Gross–Stark Units in p-adic Class Field Theory

• Gross–Stark units are canonical p-units in abelian CM-extensions of totally real fields, defined using p-adic L-functions and regulators.
• They generalize classical circular units through analytic, cohomological, and algorithmic methods, directly addressing a p-adic version of Hilbert’s 12th problem.
• Their explicit values allow effective construction of abelian extensions and refined control of class groups, underpinning advances in p-adic Stark theory.

Gross–Stark units are canonical $p$-units in abelian CM-extensions of totally real number fields, constructed as deep arithmetic avatars of $p$-adic special values or derivatives of Artin $L$-functions, and characterized by explicit relations to $p$-adic L-functions, regulators of units, and abelian class field theory. These units generalize the classical circular units and Jacobi sums, crucially implementing “Hilbert’s 12th problem” by providing a $p$-adic and arithmetic construction of abelian extensions via the values of $p$-adic L-functions and modular forms.

1. Foundational Setup and Definition

Let $k$ be a totally real field, $K/k$ a finite abelian CM-extension with Galois group $G=\mathrm{Gal}(K/k)$, and $p$ a rational prime unramified in $k$. Denote by $S_\infty$ the set of archimedean places and by $S_p$ the primes above $p$. The “minus” $S$-unit group is defined as

$$U^- = \{u \in \mathcal{O}_K[1/p]^\times : u^{1+c}=1\},$$

where $c \in G$ is complex conjugation. For a totally odd character $\chi: G \to \overline{\mathbb{Q}}_p^\times$, $U^-_\chi$ is the $\chi$-isotypic component of $U^-$, and $X_\chi$ the corresponding degree-zero subgroup of divisors of $p$-places. In the rank-one case, this space is one-dimensional:

• If $\chi(\mathfrak{p})=1$ for exactly one prime $\mathfrak{p}|p$ and $\chi(\mathfrak{p'}) \neq 1$ for other $\mathfrak{p'}|p$, then $\dim_{\overline{\mathbb{Q}}_p} U^-_\chi = 1$.

A generator $u_p^\chi \in U^-_\chi$, well-defined up to scalars and roots of unity, is called the Gross–Stark unit (Gross et al., 2023).

2. $p$-adic $L$-functions and Regulators

The relevant $p$-adic $L$-function is the Deligne–Ribet–Cassou-Noguès function $L_p(\chi\omega, s)$, where $\omega$ is the $p$-adic cyclotomic character. Under the rank-one hypothesis, $L_p(\chi\omega, 0) = 0$ and $\operatorname{ord}_{s=0} L_p(\chi\omega, s)=1$, so the first derivative $$L'_p(\chi\omega, 0)\in\overline{\mathbb{Q}}_p^\times$$ is canonical.

There are two intrinsic isomorphisms $U^-_\chi\to X_\chi$:

• The “valuation map” $$u \mapsto \sum_{\mathfrak{q}|p} f_{\mathfrak{q}}\cdot \operatorname{ord}_{\mathfrak{q}}(u)\cdot \mathfrak{q}$$.
• The “$p$-adic logarithm” $$u \mapsto \sum_{\mathfrak{q}|p} \log_p N_{K_{\mathfrak{q}}/\mathbb{Q}_p}(u)\cdot \mathfrak{q}$$.

The Gross $p$-adic regulator is

$$R_p(\chi) = \frac{ \log_p N_{K_\mathfrak{q}/\mathbb{Q}_p}(u) }{ \operatorname{ord}_\mathfrak{q}(u) }$$

for any generator $u\in U^-_\chi$ (Gross et al., 2023, Honnor, 2021).

3. Gross’s $p$-adic Stark Conjecture and Unconditional Proofs

Gross’s conjecture (1981) in the rank-one abelian case asserts

$R_p(\chi)\cdot A_p(\chi) = L'_p(\chi\omega, 0),$

where $A_p(\chi)$ is the (signed) complex $L$-value $-L(\chi, 0)$ (up to explicit constants). Normalizing so $\operatorname{ord}_\mathfrak{q}(u_p^\chi)=1$, one obtains

$$\log_p N_{K_\mathfrak{q}/\mathbb{Q}_p}(u_p^\chi) = - \frac{L'_p(\chi\omega, 0)}{L(\chi,0)},$$

or, equivalently,

$$u_p^\chi = \exp_p\left( R_p(\chi)^{-1} L'_p(\chi\omega,0) \right).$$

Dasgupta–Darmon–Pollack first proved this formula under Leopoldt’s conjecture and a nonvanishing hypothesis; Ventullo (2013) removed these, and Kakde–Ventullo (2018) gave a fully unconditional proof in the general rank-one abelian setting (Gross et al., 2023, Ventullo, 2013, Dasgupta et al., 2016).

In higher-rank, for $r > 1$ places above $p$ with $\chi(\mathfrak{p}) = 1$, the derivative $\operatorname{ord}_{s=0}L_p(\chi\omega,s)\ge r$, and the $r$-th order term is conjecturally and provenly governed by the determinant of the $r\times r$ Gross regulator matrix (Gross et al., 2023, Dasgupta et al., 2016).

4. Explicit Analytic and Cohomological Constructions

The explicit formula for Gross–Stark units involves $p$-adic integration over Shintani domains. Dasgupta provided an analytic construction using $p$-adic multiplicative integrals and measures $\mu(\mathfrak{b}, D)$ on local units, which, up to compensators and canonical powers of uniformizers, gives units in $F_\mathfrak{p}^\times$ matching the conjectural Gross–Stark units. This construction is formulated as:

$$u_{\mathfrak{p}, T}(\mathfrak{b}, D) = \epsilon(\mathfrak{b}, D, \pi)\cdot\pi^{\zeta_{R,T}(H/F, \mathfrak{b}, 0)}\cdot \prod_\mathcal{O}^\times x\, d\mu(\mathfrak{b}, D, x)$$

(Honnor, 2021).

Concurrently, the cohomological formula of Dasgupta–Spieß realizes the regulator as a cup-cap pairing in the group cohomology of units, relating Eisenstein cocycles (arising from Shintani zeta values) with local cocycles of $p$-adic logarithms and valuations. The regulator matrix

$$\mathcal{L}_{\mathrm{alg}}(\chi)_{\mathfrak{p},\mathfrak{q}} = - \frac{ \ell_\mathfrak{q}(u_{\mathfrak{p}, \chi}) }{ o_\mathfrak{q}(u_{\mathfrak{p}, \chi}) }$$

is shown to coincide (diagonally, and for $[F:\mathbb{Q}]=3$ in totality) with the analytic regulator (Honnor, 2021). This establishes a bridge between arithmetic and cohomological constructions.

5. Applications in Explicit Class Field Theory

Gross–Stark units realize abelian extensions of totally real fields “p-adically,” paralleling the role of special values of modular functions in complex multiplication for imaginary quadratic fields. The explicit values of the units at Frobenius elements allow for explicit Galois actions, furnishing class field theory extensions—addressing a $p$-adic version of Hilbert’s 12th problem.

The proof of the Brumer–Stark and “integral Gross–Stark” conjectures over $\mathbb{Z}[1/2]$ exploits Gross–Stark units as $p$-adic analogues of the canonical generators of abelian extensions. Furthermore, explicit generation results demonstrate that the maximal abelian extension of any totally real field is achieved by adjoining Gross–Stark units (across conductors) together with suitably chosen totally positive square roots (Gross et al., 2023, Dasgupta et al., 2021).

6. Algorithmic Computation and Examples

Recent advances include effective algorithms for computing $p$-adic logarithms and recovering explicit Gross–Stark units in concrete number fields, using overconvergent modular forms and reduction theory. Explicit tables for narrow Hilbert class fields of real quadratic fields and various primes $p$ showcase the computational tractability for fields of moderate discriminant and class number, relying on q-expansions, Newton polygons, and lattice reductions for unit identification (Damm-Johnsen, 2023).

A classical example is the case $k=\mathbb{Q}$ and $K=\mathbb{Q}(\sqrt{-D})$ with quadratic Dirichlet character $\chi$; then

$$u_p^\chi \sim \frac{\prod_{a\,(\mathrm{mod}\, D), \chi(a)=1} J(a,p)}{ \prod_{a, \chi(a)=-1} J(a,p)}$$

and $p$-adic logarithms of $u$ recover the $L$-derivative $L'_p(\chi\omega, 0)$ up to explicit constants (Gross et al., 2023).

7. Conceptual and Theoretical Significance

Gross–Stark units unify and refine the theory of Stark units, $p$-adic and complex $L$-functions, Shintani zeta values, and the Eisenstein cocycle framework. The proof of Gross’s conjecture in both rank-one and higher-rank cases by Dasgupta, Kakde, and Ventullo provides a cornerstone for $p$-adic Stark theory. Consequences include refined control of Fitting ideals of class groups (in the minus part) and advances toward the Equivariant Tamagawa Number Conjecture.

Moreover, extensions and refinements—such as period-ring-valued enhancements via $p$-adic Hodge theory and generalizations to Beilinson–Flach classes and adjoint Artin representations—indicate ongoing developments forming the “axiomatic exceptional zero theory” landscape in modern Iwasawa theory (Gross et al., 2023, Kashio, 2017, Rivero, 2021).

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References

• (Gross et al., 2023) Two encounters with the $p$-adic Stark conjecture
• (Honnor, 2021) Comparing Two Formulas for the Gross-Stark Units
• (Ventullo, 2013) On the rank one abelian Gross-Stark conjecture
• (Dasgupta et al., 2016) On the Gross-Stark Conjecture
• (Dasgupta et al., 2021) Brumer-Stark Units and Explicit Class Field Theory
• (Damm-Johnsen, 2023) Modular algorithms for Gross-Stark units and Stark-Heegner points
• (Kashio, 2017) On a common refinement of Stark units and Gross-Stark units