Log-Rigid Analytic Classes

• Log-rigid analytic classes are a framework in p-adic Hodge theory that encode cohomological phenomena using rigid-analytic and logarithmic structures.
• They facilitate explicit computation and comparison isomorphisms in Hyodo–Kato theory and the construction of syntomic cohomology for semistable schemes.
• These classes extend to group cohomology with applications in computing p-adic regulators, special values of L-functions, and arithmetic duality.

Log-rigid analytic classes constitute a framework within $p$-adic Hodge theory and arithmetic geometry for encoding and analyzing cohomological phenomena via rigid-analytic and logarithmic structures. Their chief utility is the explicit computation and comparison of $p$-adic cohomology theories such as Hyodo–Kato, syntomic, and crystalline cohomologies, particularly for varieties with semistable ($\log$) structure and in settings susceptible to bad reduction. They also manifest in the study of special values of $p$-adic $L$-functions and arithmetic regulator maps, framing group cohomological and analytic invariants in a rigid-analytic context.

1. Definitions and Foundational Structures

The formalism of log-rigid analytic classes emerges from the theory of log-rigid cohomology for proper strictly semistable log schemes. Given a prime $p$, a complete discrete valuation ring $V$ of mixed characteristic $(0,p)$ with residue field $k$ and fraction field $K$, the standard bases in weak formal geometry are

$$S = \mathrm{Spwf}\;W[k],\quad W^0 = \mathrm{Spf}\;W,\quad V^\# = \mathrm{Spf}\;V,$$

where $W = W(k)$ is the ring of Witt vectors.

For a strictly semistable log scheme $Y/k^0$ with horizontal divisor $D$, and any unipotent log-overconvergent isocrystal $$(\mathcal{E}, \Phi) \in \mathrm{F\text{-}Isoc}^\dagger(Y/W^0)_{\mathrm{unip}}$$, the log-rigid cohomology is defined as

$$\begin{aligned} \mathrm{R}\Gamma_{\log\text{-}\mathrm{rig}}(Y/W^0,\mathcal{E}) &:= \mathrm{R}\Gamma_{\mathrm{rig},M_D}(Y/(W^0),\mathcal{E}), \ H^i_{\log\text{-}\mathrm{rig}}(Y/W^0,\mathcal{E}) &:= H^i(\mathrm{R}\Gamma_{\log\text{-}\mathrm{rig}}(Y/W^0,\mathcal{E})), \end{aligned}$$

with the compact-support variant

$$\begin{aligned} \mathrm{R}\Gamma_{\log\text{-}\mathrm{rig},c}(Y/W^0,\mathcal{E}) &:= \mathrm{R}\Gamma_{\mathrm{rig},M_D,c}(Y/(W^0),\mathcal{E}), \ H^i_{\log\text{-}\mathrm{rig},c}(Y/W^0,\mathcal{E}) &:= H^i(\mathrm{R}\Gamma_{\log\text{-}\mathrm{rig},c}(Y/W^0,\mathcal{E})). \end{aligned}$$

These groups are computed by Čech-type complexes on dagger spaces associated to semistable charts and universal formal enlargements, with the resulting complexes inheriting Frobenius and monodromy operators and functoriality. The approach allows for explicit cohomological calculations with arithmetic significance (Ertl et al., 2020).

2. Hyodo–Kato Theory and Comparison Isomorphisms

A central role of log-rigid analytic classes is the explicit realization of Hyodo–Kato theory and its comparison isomorphisms. For a proper semistable scheme $X/V^\#$ with special fiber $Y$, the rigid Hyodo–Kato cohomology is given by

$$\mathrm{R}\Gamma_{\mathrm{HK}}(Y,\mathcal{E}) := \mathrm{R}\Gamma_{\mathrm{rig},M_D}(Y/W^0,\mathcal{E}),$$

with $$H^i_{\mathrm{HK}}(Y,\mathcal{E}) = H^i(\mathrm{R}\Gamma_{\mathrm{HK}}(Y,\mathcal{E}))$$.

The Hyodo–Kato comparison isomorphism, constructed canonically on the rigid-analytic site and independent of uniformizer (modulo the choice of a branch of the $p$-adic logarithm), asserts an equivalence between Hyodo–Kato and log-rigid compact-support cohomology: $$\psi_{\mathrm{HK},\pi}: H^i_{\mathrm{HK},c}(Y) \otimes_F K \xrightarrow{\;\cong\;} H^i_{\log\text{-}\mathrm{rig},c}(Y/V^\#).$$ This isomorphism is Frobenius- and monodromy-equivariant and respects log structures (Ertl et al., 2020, Ertl et al., 2019). Explicit Čech and de Rham complexes realize these theories, enabling concrete calculations and comparisons with log-crystalline cohomology.

3. Poincaré Duality and Cup Product Structures

Log-rigid analytic classes accommodate a canonical Poincaré duality theory for log-rigid cohomology with and without compact support. For $\dim Y = d$, and dual overconvergent isocrystal $$\mathcal{E}^\vee = \underline{\mathrm{Hom}}(\mathcal{E}, \mathcal{O}_Y)$$, the duality is formulated as: $$\mathrm{R}\Gamma_{\log\text{-}\mathrm{rig},c}(Y/W^0,\mathcal{E}) \otimes^{\mathbb L}_{F} \mathrm{R}\Gamma_{\log\text{-}\mathrm{rig}}(Y/W^0,\mathcal{E}^\vee) \xrightarrow{\;\cup\;} \mathrm{R}\Gamma_{\log\text{-}\mathrm{rig},c}(Y/W^0) \xrightarrow{\;\mathrm{Tr}\;} F(-d)[-2d],$$ where the cup product is induced by the bigraded Čech–de Rham complex and the trace is rigid-analytic, reflecting the geometry of the weak formal model. This duality establishes explicit cohomological pairings, with compatibility for the $(\varphi, N)$-module structure, and extends analogously to rigid Hyodo–Kato cohomology (Ertl et al., 2020).

4. Role in Syntomic Cohomology and Extension Groups

Log-rigid analytic classes underpin the construction of log-rigid syntomic cohomology for strictly semistable schemes over $p$-adic integer rings. The construction involves a diagram of comparison maps among rigid, $K$-specialization, and de Rham complexes: $$\begin{array}{ccccc} R\Gamma_{\mathrm{rig}} &\xrightarrow{\;\alpha\;}&R\Gamma_K &\xleftarrow{\;\beta\;}&R\Gamma_{\mathrm{dR}} \ (\phi,N) &&&& (F^\bullet) \end{array}$$ with the log-rigid syntomic complex defined as the totalization of a mapping-fiber diagram: $$\operatorname{Cone}[M_{\mathrm{rig}} \oplus F^n M_{\mathrm{dR}} \to M_{\mathrm{rig}} \oplus M_K][-1],$$ yielding cohomology groups

$$H^i_{\mathrm{syn}}(X, n) = H^i(\operatorname{Tot}[\cdots]).$$

There is a canonical interpretation of these groups as extension groups in the derived category of admissible filtered $(\phi,N)$-modules (Fontaine's category), with explicit complexes and differentials constructed from log-rigid data (Yamada, 2015). This realizes log-rigid syntomic cohomology as absolute $p$-adic Hodge cohomology, connecting syntomic regulators, $K$-theoretic Chern classes, and cycles to analytic classes.

5. Representation in Arithmetic and Group Cohomology

A recent development is the construction of log-rigid analytic classes in the group cohomology of arithmetic groups. For $\mathrm{SL}_n(\mathbb{Z})$, "log-rigid analytic classes" are defined as group cohomology classes with values in modules of log-rigid analytic functions on Drinfeld's $p$-adic symmetric domain $\Omega_p$: $$J_{E,c} \in H^{n-1}\left(\mathrm{SL}_n(\mathbb{Z}), A_c^{\mathrm{rig}, \log}(\Omega_p)\right).$$ The module $A_c^{\mathrm{rig}, \log}$ consists of functions admitting (finite) sums of rigid-analytic functions and $p$-adic logarithms of coordinate ratios, with a natural $\mathrm{SL}_n(\mathbb{Q}_p)$-action. These classes arise through a $p$-adic Poisson transform applied to topological Eisenstein cocycles, establishing a bridge between topological, group-theoretic, and analytic perspectives (Roset et al., 12 Dec 2025).

A key phenomenon is that such classes, when evaluated at special $p$-adic points associated to totally real fields $F/\mathbb{Q}$ where $p$ is inert, yield values conjecturally related to $p$-adic logarithms of Gross–Stark units. For Galois fields, it is shown that the evaluations recover $\log_p(u^{\sigma_a})$ exactly in $\mathbb{Q}_p$, confirming the conjecture in this case.

6. Computational Methods and Explicit Examples

Log-rigid analytic classes are amenable to explicit Čech, de Rham, or spectral sequence computations. In the context of strictly semistable schemes, compact-support log-rigid cohomology can be computed from the combinatorics of the irreducible components: $$E_1^{p,q} = H^q_{\log\text{-}\mathrm{rig}}(Y^{(p)}, M_D) \implies H^{p+q}_{\log\text{-}\mathrm{rig},c}(Y),$$ where $Y^{(p)}$ is the $p$-fold intersection locus, each a dagger-affinoid space. Calculations yield explicit descriptions of $p$-adic regulator maps and cycle classes, with applications to $p$-adic Beilinson and Bloch–Kato conjectures (Ertl et al., 2020).

A prototypical example is the Tate curve, whose log-rigid cohomology yields canonical basis classes for which the Frobenius and monodromy operators, as well as comparison maps to de Rham cohomology, have explicit analytic and arithmetic formulas (Ertl et al., 2019).

7. Connections and Applications in $p$-adic Arithmetic

Log-rigid analytic classes encode deep arithmetic invariants relevant to the study of $p$-adic $L$-functions, Stark-type conjectures, and the arithmetic of $p$-adic regulators. Their explicit character enables effective calculation of syntomic and motivic regulators in settings with semistable reduction, and offers tools for constructing and analyzing $p$-adic polylogarithms, Eisenstein classes, and arithmetic cycles in both the local and global $p$-adic contexts (Roset et al., 12 Dec 2025, Yamada, 2015, Ertl et al., 2020).

Their relationship with extension groups of filtered $(\phi,N)$-modules underpins structural bridges between $p$-adic Hodge theory, algebraic $K$-theory, and the values of motivic $L$-functions, establishing log-rigid analytic classes as pivotal objects in modern arithmetic geometry.