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 $\phi :R\to R^{\prime }$ is a continuous homomorphism of local rings such that ${\pi }_R={\pi }_{R^{\prime }}\circ \phi$. The category of coefficient rings will be denoted ${𝒞}$.

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

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

Two continuous representations ${\rho }_1,{\rho }_2:\mathrm{\Pi }\to {\mathrm{G}\mathrm{L}}_n(R)$ of a profinite group $\mathrm{\Pi }$ are said to be strictly equivalent if there exists $M\in {\mathrm{\Gamma }}_n(R)$ such that $M^{-1}{\rho }_{1}M={\rho }_2.$ This is stronger than the usual isomorphism of representations, since we require that the isomorphism ${\rho }_1\stackrel{\sim }{\to }{\rho }_2$ induces the identity map on residual representations.

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

Any morphism of coefficient rings $\phi :R\to R^{\prime }$ must send ${\mathrm{\Gamma }}_n(R)$ to ${\phi }_{*}[{\mathrm{\Gamma }}_n(R)]\subseteq {\mathrm{\Gamma }}_n(R^{\prime })$, so if we fix a residual deformation $\bar{\rho }$, we see that composition with ${\phi }_{*}$sends $R$-valued deformations of $\bar{\rho }$ to $R^{\prime }$-valued deformations of $\bar{\rho }$. In other words, ${\mathrm{D}\mathrm{e}\mathrm{f}}_{\bar{\rho }}$ defines a (covariant) functor ${\mathrm{D}\mathrm{e}\mathrm{f}}_{\bar{\rho }}:{𝒞}\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$.

Deformation rings

If we fix a residual representation $\bar{\rho }:\mathrm{\Pi }\to {\mathrm{G}\mathrm{L}}_n(k)$, then under certain conditions (most notably that the representation $\bar{\rho }$ admits only scalar automorphisms) one can show that the deformation functor ${\mathrm{D}\mathrm{e}\mathrm{f}}_{\bar{\rho }}$ is representable, in that there exists some coefficient ring $R_{\bar{\rho }}$ such that ${\mathrm{D}\mathrm{e}\mathrm{f}}_{\rho }(R^{\prime })\cong {\mathrm{H}\mathrm{o}\mathrm{m}}_{{𝒞}}(R_{\bar{\rho }},R^{\prime })$. Such a coefficient ring $R_{\bar{\rho }}$ is called a (universal) deformation ring for $\bar{\rho }$. Consequently, any deformation $\rho :\mathrm{\Pi }\to {\mathrm{G}\mathrm{L}}_n(R_{\bar{\rho }})$ of $\bar{\rho }$ must be of the form $\rho ={\phi }_{*}\circ {\rho }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}$ for some deformation ${\rho }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}:\mathrm{\Pi }\to {\mathrm{G}\mathrm{L}}_n(R_{\bar{\rho }})$ and some $\phi \in {\mathrm{E}\mathrm{n}\mathrm{d}}_{{𝒞}}(R_{\bar{\rho }})$. The pair $(R_{\bar{\rho }},{\rho }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}})$ is called a universal deformation of $\bar{\rho }$.

More concisely, such a pair $(R_{\bar{\rho }},{\rho }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}})$ is uniquely characterised by the following universal property: if $R$ is any coefficient ring and $\rho :\mathrm{\Pi }\to {\mathrm{G}\mathrm{L}}_n(R)$ is any deformation of $\bar{\rho }$ to $R$, then there is a unique ${𝒞}$-morphism $\phi :R_{\bar{\rho }}\to R$ such that $\rho ={\phi }_{*}\circ {\rho }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}$. 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 $\mathrm{\Pi }=G_{\mathbb{\mathbb{Q}},S}$ 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 $\epsilon :\mathrm{\Pi }\to {\mathbb{𝔽}}_p^{\times }$ denote the $p$-adic cyclotomic character given by$$\mathrm{\Pi }\twoheadrightarrow \mathrm{G}\mathrm{a}\mathrm{l}(\mathbb{\mathbb{Q}}({\zeta }_p)/\mathbb{\mathbb{Q}})\stackrel{\sim }{\to }{\mathbb{𝔽}}_p^{\times },$$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 ${\mathbb{\mathbb{Z}}}_p[[{\mathrm{\Gamma }}^{\mathrm{a}\mathrm{b}}]]$, where $\mathrm{\Gamma }$ is the maximal pro-$p$-quotient of $\mathrm{\Pi }$, and a universal deformation for $\epsilon$ is given by $${\chi }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}:G_{\mathbb{\mathbb{Q}}}\to {\mathbb{\mathbb{Z}}}_p[[{\mathrm{\Gamma }}^{\mathrm{a}\mathrm{b}}]{]}^{\times };\sigma \mapsto \hat{\epsilon }(\sigma )\varpi (\sigma ),$$where $\hat{\epsilon }:\mathrm{\Pi }\to {\mathbb{\mathbb{Z}}}_p^{\times }$ is the composition of $\epsilon$ with the Teichmüller character, and $\varpi :\mathrm{\Pi }\twoheadrightarrow {\mathrm{\Gamma }}^{\mathrm{a}\mathrm{b}}$ 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, https://books.google.com/books?id=Va-quzVwtMsC 
• Gouvêa, Fernando, "Deformations of Galois representations", IAS/Park City Mathematics Series (American Mathematical Society) 9: pp. 233-406, doi:10.1090/pcms/009/05, ISBN 978-0-8218-8691-5, https://pubs.ams.org/ebooks/pcms/009/