The modular case of Weil's cohomology dimension formula for Fuchsian orbifold groups: characteristic dividing an elliptic generator order

In 'Remarks on the Cohomology of Groups' (1964), André Weil computed, for a Fuchsian group $Γ$ of signature $(g; s₁, …, s_n)$ — the orbifold fundamental group of a closed 2-orbifold — and a finite-dimensional $Γ$-module $V$ over a field $K$ whose characteristic divides none of the elliptic orders $s_ν$, the dimension of the space of classes of parabolic cocycles (cocycles restricting to coboundaries on the cyclic subgroups generated by the elliptic generators $c_ν$), obtaining $(2g-2)d + i + i' + Σ_ν e_ν$ with $d = dim V$, $i$ and $i'$ the dimensions of the invariant subspaces of $V$ and of the dual module, and $e_ν = dim im(1 - ρ(c_ν))$ ($ρ$ denoting the action of $Γ$ on $V$); and he proved that the representation space $R ⊂ G^{2g} × ∏ W_ν$ ($W_ν$ the $s_ν$-torsion locus of $G$) is a manifold of dimension $(2g-1)d + Σ_ν e_ν$ at every $ρ$ with $i' = 0$ (for this statement $V = 𝔤$, the Lie algebra of $G$, under the adjoint action of $ρ$). The whole argument uses the hypothesis on the characteristic at a single point: it makes every cocycle parabolic. The case where char $K$ divides some $s_ν$ — the modular case — is resolved in the literature only for rank-one local systems (Li–Liu, published 2025), while the most recent treatment with general finite-dimensional modules (Popa, published 2025) still assumes invertibility of all elliptic orders. The surviving open problem: extend Weil's dimension formula, and the local structure theory of the representation variety $hom(Γ, G)$, to arbitrary finite-dimensional modules in the modular case; any correct formula must involve additional invariants beyond the classical ones, namely the Jordan data of $ρ(c_ν)$ at eigenvalue 1, the rank of the restriction map to the cyclic subgroups, and the new $H²$.

Background

A Fuchsian group is a discrete subgroup of $PSL(2, ℝ)$, the group of orientation-preserving isometries of the hyperbolic plane. A finitely generated Fuchsian group whose quotient orbifold is compact is the orbifold fundamental group of that quotient, a closed 2-dimensional orbifold of signature $(g; s₁, …, s_n)$: it admits a presentation with generators $a₁, …, a_g, b₁, …, b_g, c₁, …, c_n$ and relations $∏_α [a_α, b_α] · c₁⋯c_n = 1$ and $c_ν^{s_ν} = 1$ with integers $s_ν ≥ 2$. The $c_ν$ are the elliptic generators (each has finite order $s_ν$); $g$ is the underlying surface genus and $n$ the number of cone points. Example: the $(2, 3, 7)$ triangle group has signature $(0; 2, 3, 7)$.

Group cohomology enters as follows. Let $V$ be a finite-dimensional right module over the group algebra $KΓ$ (equivalently, a representation of $Γ$ by linear maps on a $K$-vector space $V$, written $v ↦ v|γ$). A 1-cocycle is a map $u: Γ → V$ satisfying $u(gh) = u(g)|h + u(h)$ for all $g, h$; cocycles of the form $u(g) = v - v|g$ for a fixed $v ∈ V$ are coboundaries, and the quotient of cocycles by coboundaries is the cohomology group $H¹(Γ, V)$. In Weil's set-up one takes $G$ a matrix group (algebraic group) over $K$ with Lie algebra $𝔤$, and a representation $ρ: Γ → G$; the central example is $V = 𝔤$ with the adjoint action $X ↦ ρ(γ) X ρ(γ)^{-1}$. Weil's fundamental observation is that the Zariski tangent space at $ρ$ to the space of representations $hom(Γ, G)$ identifies with the space of 1-cocycles $Z¹(Γ, 𝔤)$, so cohomology computes infinitesimal deformations: $H¹(Γ, 𝔤)$ is the tangent space to deformations modulo infinitesimal conjugacy, and $H¹ = 0$ implies local rigidity (every nearby representation is conjugate to $ρ$).

Weil calls a cocycle parabolic if its restriction to each cyclic subgroup $⟨c_ν⟩$ generated by an elliptic generator is a coboundary. The parabolic condition is what the torsion relations $c_ν^{s_ν} = 1$ impose on first-order deformations, so parabolic cohomology — the classes of parabolic cocycles, the kernel of the restriction map $H¹(Γ, V) → ⊕_ν H¹(⟨c_ν⟩, V)$ — is the natural object in the deformation problem. The algebraic hinge of Weil's paper is: if the characteristic of $K$ does not divide $s_ν$, then for any module $V$, $H¹(⟨c_ν⟩, V) = 0$, because the identity $ker(1 + ρ(c_ν) + ⋯ + ρ(c_ν)^{s_ν - 1}) = im(1 - ρ(c_ν))$ holds (writing $ρ(c_ν)$ for the action of the generator $c_ν$ on $V$; the polynomial $x^{s_ν} - 1$ is then separable, so $ρ(c_ν)$ acts semisimply). Consequently every cocycle is automatically parabolic, and the parabolic condition can be ignored.

Under this hypothesis Weil proves two results for $d = dim_K V$, $i = dim V^{Γ}$ (the invariant subspace), $i'$ the dimension of the invariants in the dual module, and $e_ν = dim_K im(1 - ρ(c_ν))$ (the dimension of the 'moving part' at the cone point): first, the space of classes of parabolic cocycles has dimension $(2g-2)d + i + i' + Σ_ν e_ν$; second, for $V = 𝔤$ under the adjoint action of $ρ$ (so $d = dim_K 𝔤$), the representation space $R = \{(g_α, h_α, x_ν) ∈ G^{2g} × ∏_ν W_ν : ∏_α [g_α, h_α] · ∏_ν x_ν = 1\}$, where $W_ν = \{x ∈ G : x^{s_ν} = 1\}$ is the $s_ν$-torsion locus, is at $ρ$ a manifold of dimension $(2g-1)d + Σ_ν e_ν$ whenever $i' = 0$ (a transversality condition: the single long relation cuts $G^{2g} × ∏ W_ν$ transversally). For a closed surface group ($n = 0$) these recover the classical values: $dim H¹(Γ, 𝔤) = (2g-2)d + 2i$ and $dim R = (2g-1)d$.

Everything changes when char $K$ divides some $s_ν$ — call this the modular case. First, the cyclic group algebra $K⟨c_ν⟩ = K[x]/(x^{s_ν} - 1)$ is then non-semisimple: writing $s_ν = p^k s'_ν$ with $p ∤ s'_ν$, the identity $x^{s_ν} - 1 = (x^{s'_ν} - 1)^{p^k}$ shows the algebra acquires nilpotents (it is local exactly when $s_ν$ is a power of $p$), and the local cohomology $H¹(⟨c_ν⟩, V) = ker(1 + ρ(c_ν) + ⋯ + ρ(c_ν)^{s_ν-1}) / im(1 - ρ(c_ν))$ can be nonzero; for the trivial module it is one-dimensional whenever char $K$ divides $s_ν$. This group depends on the Jordan block structure of $ρ(c_ν)$ at eigenvalue 1 (the dimensions of the kernels of the powers of $(1 - ρ(c_ν))$), not merely on the dimension of the fixed subspace, so Weil's term $e_ν$ has no direct substitute. Second, $H²(⟨c_ν⟩, V)$ can be nonzero and the vanishing $H²(Γ, V) = 0$ — used throughout the classical theory, e.g. in Popa's trace-formula paper via a free-product argument of Swan — can fail, so the Euler-characteristic dimension count breaks. Third, in characteristic $p$ dividing $s_ν$ one has the polynomial identity $x^{s_ν} - 1 = (x^{s_ν/p} - 1)^p$, so the torsion locus $W_ν$ is a non-reduced scheme (its coordinate ring has nilpotent elements; its underlying variety only sees the prime-to-$p$ part of the order), and the ambient geometry of the constrained model $R ⊂ G^{2g} × ∏ W_ν$ degenerates.

The later framework in which the question now sits: Lubotzky and Magid (1985) developed the general theory of representation varieties of finitely generated groups (VAR-equivalence, their notion of equivalence of representations via isomorphism of the attached representation varieties, semisimple points, dimension theory); Goldman and Millson (1988) described the singularities of representation varieties at points where the cup product $H¹ × H¹ → H²$ does not vanish (quadratic singularities); Larsen and Lubotzky (2012) surveyed representation varieties of Fuchsian groups specifically; Porti (2022, 2025) computed dimensions and the reducible-point singularities of character varieties of 2-orbifold groups. All of this work is in characteristic 0 or under invertibility of the elliptic orders. On the arithmetic side, Li and Liu (2021, published 2025) computed $dim H¹$ of orbifold groups over an arbitrary algebraically closed field — including the modular case — but only for rank-one local systems (one-dimensional representations), and Popa's trace formula for Hecke operators on $H¹(Γ, V)$ (published 2025), the most recent treatment with general finite-dimensional modules, explicitly assumes throughout that the orders of elliptic elements are invertible in the coefficient field.

Problem Statement

Let $K$ be a field of characteristic $p ≥ 0$ and let $Γ$ be the orbifold group of signature $(g; s₁, …, s_n)$, i.e. the group with generators $a₁, …, a_g, b₁, …, b_g, c₁, …, c_n$ and relations $∏_α [a_α, b_α] · c₁⋯c_n = 1$, $c_ν^{s_ν} = 1$ ($s_ν ≥ 2$). Let $V$ be a finite-dimensional right $KΓ$-module; the central case is $V = 𝔤$, the Lie algebra of an algebraic group $G$ over $K$, acted on through the adjoint action of a representation $ρ: Γ → G$. Assume the modular case: $p$ divides at least one $s_ν$. (The cusped finite-covolume variant may also be addressed on the same terms; there the group is a free product of the cyclic subgroups with a free group, so the analysis decomposes through Swan's Mayer–Vietoris sequence for free products, and the single long relation that couples the cone points in the compact case is absent. The compact case is the core.) Both of the following questions are posed. A further goal is to delimit precisely which additional invariants any correct dimension formula must involve, without producing the expression itself.

(1) Dimension formula. Determine $dim_K H¹(Γ, V)$ as a closed-form expression in representation-theoretic data of the module $V$ (equivalently, of the representation of $Γ$ on $V$) — the modular-case extension of Weil's formula $(2g-2)d + i + i' + Σ_ν e_ν$ for the classes of parabolic cocycles. Here 'determine' means an expression in invariants of the module, not an algorithm; 'invariants of the module' are quantities determined by the representation of $Γ$ on $V$ that do not invoke the cohomology being computed, so the expression may not use $dim_K H¹(Γ, V)$, the dimension of the parabolic cohomology, or any quantity defined from them as a term: $dim H¹(Γ, V)$ is of course computable by finite linear algebra for any explicitly given module, but no expression in terms of the classical invariants ($d$, $i$, $i'$, the $e_ν$) — or of any published larger list of invariants — is known when $p$ divides some $s_ν$. The answer must in particular give the rank of the restriction map $H¹(Γ, V) → ⊕_ν H¹(⟨c_ν⟩, V)$ (so that, by rank–nullity, the dimension of the parabolic cohomology is $dim_K H¹(Γ, V)$ minus this rank), and is expected to involve the local groups $H¹(⟨c_ν⟩, V) = ker(1 + c_ν + ⋯ + c_ν^{s_ν - 1}) / im(1 - c_ν)$, where $c_ν$ acts on $V$ through the module structure; these depend on the Jordan structure of the action of $c_ν$ at eigenvalue 1, together with a correction term reflecting the possible non-vanishing of $H²(Γ, V)$.

(2) Local structure. Determine the local structure of the representation variety $R = hom(Γ, G) ⊂ G^{2g+n}$ at $ρ$ in the modular case. This comprises three required parts: a smoothness criterion replacing Weil's condition $i' = 0$; the local dimension at smooth points; and the singularity type at singular points, where the singularity type means the isomorphism class of the completed local ring $\widehat{𝒪}_{R,ρ}$ of $R$ at $ρ$.

References

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