Drinfeld Quotient: Frameworks & Applications

• Drinfeld quotient is a family of procedures applied to Drinfeld modules, p-adic spaces, and Hopf algebras that preserve rich moduli and cohomological structures.
• In the modular-scheme context, it ensures regularity by quotienting Drinfeld modular schemes with admissible subgroup actions, guided by deformation theory and invariant methods.
• In p-adic geometry and Hopf algebra settings, it yields quotient stacks and surjective algebra maps that enable explicit cohomological computations and tensor subcategory classifications.

Searching arXiv for papers on “Drinfeld quotient” and closely related uses of the term. “Drinfeld quotient” is not a single universally fixed construction. In current arXiv usage, the term refers to several quotient procedures attached to objects introduced by Drinfeld: quotients of Drinfeld modular schemes by admissible finite group actions, Hopf algebra quotients of the Drinfeld double of a finite group scheme, and quotient stacks of Drinfeld upper half spaces by arithmetic groups (Kondo et al., 2017, Arreola et al., 31 Mar 2026, Yi, 3 Oct 2025). In each case, the quotient is designed to preserve a highly structured moduli, tensor-categorical, or cohomological framework, and the principal results concern regularity, classification, or explicit computation rather than the mere existence of an orbit space.

1. Terminological scope

The expression “Drinfeld quotient” is best understood contextually. In the theory of Drinfeld modules, it denotes quotients of regular affine moduli schemes by admissible subgroups of automorphism groups of level structures, with regularity as the central issue (Kondo et al., 2017). In Hopf algebra theory, it denotes surjective Hopf algebra maps from a Drinfeld double $D(G)$ to a smaller quasitriangular Hopf algebra $D$, together with the classification of the resulting quotient pairs (Arreola et al., 31 Mar 2026). In $p$-adic geometry, it denotes quotient stacks such as $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$ and $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$, which carry moduli interpretations in terms of special formal $\mathcal{O}_D$-modules (Yi, 3 Oct 2025).

Setting	Quotient object	Main structural result
Drinfeld modular schemes	$M_{A,d}(N)_U/H$	existence and regularity for admissible $H$
Drinfeld double of a finite group scheme	$D(G)\twoheadrightarrow D(K,H,B)$	classification of quotient pairs
Drinfeld upper half space	$[\mathcal{H}^{n-1}_K/G]$	explicit $D$0-adic and $D$1-adic pro-étale cohomology

This multiplicity of meanings is not accidental. The common theme is that a Drinfeld object is first equipped with a symmetry or level datum and is then quotiented in a way that remains compatible with deformation theory, braided tensor structure, or equivariant cohomology. A plausible implication is that the phrase is best treated as a family resemblance term rather than a single technical definition.

2. Quotients of Drinfeld modular schemes

In the modular-scheme setting, one starts with a smooth projective geometrically irreducible curve $D$2 over $D$3, a closed point $D$4, and the coefficient ring

$D$5

which is a Dedekind domain with fraction field $D$6. A rank $D$7 Drinfeld $D$8-module over an $D$9-scheme $p$0 is a ring homomorphism

$p$1

whose images are additive polynomials with the prescribed linear coefficient and rank condition. For a finitely generated torsion $p$2-module $p$3, Kondo and Yasuda define a level $p$4 structure as an $p$5-module homomorphism

$p$6

such that for every $p$7, the effective Cartier divisor

$p$8

on $p$9 is a closed subscheme of $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$0 (Kondo et al., 2017). The case

$$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$1

recovers Drinfeld’s full level $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$2 structure.

Fixing rank $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$3, a finitely generated torsion $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$4-module $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$5, and an open subscheme $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$6, the moduli functor

$$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$7

sends $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$8 to isomorphism classes of rank $$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(\mathcal{O}_K)]$$9 Drinfeld $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$0-modules over $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$1 with level $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$2 structure. Proposition 4.2.1 gives representability and regularity in two cases: if $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$3 and $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$4 is any open, or more generally if $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$5 is nonempty and $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$6 is open. In these situations the functor is representable by a regular affine $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$7-scheme $[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$8 (Kondo et al., 2017).

The automorphism group

$[\mathcal{H}^{n-1}_K/\operatorname{GL}_n(K)]$9

acts on level structures by precomposition $\mathcal{O}_D$0 and hence on $\mathcal{O}_D$1. The decisive notion is that of an admissible subgroup. For a $\mathcal{O}_D$2-primary module

$\mathcal{O}_D$3

an admissible subgroup $\mathcal{O}_D$4 is defined by congruence conditions on matrix entries:

$\mathcal{O}_D$5

Globally, admissibility is imposed prime by prime through the primary decomposition of $\mathcal{O}_D$6 (Kondo et al., 2017).

The main regularity theorem states that if $\mathcal{O}_D$7 with $\mathcal{O}_D$8 generated by at most $\mathcal{O}_D$9 elements, $M_{A,d}(N)_U/H$0, and $M_{A,d}(N)_U/H$1, and if $M_{A,d}(N)_U/H$2 is chosen so that $M_{A,d}(N)_U/H$3 satisfies the representable-regular hypotheses above, then for every admissible subgroup $M_{A,d}(N)_U/H$4 acting trivially on $M_{A,d}(N)_U/H$5, the quotient

$M_{A,d}(N)_U/H$6

exists as a scheme and is regular (Kondo et al., 2017).

3. Regularity mechanism, congruence quotients, and Hecke-theoretic role

The proof of regularity is explicitly modeled on Katz–Mazur’s treatment of elliptic modular curves, with additional modular invariant theory from Dickson. The strategy is local. First, one reduces to one prime at a time via the primary decomposition of $M_{A,d}(N)_U/H$7. Away from a prime $M_{A,d}(N)_U/H$8 in the support of $M_{A,d}(N)_U/H$9, the quotient map is étale and therefore preserves regularity. At $H$0, one passes to completed local rings that are universal deformation rings of formal $H$1-modules with level $H$2, and the invariant rings depend only on the height of the formal $H$3-module. The argument then reduces to the supersingular case of maximal height $H$4, standardizes the level to $H$5, and analyzes a filtration of the admissible group by normal subgroups $H$6 and $H$7 (Kondo et al., 2017).

At the completed local level, Katz–Mazur’s proposition on invariants of complete local regular rings under groups acting trivially on residue fields is used iteratively, with explicit regular parameters described by additive polynomials $H$8. The final step identifies a residual linear action of Levi factors

$H$9

on a $D(G)\twoheadrightarrow D(K,H,B)$0-span of parameters. Dickson’s theorem then yields an invariant ring that is again a formal power series ring in homogeneous invariant generators, hence regular (Kondo et al., 2017). The regularity of the quotient is therefore not a formal corollary of finite generation; it is a consequence of a precise deformation-theoretic and invariant-theoretic analysis.

The framework covers the standard congruence-type subgroups for

$D(G)\twoheadrightarrow D(K,H,B)$1

The groups

$D(G)\twoheadrightarrow D(K,H,B)$2

and

$D(G)\twoheadrightarrow D(K,H,B)$3

are admissible, so the quotients

$D(G)\twoheadrightarrow D(K,H,B)$4

are regular (Kondo et al., 2017). More generally, parabolic subgroups $D(G)\twoheadrightarrow D(K,H,B)$5 attached to partitions $D(G)\twoheadrightarrow D(K,H,B)$6 are admissible; these are precisely the parabolics arising in Hecke correspondences.

This regularity is a foundational input for the construction of finite correspondences and Hecke operators. For example, with $D(G)\twoheadrightarrow D(K,H,B)$7 prime to a finite prime $D(G)\twoheadrightarrow D(K,H,B)$8, setting

$D(G)\twoheadrightarrow D(K,H,B)$9

the admissible group $[\mathcal{H}^{n-1}_K/G]$0 produces finite flat maps

$[\mathcal{H}^{n-1}_K/G]$1

and hence the $[\mathcal{H}^{n-1}_K/G]$2-th Hecke operator

$[\mathcal{H}^{n-1}_K/G]$3

(Kondo et al., 2017). In this sense, the “Drinfeld quotient” is not merely a quotient construction but a device that makes Hecke actions available on regular schemes.

4. Quotient stacks of Drinfeld spaces

In a distinct $[\mathcal{H}^{n-1}_K/G]$4-adic setting, the Drinfeld upper half space of dimension $[\mathcal{H}^{n-1}_K/G]$5 over a finite extension $[\mathcal{H}^{n-1}_K/G]$6 is

$[\mathcal{H}^{n-1}_K/G]$7

It is smooth, quasi-Stein, and carries a natural action of $[\mathcal{H}^{n-1}_K/G]$8, with $[\mathcal{H}^{n-1}_K/G]$9 as a compact open “level-0” stabilizer. The quotient stacks

$D$00

are stacks on the pro-étale site of adic spaces over $D$01 (Yi, 3 Oct 2025).

These stacks have a moduli interpretation. Let $D$02 be the central division algebra over $D$03 of invariant $D$04, with ring of integers $D$05. Then

$D$06

where $D$07 classifies special formal $D$08-modules up to isomorphism and $D$09 classifies the same objects up to quasi-isogeny (Yi, 3 Oct 2025). The passage to these quotient stacks is obtained by descending from the infinite-level Drinfeld tower and using the Faltings–Scholze–Weinstein equivalence between the Drinfeld and Lubin–Tate towers.

The basic computational tool is an equivariant descent spectral sequence:

$D$10

with $D$11 and $D$12 (Yi, 3 Oct 2025). For $D$13, Schneider–Stuhler’s description gives

$D$14

For $D$15-adic coefficients, Colmez–Dospinescu–Nizioł obtain a strictly exact sequence

$D$16

From these inputs the quotient-stack cohomology is computed explicitly. For $D$17-adic coefficients,

$D$18

and

$D$19

For $D$20-adic coefficients, the isomorphism stack satisfies

$D$21

while the isogeny stack satisfies

$D$22

with $D$23 (Yi, 3 Oct 2025).

Here the quotient is intrinsically stack-theoretic. The isotropy groups encode automorphisms or quasi-isogenies of the underlying special formal $D$24-modules, and the resulting cohomology reflects those isotropy factors directly. A plausible implication is that, in this setting, “Drinfeld quotient” should be read as an equivariant moduli object rather than as a coarse analytic quotient.

5. Hopf algebra quotients of the Drinfeld double

A third usage concerns the Drinfeld double of a finite group scheme. Let $D$25 be a finite group scheme over an algebraically closed field $D$26 of characteristic $D$27, with coordinate algebra $D$28 and dual Hopf algebra $D$29. The Drinfeld double is

$D$30

with product

$D$31

where $D$32 is the left coadjoint action (Arreola et al., 31 Mar 2026). The representation category

$D$33

is a finite non-degenerate ribbon braided tensor category.

A Hopf algebra quotient pair of $D$34 is a finite-dimensional Hopf algebra $D$35 together with a surjective Hopf algebra homomorphism

$D$36

The classification theorem states that every such quotient is of the form

$D$37

where $D$38 are normal subgroup schemes that centralize each other and

$D$39

is a $D$40-equivariant Hopf algebra map (Arreola et al., 31 Mar 2026). The quotient map is

$D$41

and its kernel is generated by $D$42 together with

$D$43

Thus equivalence classes of quotient pairs are in bijection with triples $D$44 of this type.

The quotient inherits quasitriangular and ribbon structure from $D$45. Its universal $D$46-matrix and ribbon element are the images under $D$47 and $D$48 of those for $D$49:

$D$50

$D$51

The categorical counterpart is that the assignment

$D$52

gives a bijection between such triples and tensor subcategories of $D$53 (Arreola et al., 31 Mar 2026).

The same paper determines centralizers and criteria for symmetry or non-degeneracy. Writing

$D$54

the Müger centralizer is

$D$55

Moreover, $D$56 is symmetric precisely when $D$57 and

$D$58

and it is non-degenerate precisely when $D$59 and the convolution map

$D$60

is a Hopf algebra isomorphism (Arreola et al., 31 Mar 2026). In characteristic $D$61, this recovers the Naidu–Nikshych–Witherspoon classification; in positive characteristic, semisimplicity typically fails, but the Hopf-quotient classification still works uniformly.

In this context, a “Drinfeld quotient” is emphatically not a geometric quotient of a space. It is a quotient in the Hopf-algebraic sense, carrying enough structure to control the tensor subcategory lattice of $D$62 and the behavior of simple and projective objects.

6. Conceptual comparison and recurrent misconceptions

The three principal uses of “Drinfeld quotient” differ in both ambient category and intended output. In the modular-scheme setting, the quotient is an affine categorical quotient by a finite group action, and the central theorem is that regularity survives for admissible subgroups under explicit support hypotheses on the level module (Kondo et al., 2017). In the Drinfeld-space setting, the quotient is a stack in the pro-étale topology, and its importance lies in a moduli interpretation and in explicit cohomological formulas (Yi, 3 Oct 2025). In the Drinfeld-double setting, the quotient is a surjective Hopf algebra map whose classification controls braided tensor subcategories and Müger centralizers (Arreola et al., 31 Mar 2026).

A common misconception is to treat all of these quotients as variants of coarse orbit spaces. That is inaccurate. The quotient stack

$D$63

retains isotropy and classifies objects up to quasi-isogeny rather than collapsing stabilizers. Likewise, the quotient $D$64 is algebraic and braided, not geometric. Even in the modular-scheme setting, the regular quotient theorem is not a statement about arbitrary subgroup actions, but about admissible subgroups under the decomposition

$D$65

with $D$66 in the representable-regular range of Proposition 4.2.1 (Kondo et al., 2017).

A second misconception is that regularity or explicit computability automatically extends to compactifications or all coefficient systems. The modular-scheme paper explicitly focuses on open moduli and does not assert extensions to compactifications or stack-theoretic settings (Kondo et al., 2017). The quotient-stack computations are for pro-étale cohomology with coefficients $D$67 or $D$68, not for arbitrary theories (Yi, 3 Oct 2025). The Drinfeld-double classification is robust in all characteristics, but the representation category is typically non-semisimple for $D$69 (Arreola et al., 31 Mar 2026).

Taken together, these works show that “Drinfeld quotient” is a structurally rich but context-dependent notion. In arithmetic geometry it furnishes regular moduli schemes and finite flat Hecke correspondences; in local $D$70-adic geometry it produces quotient stacks whose cohomology can be computed explicitly; and in Hopf algebra and tensor-category theory it yields a classification of quasitriangular quotients and of tensor subcategories of Drinfeld centers (Kondo et al., 2017, Yi, 3 Oct 2025, Arreola et al., 31 Mar 2026).