Coh Zeta Function Analysis

• Coh zeta function is a generating series that enumerates finite modules over a commutative ring with weights given by automorphism groups and module cardinalities.
• It connects arithmetic module enumeration to classical q-series identities, exemplified by finitized versions of Andrews–Gordon and Bressoud polynomials in quadratic-order settings.
• Recent developments leverage Möbius inversion on submodule posets to transform module counting into explicit q-hypergeometric sums, particularly addressing inert cases.

Searching arXiv for papers directly using the term and nearby related zeta constructions. {"^^^^^^^^3query3 zeta function\"^^^^^^^^3 OR ti:\3^^^^^^^^"Coh zeta\"","max_results":^^^^3all:\3query3^^^^} The Coh zeta function is a Cohen–Lenstra style generating series attached to finite modules over a commutative ring. For a commutative ring PRESERVED_PLACEHOLDER_^^^^3query3^^^^, it is defined by

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^

where the sum runs over isomorphism classes of finite PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^-modules $Q$. In the quadratic-order setting developed for local orders over $k[[T]]$, the theory is closely tied to a finitized version obtained from Quot zeta functions, and its recent formulation connects the arithmetic of finite modules to classical $q$-series identities of Andrews–Gordon and Bressoud type (&&&^^^^3query3^^^^&&&).

^^^^3all:\3^^^^. Definition and formal framework

The basic companion object is the Quot zeta function of an $R$-module $M$,

$\zeta_M^R(s) := \sum_{L \subseteq_R M} (M:L)^{-s},$

where $L$ runs over finite-index PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^-submodules of PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^. The finitized Coh zeta function is defined by framing: PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^ A key theorem recalled in the inert-quadratic-order work states that, if the Coh zeta function is well-defined, then

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^

coefficient-wise as formal Dirichlet series (&&&^^^^3query3^^^^&&&).

This formalism makes the Coh zeta function a module-enumerating analogue of more classical Euler-product constructions. The weights are determined by automorphism groups and module cardinalities rather than by point counts, cycle lengths, or subgroup indices. In that sense, the function is not merely a counting series: it records representation-theoretic symmetry through the factor PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4, and it is precisely this Cohen–Lenstra style weighting that underlies the term “Coh.”

^^^^3 OR ti:\3^^^^. Quadratic orders and the three-family correspondence

The modern theory is organized around three families of non-maximal quadratic orders over a local field PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5. These correspond to the three quadratic-extension types—ramified, split, and inert—and are treated uniformly through finitized Coh zeta functions (&&&^^^^3query3^^^^&&&).

Family	Order	Associated PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6-series pattern
Ramified	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7	Andrews–Gordon
Split	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8	Bressoud with sign twist
Inert	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9	PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3query3^^^^-deformed Bressoud, conjecturally

For odd PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^, the split family is isomorphic to PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^. The inert family is the one studied as the missing case in the trilogy. The known finitized formulas recalled for the first two families are

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^

and

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4

with PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5.

The classical finite PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6-series polynomials entering these identities are the finitized Andrews–Gordon and Bressoud polynomials,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7

Their PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8 specializations are finitized versions of the classical central Andrews–Gordon and Bressoud identities, with infinite limits

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9

and

$Q$^^^^3query3^^^^

3. Inert quadratic orders and the main conjecture

For the inert family, the central conjecture is

$Q$^^^^3all:\3^^^^

This identifies the inert finitized Coh zeta function with the direct $Q$^^^^3 OR ti:\3^^^^-deformation of the Bressoud polynomial, without the sign twist that appears in the split case (&&&^^^^3query3^^^^&&&).

The structural explanation proposed in the same work uses quadratic twisting. For odd $Q$3, if $Q$4 is a nonsquare, then the inert order $Q$5 is isomorphic to the quadratic twist

$Q$6

whereas the twist of the ramified order,

$Q$7

is isomorphic to the ramified order itself. This gives a conceptual reason for the symmetry

$Q$8

This suggests that the three-family correspondence is not merely formal. A plausible implication is that ramified, split, and inert Coh zeta functions are best viewed as a single deformation-theoretic pattern, with the sign behavior governed by the quadratic-extension type.

4. Möbius inversion on submodule posets

A principal methodological development is a new computation of saturation zeta functions by Möbius inversion on the poset of submodules. For an inclusion $Q$9 of finite rings and a finite $k[[T]]$^^^^3query3^^^^-module $k[[T]]$^^^^3all:\3^^^^, the paper proves

$k[[T]]$^^^^3 OR ti:\3^^^^

where $k[[T]]$3 is the Möbius function of the submodule poset (&&&^^^^3query3^^^^&&&).

This formula isolates saturation contributions and turns the problem into explicit combinatorics on module lattices. In the DVR-quotient setting, Hall polynomials enter through a lemma of the form

$k[[T]]$4

with $k[[T]]$5 denoting Hall polynomials.

The significance of this step is computational rather than merely formal. Earlier techniques were effective in the ramified and split cases but encountered obstacles for inert orders. The poset-based inversion method converts module counting into explicit $k[[T]]$6-hypergeometric sums and thereby makes the inert case tractable.

5. Explicit formulas, specializations, and evidence

The main technical theorem gives explicit formulas for $k[[T]]$7 in all three families. In the inert case it yields

$k[[T]]$8

and, after specializing to $k[[T]]$9,

$q$^^^^3query3^^^^

(&&&^^^^3query3^^^^&&&).

This is compared with the expected $q$^^^^3all:\3^^^^ specialization of the conjecture,

$q$^^^^3 OR ti:\3^^^^

For the already known cases, the corresponding specializations are

$q$3

The strongest explicit evidence is obtained for the simplest inert order

$q$4

There the paper derives the first explicit formulas for the inert finitized Coh zeta function at all $q$5, as well as a normalized double-sum formula. It also proves a combinatorial identity counting $q$6-codimensional $q$7-subspaces $q$8 satisfying $q$9. The cumulative evidence cited consists of exact formulas for the $R$^^^^3query3^^^^ specialization for all $R$^^^^3all:\3^^^^, a complete explicit computation for $R$^^^^3 OR ti:\3^^^^, consistency with the ramified/split trilogy and twist symmetry, numerical verification against the predicted $R$3-deformed Bressoud sums, and the derivation of new $R$4-series identities from algebraic module-counting.

6. Relation to nearby zeta-function notions

The Coh zeta function should be distinguished from the cotype zeta function. For $R$5, the cotype zeta function is a multivariable refinement of the subgroup-growth zeta function that records the invariant factors of $R$6 for a sublattice $R$7, and its analysis leads to asymptotics for sublattices of bounded corank (Chinta et al., 2017). A closely related but different construction is the cotype zeta function for subrings of $R$8, where finite-index subrings are counted according to the elementary divisors of the quotient $R$9 and the local factors are computed by $M$^^^^3query3^^^^-adic integration (Chimni et al., 2020).

It should also be separated from the pure non-abelian zetas attached to curves over finite fields. Those zeta functions are built from moduli spaces of semi-stable vector bundles and use weights of the form

$M$^^^^3all:\3^^^^

so their defining structure is cohomological and geometric rather than Cohen–Lenstra style module enumeration (Weng, 2012).

These comparisons clarify the role of the Coh zeta function within the broader zeta-function landscape. It is neither a cotype-counting refinement nor a bundle-theoretic cohomological zeta. Its defining feature is the weighted enumeration of finite $M$^^^^3 OR ti:\3^^^^-modules, and in the quadratic-order case this enumeration appears to organize itself according to deep $M$3-series identities.