

In summary, Bouis and Gazda successfully construct a $q$-deformed regulator that unifies $p$-adic phenomena across primes, explicitly links cyclotomic units to $q$-polylogarithms, and lays the groundwork for a deeper understanding of the arithmetic of number fields through the lens of Habiro rings and prismatic geometry.