---
title: 'Lubin-Tate Space: Definition, Applications, and Significance'
url: https://www.emergentmind.com/topics/lubin-tate-space
type: topic
---

# Lubin-Tate Space: Definition, Applications, and Significance

Lubin–Tate space is the formal and rigid-analytic moduli space of deformations of a one-dimensional formal $\mathcal O_F$-module of fixed height, usually equipped with a quasi-isogeny to a fixed special fiber and, at finite levels, a Drinfeld level structure. If $F$ is a non-archimedean local field with ring of integers $\mathcal O_F$, uniformizer $\pi$, and residue field of cardinality $q$, and if $\Sigma$ is a one-dimensional formal $\mathcal O_F$-module of height $h$ over an algebraic closure of the residue field, then its deformation functor is represented by a formal scheme of relative dimension $h-1$. Its rigid generic fiber is an open unit polydisc of dimension $h-1$. The resulting tower, with its commuting actions of $\mathrm{GL}_h(F)$, the division algebra of invariant $1/h$, and the Weil group $W_F$, is a central local-geometric object in the realization of local Langlands and Jacquet–Langlands correspondences.

## 1. Moduli problem and universal deformation

A one-dimensional formal $\mathcal O_F$-module over a ring $R$ is a one-dimensional commutative formal group law
\[
F(X,Y)=X+Y+\cdots
\]
equipped with endomorphisms $[a]_F(X)$ for $a\in\mathcal O_F$, compatible with the $\mathcal O_F$-module structure. Formal addition is denoted $X+_F Y$.

In equal characteristic, with
\[
F=k((\pi)),\qquad \mathcal O_F=k[[\pi]],\qquad |k|=q,
\]
a formal module has height $h$ if
\[
[\pi]_F(X)=f(X^{q^h})
\]
for a power series $f$ satisfying $f'(0)\neq 0$. Fix a formal module $\Sigma/\overline{k}$ of height $h. A deformation of $\Sigma$ over a complete local $\widehat{\mathcal O}_F$-algebra is a formal $\mathcal O_F$-module together with an identification of its special fiber with $\Sigma$.

The deformation functor is represented by a universal deformation ring
\[
\mathcal A\simeq \widehat{\mathcal O}_F[[u_1,\ldots,u_{h-1}]].
\]
The level-zero formal Lubin–Tate space is therefore
\[
X(1)=\operatorname{Spf}\mathcal A,
\]
and its rigid generic fiber
\[
\mathfrak X(1)
\]
is the open unit polydisc of dimension $h-1$. In mixed characteristic, the corresponding formal deformation space is often written
\[
\widehat{\mathcal O}^{\mathrm{nr}}[[x_1,\ldots,x_{h-1}]],
\]
and its analytic generic fiber has the same relative dimension.

In equal characteristic one may choose an explicit universal deformation with
\[
X+_{F^{\mathrm{univ}}Y=X+Y,\qquad [\zeta]_{F^{\mathrm{univ}}}(X)=\zeta X\quad(\zeta\in k),
\]
and
\[
[\pi]_{F^{\mathrm{univ}}}(X)
=\pi X+u_1X^q+\cdots+u_{h-1}X^{q^{h-1}}+X^{q^h}.
\]
This coordinate presentation is particularly useful for constructing explicit affinoids and computing their reductions.

A related formulation uses a formal $\mathcal O$-module $H$ over an algebraically closed residue field and a quasi-isogeny
\[
\rho:H'\to H.
\]
The resulting formal scheme decomposes into components indexed by the $\mathcal O$-height of the quasi-isogeny:
\[
\widehat{\mathcal M}=\coprod_{i\in\mathbf Z}\widehat{\mathcal M}_i.
\]
Each component is noncanonically isomorphic to the height-zero component, and has relative dimension $h-1$. This quasi-isogeny formulation is the standard Rapoport–Zink interpretation of the Lubin–Tate deformation problem.

## 2. Drinfeld levels and the Lubin–Tate tower

For $n\geq 1$, a Drinfeld level $\pi^n$ structure on a formal module $F$ over a complete local ring $A$ is an $\mathcal O_F$-module homomorphism
\[
\phi:(\pi^{-n}\mathcal O_F/\mathcal O_F)^{\oplus h}\longrightarrow \mathfrak m_A
\]
such that
\[
\prod_{x\in(\pi^{-n}\mathcal O_F/\mathcal O_F)^{\oplus h}}
\bigl(X-\phi(x)\bigr)
\]
divides $[\pi^n]_F(X)$. The images of the standard basis vectors form a Drinfeld basis
\[
X_1^{(n)},\ldots,X_h^{(n)}
\]
of the $\pi^n$-torsion.

The moduli functor of triples
\[
(F,\iota,\phi),
\]
where $F$ is a deformation of $\Sigma$, $\iota$ identifies its special fiber with $\Sigma$, and $\phi$ is a Drinfeld level structure, is represented by a regular local ring
\[
\mathcal A(\pi^n)
\]
of relative dimension $h-1. The corresponding formal and rigid spaces are
\[
X(\pi^n)=\operatorname{Spf}\mathcal A(\pi^n),
\qquad
\mathfrak X(\pi^n)=X(\pi^n)^{\mathrm{rig}}.
\]

The transition maps are finite:
\[
\mathfrak X(\pi^{n+1})\longrightarrow\mathfrak X(\pi^n),
\]
and satisfy
\[
[\pi]_{F^{\mathrm{univ}}}\bigl(X_i^{(n+1)}\bigr)=X_i^{(n)}.
\]
The Lubin–Tate tower is
\[
\{\mathfrak X(\pi^n)\}_{n\geq 0}.
\]

At finite level, changing the Drinfeld basis gives a right action of
\[
\mathrm{GL}_h(\mathcal O_F/\pi^n).
\]
If $M=(a_{ij})$, its action on coordinate functions is
\[
M\bigl(X_i^{(n)}\bigr)
=
[a_{i1}]_{F^{\mathrm{univ}}}\bigl(X_1^{(n)}\bigr)
+_{F^{\mathrm{univ}}}\cdots
+_{F^{\mathrm{univ}}}
[a_{ih}]_{F^{\mathrm{univ}}}\bigl(X_h^{(n)}\bigr).
\]
The map
\[
\mathfrak X(\pi^n)\longrightarrow\mathfrak X(1)
\]
is Galois with group $\mathrm{GL}_h(\mathcal O_F/\pi^n)$.

At infinite level, the inverse limit of the finite-level tower is a perfectoid space. A point is represented by a deformation together with a trivialization of its rational Tate module. In the height-two case, the infinite-level space classifies triples
\[
(G,\rho,\alpha),
\]
where $\alpha:\mathbf Z_p^2\to T(G)$ is a full Tate-module basis. More generally, the infinite-level tower carries a $\mathrm{GL}_h(F)$-action through changes of level structure.

The infinite-level geometry can also be described using the determinant morphism to the height-one Lubin–Tate tower. In suitable coordinates, the completed coordinate ring is expressed using compatible $q$-power roots of variables and a determinant-like alternating series $\delta$; the perfectoid space is locally cut out by equations of the form
\[
\delta(X)=t,
\]
where $t$ is a compatible system of height-one Lubin–Tate torsion points.

## 3. Group actions, components, and period maps

Let
\[
\mathcal O_B=\operatorname{End}(\Sigma),\qquad
B=\mathcal O_B\otimes_{\mathcal O_F}F.
\]
Then $B$ is the central division algebra over $F$ of invariant $1/h$. If $E/F$ is the unramified extension of degree $h$, $\mathcal O_B$ is generated by $\mathcal O_E$ and an element $\Phi$ satisfying
\[
\Phi^h=\pi,\qquad \Phi\alpha=\alpha^q\Phi.
\]
The group $\mathcal O_B^\times$ acts through automorphisms of the framing object. After descent and base change, the Weil group $W_F$ acts on the tower. Consequently, the cohomology carries commuting actions, schematically,
\[
\mathrm{GL}_h(F)\times B^\times\times W_F.
\]

The connected components are controlled by a determinant relation. For suitable $(g,b,\tau)$, the action on components is given by
\[
(g,b,\tau)\longmapsto
\det(g)\,\operatorname N_{B/F}(b)^{-1}\,\chi(\tau)^{-1},
\]
where $\chi$ is the local reciprocity character. Equivalent formulations use
\[
\det(g)^{-1}\operatorname{Nrd}_{B/F}(b)\operatorname{Art}_F^{-1}(\tau).
\]
The scalar subgroup acts trivially in the corresponding product action.

The Gross–Hopkins period morphism is an étale, surjective, equivariant map
\[
\Phi:X_0^{\mathrm{rig}}\longrightarrow \mathbf P^{h-1},
\]
constructed from global sections of the Lie-algebra line bundle of the universal deformation. It identifies the natural equivariant line bundles with pullbacks:
\[
(\mathcal M_0^s)^{\mathrm{rig}}
\simeq
\Phi^*\mathcal O_{\mathbf P^{h-1}}(s).
\]
At level $m$, with $\Phi_m=\Phi\circ\pi_{m,0}$,
\[
(\mathcal M_m^s)^{\mathrm{rig}}
\simeq
\Phi_m^*\mathcal O_{\mathbf P^{h-1}}(s).
\]

The period map is not always injective on natural critical discs. In height $2$, for example, the Gross–Hopkins period map is not injective on the first critical disc
\[
\Delta_0=\{|u|\leq|\pi|^{1/(q+1)}\}.
\]
Nevertheless, analytic group actions can be constructed directly on such discs. A level-zero subgroup of a nonsplit torus acts analytically on $\Delta_0$, and suitable wide-open rigid groups act analytically on larger critical discs associated with quasi-canonical liftings [1503.03580].

The action on spaces of global sections produces locally analytic representations. For all levels $m$ and integers $s$, the action of $\operatorname{Aut}(H_0)$ on
\[
M_m^s=H^0\left(X_m^{\mathrm{rig}},(\mathcal M_m^s)^{\mathrm{rig}}\right)
\]
is locally $K$-analytic. At level zero, the locally finite vectors are precisely the pullbacks of algebraic sections on projective space:
\[
(M_0^s)^{\mathrm{lf}}
\simeq
H^0\left(\mathbf P^{h-1}_{\widehat K},\mathcal O_{\mathbf P^{h-1}}(s)\right)
\]
for $s\geq 0$, and vanish for $s<0$. At positive level,
\[
(M_m^s)^{\mathrm{lf}}
\simeq
K_m\otimes_K(M_0^s)^{\mathrm{lf}},
\]
where $K_m$ is the $m$-th Lubin–Tate extension [1711.06858].

## 4. Affinoids, reductions, and explicit geometry

A major line of research constructs explicit affinoid subspaces of Lubin–Tate towers and computes their reductions. These affinoids are often centered at canonical or CM points and are designed so that their formal models have smooth Artin–Schreier-type special fibers.

In equal characteristic, the “too supersingular” region is defined by valuation inequalities
\[
v(u_i)\geq
\frac{q^h-q^i}{q^h-1}.
\]
After scaling torsion coordinates by a suitable element $\varpi$, the level-$\pi$ reduction becomes
\[
\mu(V_1,\ldots,V_h)^{q-1}=1,
\]
where $\mu$ is the Moore determinant. This is a Deligne–Lusztig-type reduction.

At level $\pi^2$, an affinoid associated with an unramified canonical point and a subgroup
\[
K_{x,2}=1+\mathfrak p_E^2+\mathfrak p_E C
\]
has an étale cover with group
\[
H\simeq\mathbf F_{q^h}.
\]
Suitable invariant coordinates yield an Artin–Schreier equation
\[
Y^{q^h}-Y=d(V_1,\ldots,V_{h-1}),
\]
and the reduction is the smooth affine hypersurface
\[
\det\left(
\begin{matrix}
V_1^{q^h}-V_1 & \cdots & V_h^{q^h}-V_h\\
1 & V_1^q & \cdots & V_{h-1}^q\\
0 & 1 & \cdots & V_{h-2}^{q^2}\\
\vdots & & \ddots & \vdots\\
0 & \cdots & 1 & V_1^{q^{h-1}}
\end{matrix}
\right)=0.
\]
For $h=2$, this becomes, after a change of variables, a disjoint union of Hermitian curves
\[
Y+Y^q=V^{q+1}.
\]
The associated cohomology is conjecturally concentrated in the middle degree for primitive Artin–Schreier characters, with Frobenius eigenvalue of absolute size
\[
q^{h(h-1)/2}
\]
[1001.3226].

In the tame simple-supercuspidal setting, CM-centered affinoids in the Lubin–Tate perfectoid space have reductions equal to perfections of quadratic Artin–Schreier varieties:
\[
z^q-z=\sum_{1\leq i\leq j\leq n-1}y_iy_j.
\]
Their middle cohomology carries the nontrivial Artin–Schreier characters, and Frobenius acts through quadratic Gauss sums. These varieties provide the geometric input for realizing simple supercuspidal local Langlands and Jacquet–Langlands correspondences [1308.1276].

The wild case requires modified points that are generally not CM points. Correction terms encode the wild extension appearing in the Langlands parameter. The resulting reduction has the form
\[
z^{p^m}-z
=
y^{p^e+1}
-\frac1{n'}\sum_{1\leq i\leq j\leq n-2}y_i y_j.
\]
Its cohomology realizes primitive Weil representations whose inertia action factors through a noncommutative semidirect product with Heisenberg-type structure [1603.04693].

For totally ramified essentially tame parameters with minimal jump $\nu$, affinoids $Z_\nu$ have reductions that are Artin–Schreier covers of the hyperplane
\[
y_1+\cdots+y_n=0.
\]
For odd $\nu$, the defining equation is a quadratic Artin–Schreier equation. For even $\nu$, the reduction involves a Heisenberg-type algebraic group and a Lang torsor. When $(n,\nu)=1$, the middle cohomology realizes the relevant $\mathrm{GL}_n(K)$, $D^\times$, and $W_K$ representations [1609.02524].

In height two and odd equal characteristic, affinoids around unramified CM points reduce to curves such as
\[
X^{q^2}-X-\bigl(Y^{q(q+1)}-Y^{q+1}\bigr)^{q^{k-1}}=0,
\]
while ramified CM affinoids reduce, in the relevant odd levels, to
\[
a^q-a=s^{q^{m+1}}.
\]
Their cohomology injects into the cohomology of the Lubin–Tate curve through Huber’s formal nearby-cycle comparison. This yields a purely local realization of the non-abelian Lubin–Tate theory for $\mathrm{GL}_2$ in odd equal characteristic [1604.08857].

## 5. Cohomology and local Langlands

The compactly supported étale cohomology of a fixed component of the Lubin–Tate tower has a constrained range. For a height-$n$ component,
\[
H_c^i\neq 0
\quad\Longrightarrow\quad
n-1\leq i\leq 2(n-1),
\]
whereas ordinary cohomology satisfies
\[
H^i\neq 0
\quad\Longrightarrow\quad
0\leq i\leq n-1.
\]
At infinite level, the direct limit of cohomology carries a smooth action of
\[
\mathrm{GL}_n(F)\times D^\times\times W_F.
\]

The cohomology realizes the local Langlands and Jacquet–Langlands correspondences. Suitable isotypic components correspond simultaneously to irreducible admissible representations of $\mathrm{GL}_n(F)$, representations of $D^\times$, and $n$-dimensional Weil–Deligne representations. The resulting theorem was established using global methods by Boyer in positive characteristic and Harris–Taylor in mixed characteristic; explicit affinoid constructions seek local geometric realizations of particular representations.

For simple supercuspidal representations in the tame case, compact induction from the cohomology of affinoid reductions produces
\[
\operatorname{Hom}_{\mathrm{GL}_n(K)}(\Pi,\pi)
\simeq
\operatorname{JL}(\pi)\otimes\operatorname{LL}(\pi)
\]
for essentially simple supercuspidal $\pi$, and vanishes for the other smooth irreducible supercuspidal representations considered in the construction [1308.1276]. The wild construction gives the analogous statement for essentially simple supercuspidals whose Weil parameters are not induced from characters of degree-$n$ extensions [1603.04693].

The trace of group actions can also be computed geometrically. If $\gamma=(g,b)$ consists of regular elliptic elements satisfying the component-preserving condition
\[
v_p(\det g)+v_p(N_{D/F}^{\mathrm{rd}}(b))=0,
\]
then
\[
\operatorname{Tr}\left(
\gamma\mid H_c^\ast(\mathcal M_{j,K}\times\mathbf C_p,\mathbf Q_\ell)
\right)
=
\#\operatorname{Fix}\left(
\gamma\mid\mathcal M_{j,K}\times\mathbf C_p
\right).
\]
The fixed-point count can be expressed using stable conjugacy and orbital integrals. Fargues’s locally finite cell decomposition, together with the metric geometry of the Bruhat–Tits building, provides an exhaustion by compact analytic domains whose boundary contributes no alternating trace. Mieda’s Lefschetz theorem then yields the formula [1203.2539].

A global proof of integral freeness shows that, for $p\neq l$, the cohomology groups
\[
H_{K,\overline{\mathbf Z}_l,h}^i
\]
are torsion-free for $0\leq i\leq h-1$. The proof uses Kottwitz–Harris–Taylor Shimura varieties, Newton stratifications, Harris–Taylor local systems, the two integral perverse $t$-structures $p$ and $p+$, and mirabolic derivatives. The result gives torsion-free $\overline{\mathbf Z}_l$-lattices compatible with the $\mathrm{GL}_h(K)$ and $D^\times$ actions [1309.1946; 1611.02082].

More recent work compares compactly supported de Rham cohomology of Lubin–Tate and Drinfeld towers. At infinite level, the compactly supported de Rham cohomology groups are isomorphic as representations of
\[
\mathrm{GL}_{d+1}(L)\times D^\times,
\]
where $D$ has invariant $1/(d+1)$ [2503.09352].

## 6. Perfectoid quotients and representation-theoretic geometry

The infinite-level Lubin–Tate space admits important quotients by subgroups of $\mathrm{GL}_n(F)$. For height two over $\mathbf Q_p$, the quotient by the upper triangular Borel
\[
B(\mathbf Q_p)
\]
exists as a perfectoid space. The construction first establishes perfectoidness of
\[
M_\infty^{(0)}/B(\mathbf Z_p)
\]
and then uses the Gross–Hopkins period map and induced quotient constructions to obtain
\[
M_\infty/B(\mathbf Q_p).
\]
The quotient is quasi-compact and partially proper, and the induced Gross–Hopkins map has affinoid perfectoid fibers [1611.03685].

The quotient by the maximal parabolic
\[
P_{n-1,1}(K)
\]
also exists as a perfectoid space for arbitrary finite $K/\mathbf Q_p$ and arbitrary $n$. It has dimension $n-1$, the Gross–Hopkins map admits local sections, and the projection
\[
M_1\longrightarrow M_1/P_{n-1,1}(K)
\]
is a $P_{n-1,1}(K)$-torsor. The construction uses anticanonical towers, canonical subgroups, Harris–Taylor Shimura varieties, and Hodge–Tate period maps [1812.08203].

This perfectoid quotient has consequences for Scholze’s mod-$p$ cohomological functor. If $\pi$ is induced from a parabolic contained in $P_{n-1,1}(K)$, then
\[
\mathcal S^i(\pi)=0
\qquad\text{for }i>n-1.
\]
For $\mathrm{GL}_2(\mathbf Q_p)$, this gives vanishing in degree $2$ for principal series and twists of the Steinberg representation. The result is obtained by replacing the equivariant sheaf attached to a parabolically induced representation with a local system on the perfectoid quotient.

The perfectoidness result is essentially optimal. For other maximal parabolics $P_{n-d,d}$ with $d>1$, the corresponding quotient is not perfectoid because it contains a genuine projective-space direction
\[
\mathbf P_C^{d-1}.
\]
Thus
\[
M_1/P_{n-d,d}(K)
\text{ is perfectoid precisely in the case }d=1.
\]

## 7. Related theories and arithmetic applications

The term “Lubin–Tate theory” also appears in chromatic homotopy theory. There, a one-dimensional smooth formal group of height $h$ determines Morava $E$-theory with coefficient ring
\[
E_0\simeq W(k)[[u_1,\ldots,u_{h-1}]],
\qquad
E_*\simeq E_0[\beta^{\pm1}],
\]
and Morava $K$-theory is obtained by reduction. The Lubin–Tate deformation ring supplies the coefficient ring and formal group law, while power operations and Hecke Lie algebras are used to compute the $E$-theory of configuration spaces. These configuration spaces are not Lubin–Tate moduli spaces; rather, Lubin–Tate theory supplies the chromatic cohomology theory applied to them [1908.11321].

Lubin–Tate spaces also support intersection theory. For a height-$2h$ formal module, special cycles associated with quadratic endomorphism fields are middle-dimensional. Their intersection multiplicities are governed by invariant polynomials and resultants. If $R$ is the resultant of the corresponding invariant polynomials, the large-level intersection formula is
\[
\operatorname{Int}_n(\gamma,g)
=
|R|^{-1}.
\]
At finite level, the intersection is expressed through an integral of $|R(g)|$ over a principal congruence subgroup. These formulas lead to biquadratic fundamental lemmas relating orbital integrals and derivatives of orbital integrals to intersection multiplicities [2010.07365].

The same intersection numbers admit a non-archimedean analytic interpretation through Green $\delta$-forms. If a special cycle is a complete intersection defined by a regular sequence and $\phi$ is its valuation-theoretic distance function, then
\[
\gamma=(-1)^{r-1}\phi(d'd''\phi)^{r-1}
\]
is a Green $\delta$-form, satisfying
\[
\delta_Z+d'd''[\gamma]=0.
\]
For Lubin–Tate special cycles, integrating this form over the complementary cycle gives the formal intersection length and ultimately the inverse absolute value of a resultant [2112.10018].

Finally, Lubin–Tate formal groups govern generalized Coleman power series. Given two Lubin–Tate formal group laws $F_L$ and $F_K$, a power series $r(x)$ interpolating an $F_L$-norm-compatible sequence in the $F_K$-Lubin–Tate tower satisfies
\[
r(x)+_L r(x+_Kz_1)+_L\cdots+_Lr(x+_Kz_{q_K-1})
=
[a]_L\bigl(r([\varpi_K]_K(x))\bigr).
\]
Applying the formal logarithm of $F_L$ converts this nonlinear compatibility equation into an eigenspace equation for Coleman’s trace operator:
\[
\mathcal L_{F_K}(\log_{F_L}r)=a\,\log_{F_L}r.
\]
Away from the exceptional value
\[
a=\frac{q_K}{\varpi_L^{f_{K/L}}},
\]
the generalized Coleman series are controlled by trace-zero or trace-eigenvector spaces [2410.10029].

The Lubin–Tate space therefore links deformation theory, rigid and perfectoid geometry, non-archimedean intersection theory, locally analytic representation theory, local Langlands correspondences, $(\varphi,\Gamma)$-modules, and chromatic homotopy theory. Its defining moduli problem is elementary in formulation—deformations of a one-dimensional formal module—but its tower carries the geometric and cohomological structures through which these apparently distinct theories interact.

Source: https://www.emergentmind.com/topics/lubin-tate-space