Deformation ring

In mathematics, a deformation ring is a ring that controls liftings of a representation of a profinite group (usually a Galois group) from a finite field to a local ring. In particular, for any such lifting problem there is often a universal deformation ring that classifies all such liftings, and whose spectrum is the universal deformation space.

A key step in Wiles's proof of the modularity theorem was to study the relation between universal deformation rings and Hecke algebras.

Coefficient rings and the deformation functor

For a (fixed) finite field $k$, a coefficient ring for $k$ is a commutative Noetherian local ring $(R,{\mathfrak {m}})$ which is complete in the ${\mathfrak {m}}$-adic topology, whose residue field is identified with $k$. Write $\pi _{R}:R\twoheadrightarrow k$ for the quotient map. A morphism of coefficient rings $\varphi :R\to R'$ is a continuous homomorphism of local rings such that $\pi _{R}=\pi _{R'}\circ \varphi$. The category of coefficient rings will be denoted ${\mathcal {C}}$.

For example, if $k=\mathbb {F} _{p}$ is the finite field with $p$ elements, then the ring of $p$-adic integers $\mathbb {Z} _{p}$ as well as its associated formal power series ring $\mathbb {Z} _{p}[\![X]\!]$ are both coefficient rings for $k$, and the inclusion map $\mathbb {Z} _{p}\to \mathbb {Z} _{p}[\![X]\!]$ is a morphism of coefficient rings.

The $\pi _{R}$ maps induce maps $(\pi _{R})_{*}:\mathrm {GL} _{n}(R)\to \mathrm {GL} _{n}(k)$, whose kernel is denoted $\Gamma _{n}(R)$. Thus, any group representation $\rho :G\to \mathrm {GL} _{n}(R)$ gives us a related $k$-valued representation ${\overline {\rho }}:G\to \mathrm {GL} _{n}(k)$ via ${\overline {\rho }}:=\pi _{R}\circ \rho$. Such ${\overline {\rho }}$ is called the residual representation of $\rho$.

Two continuous representations $\rho _{1},\rho _{2}:\Pi \to \mathrm {GL} _{n}(R)$ of a profinite group $\Pi$ are said to be strictly equivalent if there exists $M\in \Gamma _{n}(R)$ such that ${\textstyle M^{-1}\rho _{1}M=\rho _{2}.}$ This is stronger than the usual isomorphism of representations, since we require that the isomorphism $\rho _{1}{\stackrel {\sim }{\rightarrow }}\rho _{2}$ induces the identity map on residual representations.

A deformation of ${\overline {\rho }}:\Pi \to \mathrm {GL} _{n}(k)$ to a coefficient ring $R$ is a strict equivalence class of continuous lifts of ${\overline {\rho }}$ through the reduction map $(\pi _{R})_{*}:\mathrm {GL} _{n}(R)\to \mathrm {GL} _{n}(k)$. Informally, any continuous lift ${\displaystyle \rho$ :\Pi \to \mathrm {GL} _{n}(R)} may be referred to as a deformation of ${\overline {\rho }}$, but in this sense two deformations are considered the same if and only if the underlying representations are strictly equivalent. Let $\mathrm {Def} _{\overline {\rho }}(R)$ denote the set of deformations of ${\overline {\rho }}$ to $R$.

Any morphism of coefficient rings $\varphi :R\to R'$ must send $\Gamma _{n}(R)$ to $\varphi _{*}[\Gamma _{n}(R)]\subseteq \Gamma _{n}(R')$, so if we fix a residual deformation ${\overline {\rho }}$, we see that composition with $\varphi _{*}$sends $R$-valued deformations of ${\overline {\rho }}$ to $R'$-valued deformations of ${\overline {\rho }}$. In other words, $\mathrm {Def} _{\overline {\rho }}$ defines a (covariant) functor $\mathrm {Def} _{\overline {\rho }}:{\mathcal {C}}\to \mathbf {Set}$.

Deformation rings

If we fix a residual representation ${\overline {\rho }}:\Pi \to \mathrm {GL} _{n}(k)$, then under certain conditions (most notably that the representation ${\overline {\rho }}$ admits only scalar automorphisms) one can show that the deformation functor $\mathrm {Def} _{\overline {\rho }}$ is representable, in that there exists some coefficient ring $R_{\overline {\rho }}$ such that $\mathrm {Def} _{\rho }(R')\cong \mathrm {Hom} _{\mathcal {C}}(R_{\overline {\rho }},R')$. Such a coefficient ring $R_{\overline {\rho }}$ is called a (universal) deformation ring for ${\overline {\rho }}$. Consequently, any deformation ${\displaystyle \rho$ :\Pi \to \mathrm {GL} _{n}(R_{\overline {\rho }})} of ${\overline {\rho }}$ must be of the form $\rho =\varphi _{*}\circ \rho _{\mathrm {univ} }$ for some deformation $\rho _{\mathrm {univ} }:\Pi \to \mathrm {GL} _{n}(R_{\overline {\rho }})$ and some $\varphi \in \mathrm {End} _{\mathcal {C}}(R_{\overline {\rho }})$. The pair $(R_{\overline {\rho }},\rho _{\mathrm {univ} })$ is called a universal deformation of ${\overline {\rho }}$.

More concisely, such a pair $(R_{\overline {\rho }},\rho _{\mathrm {univ} })$ is uniquely characterised by the following universal property: if $R$ is any coefficient ring and ${\displaystyle \rho$ :\Pi \to \mathrm {GL} _{n}(R)} is any deformation of ${\overline {\rho }}$ to $R$, then there is a unique ${\mathcal {C}}$-morphism $\varphi :R_{\overline {\rho }}\to R$ such that $\rho =\varphi _{*}\circ \rho _{\mathrm {univ} }$. As a consequence of this universal property, deformation rings are unique up to unique isomorphism.

Example in dimension 1

For $p$ an odd prime number and $\Pi =G_{\mathbb {Q} ,S}$ the absolute Galois group of the rational numbers with ramification restricted to $S=T\cup \{p\}$, where $T$ is the set of primes dividing $p-1$. Let ${\displaystyle \varepsilon$ :\Pi \to \mathbb {F} _{p}^{\times }} denote the $p$-adic cyclotomic character given by$$\Pi \twoheadrightarrow \mathrm {Gal} (\mathbb {Q} (\zeta _{p})/\mathbb {Q} )\,{\stackrel {\sim }{\to }}\,\mathbb {F} _{p}^{\times },$$with $\zeta _{p}=e^{2\pi i/p}$ a primitive $p$th root of unity. One can show in this case that deformation ring is exactly the completed group ring $\mathbb {Z} _{p}[\![\Gamma ^{\mathrm {ab} }]\!]$, where $\Gamma$ is the maximal pro-$p$-quotient of $\Pi$, and a universal deformation for $\varepsilon$ is given by $$\chi _{\mathrm {univ} }:G_{\mathbb {Q} }\to \mathbb {Z} _{p}[\![\Gamma ^{\mathrm {ab} }]\!]^{\times };\qquad \sigma \mapsto {\hat {\varepsilon }}(\sigma )\varpi (\sigma ),$$where ${\hat {\varepsilon }}:\Pi \to \mathbb {Z} _{p}^{\times }$ is the composition of $\varepsilon$ with the Teichmüller character, and ${\displaystyle \varpi$ :\Pi \twoheadrightarrow \Gamma ^{\mathrm {ab} }} is the quotient map.

See also

• Deformation (mathematics)
• Galois module
• Galois representation

References

• Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn, eds. (1997), Modular forms and Fermat's last theorem, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94609-2, MR 1638473
• Gouvêa, Fernando, "Deformations of Galois representations", IAS/Park City Mathematics Series, vol. 9, American Mathematical Society, pp. 233–406, doi:10.1090/pcms/009/05, ISBN 978-0-8218-8691-5{{citation}}: CS1 maint: url-status (link)