Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$

Abstract: Starting from a topological treatment of the Eisenstein class of a torus bundle, we define log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$. These are group cohomology classes for $\mathrm{SL}_n(\mathbb{Z})$ valued on log-rigid analytic functions on Drinfeld's $p$-adic symmetric domain. Such classes can be evaluated at points attached to totally real fields of degree $n$ where $p$ is inert. We conjecture that these values are $p$-adic logarithms of Gross--Stark units in the narrow Hilbert class field of totally real fields. We provide evidence for the conjecture by comparing our constructions to $p$-adic $L$-functions. In addition, we prove it in certain situations where the totally real field is Galois over $\mathbb{Q}$, as a consequence of the fact that in this case there is a conjugate of a Gross--Stark unit in $\mathbb{Q}_p$.

• The paper develops a cohomological construction of Eisenstein classes from torus bundles, generalizing modular units for higher-rank SLₙ(ℤ).
• It employs analytic techniques like Mathai-Quillen forms and Poisson kernels to represent and lift these classes into the log-rigid analytic setting.
• Results connect the analytic classes to Gross-Stark units, offering a modular framework for explicit class field theory over totally real fields.

Eisenstein Classes, Torus Bundles, and Log-Rigid Analytic Classes for $\mathrm{SL}_n(\mathbb{Z})$

Introduction and Context

The paper "Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$" (2512.11514) by Martí Roset and Peter Xu addresses the construction of cohomology classes relevant to the arithmetic of totally real fields, particularly in the context of the Gross-Stark conjecture and $p$-adic interpolation of special values of $L$-functions. The work sits at the confluence of several pivotal themes in arithmetic geometry and the algebraic theory of automorphic forms: Eisenstein cohomology, the explicit class field theory of abelian extensions of totally real fields, rigid analytic geometry, and regulators for $p$-adic $L$-values.

Eisenstein Classes and Torus Bundles

The authors initiate their construction from a topological description of the Eisenstein class of a torus bundle. This arises from viewing locally symmetric spaces attached to $\mathrm{SL}_n(\mathbb{R})$ and considering quotient torus bundles, generalizing the classical framework where Siegel units and modular units for $\mathrm{SL}_2(\mathbb{Z})$ play a central role. The Eisenstein class is defined by a canonical lift in relative singular cohomology, characterized by two properties: being a lift of a class supported on torsion points (i.e., fibers at $c$-torsion) and invariance under pushforwards by specific endomorphisms.

The framework developed by Bergeron, Charollois, and García [BCG20] is central, and the authors leverage the associated Thom isomorphisms for torus bundles to define classes in both singular and group cohomology. This construction underlies the extension from the case $n=2$ (modular curves) to general $n$, bringing higher-rank locally symmetric spaces and their toric fibrations into consideration.

Differential Form Representatives

Refining topological definitions, the authors employ Mathai-Quillen forms and their regularized averages to obtain explicit differential forms representing the Eisenstein class. Here, analytic techniques such as the construction of closed equivariant forms on vector bundles and averaging over lattices are used to bridge the gap between discrete cohomology and analytic invariants. This technical machinery is critical for connecting the topological Eisenstein class with periods of $L$-functions.

Group Cohomology and Distribution-valued Classes

The topological Eisenstein class is transferred into group cohomology, producing classes for the integral congruence subgroup $\mathrm{SL}_n(\mathbb{Z})$ valued in spaces of (total mass zero) measures on the set $X = \mathbb{Z}^n - p\mathbb{Z}^n$. Shapiro’s lemma, co-induction, and corestriction are systematically exploited to organize pullbacks of Eisenstein classes by torsion sections into a compatible system, producing what can be regarded as a higher-rank Eisenstein cocycle.

A significant technical achievement is the existence and uniqueness (up to torsion) of a lift of this class to the total mass zero distribution module, ensuring well-posedness of the ensuing analytic constructions.

Drinfeld’s Symmetric Domain and Log-Rigid Analytic Classes

The bridge to $p$-adic analytic geometry is constructed by considering Drinfeld's $p$-adic symmetric domain $\mathfrak X_p$ of dimension $n-1$—a natural generalization of the $p$-adic upper half-plane. The key analytic tool is the construction of a $p$-adic Poisson kernel, providing an equivariant morphism from measures on $X$ (of total mass zero) to a space $A_c$ of log-rigid analytic functions on $\mathfrak X_p$. In this setup, log-rigid analytic functions admit formulae as sums of rigid analytic parts and finitely supported $p$-adic logarithms of ratios of hyperplane coordinates.

Applying this machinery, the Eisenstein group cohomology class yields a "log-rigid analytic class"

$\mathrm{JE, c} \in H^{n - 1}(\mathcal T, A_c),$

where $\mathcal T$ denotes $\mathrm{SL}_n(\mathbb{Z})$ or an associated congruence subgroup.

Arithmetic Applications: Values at CM Points and the Gross-Stark Conjecture

A central theme is the evaluation of these log-rigid analytic classes at points in $\mathfrak X_p$ attached, via adelic data, to totally real fields $F$ of degree $n$ in which $p$ is inert. The stabilization property of these points mimics the role of CM points on modular curves but in higher rank and in the $p$-adic analytic setting.

At these special points, the authors establish, up to local trace, that the values of the log-rigid class coincide with the logarithmic derivatives at zero of $p$-adic $L$-functions (partial zeta functions with Euler factor at $p$ removed), which, according to the Gross-Stark conjecture (rank one, now a theorem [DDP11] [Ven15]), compute the $p$-adic logarithm of Gross-Stark units in the narrow Hilbert class field $H$ of $F$.

Key numerical result: For $n \geq 2$,

$$\mathrm{Tr}_{F_p/\mathbb{Q}_p} \mathrm{JE,c}[T] = \mathrm{Tr}_{F_p/\mathbb{Q}_p} \log_p(u^{\sigma_a})$$

where $u$ is a Gross-Stark unit, and $\sigma_a$ the Frobenius at an ideal associated to the data defining the point $T$.

Moreover, in the case when $F/\mathbb{Q}$ is Galois and the ideal is Galois-stable, equality (without the trace) is proven, thus producing explicit Gross-Stark units in this setting.

Theoretical and Practical Implications

From a theoretical perspective, the construction provides a purely cohomological and analytic recipe for Gross-Stark units, lifting major parts of the explicit class field theory program for totally real fields into the field of locally symmetric spaces and their cohomology. The theory delivers a modular/automorphic description of the units and their $p$-adic regulators, akin to the role played by CM theory, modular units, and complex multiplication in the arithmetic of imaginary quadratic fields.

On the practical side, these constructions—especially in the Galois case, where the units themselves and not just their $p$-adic logarithms are recovered—promise more explicit computational frameworks for generating abelian extensions of totally real fields, echoing the aims of Hilbert's twelfth problem.

Comparison with Existing Literature

The constructions generalize both the topological framework of Eisenstein cocycles for $\mathrm{GL}_n$ (Sczech, Charollois-Dasgupta-Greenberg-Spiess, Bannai et al.) and recent work on $p$-adic and rigid analytic modular cocycles ($n=2$, Darmon, Pozzi, Vonk). By organizing Eisenstein class pullbacks in compatible group cohomology systems and lifting them to analytic classes on $\mathfrak X_p$, the paper systematically extends the motivic and topological content of earlier constructions into the $p$-adic, analytic, and automorphic regimes relevant for totally real fields of arbitrary degree.

A noteworthy aspect is the use of symbol complexes, Poisson kernels, and regularized period integrals, which allow both explicit formulae and conceptual clarity in connecting cohomology and special values.

Speculations on Future Directions

Should the principal conjectures be resolved in general, one may expect a complete automorphic description of modular/gross-Stark units for arbitrary totally real fields, with extensions to ray class fields and higher-rank settings. The framework may further interface with $p$-adic Hodge theory, $K$-theory, and non-abelian generalizations (e.g., via rigid/meromorphic cocycles for orthogonal and other groups).

Computational advances in the calculation of Eisenstein cocycles and explicit $p$-adic periods would directly enable the tabulation and arithmetic use of higher-rank Stark units, paralleling the concrete realization of elliptic and Stark–Heegner points and their explicit algorithms in the quadratic setting.

Conclusion

This work provides a comprehensive construction of log-rigid analytic group cohomology classes for $\mathrm{SL}_n(\mathbb{Z})$ valued in $p$-adic analytic functions, encapsulating and extending the arithmetic of Gross-Stark units via cohomological and analytic means. The approach unifies the topological, analytic, and arithmetic perspectives and achieves strong results regarding the explicit realization and $p$-adic evaluation of these units under the inert prime and Galois conditions. The broader impact lies in offering a robust, higher-rank, automorphic framework for explicit class field theory over totally real fields, with clear trajectories for both theoretical and computational developments.