Galois Deformation Rings

In Galois Representations we talked about obtaining continuous Galois representations for example from the $\ell$ -adic etale cohomology of algebraic varieties, and hinted at being able to obtain such Galois representations from modular forms as well. While we postpone the discussion of how to obtain such a Galois representation to some future blog post (hopefully), we now mention the very important topic of modularity – which investigates, given some Galois representation, whether it comes from a modular form, and furthermore whether it provides some other information about the modular form that it comes from.

The topic of modularity is composed of two parts. The first is residual modularity – where we are given a Galois representation over a finite field (we call such a Galois representation a residual representation, in reference to the finite field being the residue field of some other ring) and figure out whether it comes from a modular form (in which case we also say that it is modular). The second part is modularity lifting, where, given a residual representation we know to be modular, we figure out whether it “lifts” to a Galois representation over $\mathbb{Q}_{\ell}$ .

In this post, we focus only on one small ingredient of the approach to proving modularity lifting. Proofs of modularity lifting rely on “R=T” theorems, where R refers to a Galois deformation ring and T comes from a (localization of) a Hecke algebra (see also Hecke Operators). The small ingredient we will focus on in this post is the R, the Galois deformation ring.

A “deformation” in our context is an equivalence class of “lifts” and before we give the precise definitions we give a little bit of intuition about why we are interested in lifts. Roughly, in our context, a lift of some field $\overline{R}$ is a local ring $R$ such that $\overline{R}$ is the residue field of $R$ , i.e. $\overline{R}=R/\mathfrak{m}$ where $\mathfrak{m}$ is the unique maximal ideal of $R$ (since $R$ is a local ring by definition it has a unique maximal ideal).

So now for the intuition. Consider the real numbers $\mathbb{R}$ . The “dual numbers” are defined to be $\mathbb{R}[x]/(x^{2})$ . Its elements are of the form $a+bx$ where $a$ and $b$ are real numbers. We can consider $x$ here to be an “infinitesimal element”. So we may think of an element of the dual numbers to be a number, given by $a$ , but with a “tangent vector” given by the number $b$ . Another way to think about it is that is at “position $a$ “, but it also has a “velocity $b$ “. It’s like numbers, but with a little “wiggle”. Now that we know about the dual numbers $\mathbb{R}[x]/(x^{2})$ , what about elements of $\mathbb{R}[x]/(x^{3})$ ? We may think of such an element, which is of the form $a+bx+cx^{2}$ , to be a position “ $a$ “, with “velocity $b$ “, and “acceleration $c$ “, a kind of “higher wiggle”.

If we continue including higher and higher derivatives, then we have something whose elements are formal power series $a+bx+cx^2+dx^3+\ldots$ . This is the ring $\mathbb{R}[[x]]$ , which is the inverse limit of the rings $\mathbb{R}/(x^{n})$ . Now the ring $\mathbb{R}[[x]]$ is a local ring with maximal ideal $(x)$ , and modding out by this maximal ideal gives $\mathbb{R}$ . So this power series ring is a lift of $\mathbb{R}$ , kind of numbers with “higher wiggles”. This is what the term “deformation” is supposed to bring to mind.

We now give more precise definitions. Let $F$ be a finite extension of $\mathbb{Q}$ , and let $k$ be a finite field. A Galois representation $\overline{\rho}:\text{Gal}(\overline{F}/F)\to \text{GL}_{2}(k)$ is also called a residual representation. Now let $W(k)$ be the ring of Witt vectors of $k$ ; for example, if $k=\mathbb{F}_{p}$ , then $W(k)=\mathbb{Z}_{p}$ . A lift, or framed deformation of the residual representation $\overline{\rho}$ is a Galois representation $\overline{\rho}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_{n}(A)$ where $A$ is a complete Noetherian local $W(k)$ -algebra, such that modding out by the unique maximal ideal of $A$ gives the residual representation $\overline{\rho}$ . A deformation of $\overline{\rho}$ is an equivalence class of lifts of $\overline{\rho}$ , where two lifts are considered equivalent if they are conjugates under the kernel of the modding out map.

Consider the functor $\text{Def}_{\overline{\rho}}^{\Box}$ from the category of complete Noetherian local $W(k)$ -algebras to the category of sets, which assigns to a complete Noetherian local $W(k)$ -algebra $A$ the set of all its lifts. This functor happens to be representable, i.e. there is a Galois representation $\overline{\rho}:\text{Gal}(\overline{F}/F)\to \text{GL}_{n}(R_{\overline{\rho}}^{\Box})$ over some ring $R_{\overline{\rho}}^{\Box}$ called the universal framed deformation ring, such that the lifts of $\overline{\rho}$ are given by maps from the Galois deformations to the universal Galois deformation.

We can also do the same for deformations instead of framed deformations, as long as our residual representation satisfies a condition called “Schur’s condition”.

We can also impose conditions on our deformations – for instance, we may want to consider only lifts with a certain fixed determinant. These conditions are also called deformation problems and they are important because it is conjectured that Galois representations coming from modular forms have certain properties, and we want to match up these Galois representations with modular forms.

Roughly, the way these are matched up goes in the following manner. We have said above that deformations of a certain fixed Galois representation $\overline{\rho}$ to $A$ , possibly with some conditions, correspond to maps $R_{\overline{\rho},\mathrm{conditions}}\to A$ . We state that, given an isomorphism between the complex numbers and the p-adic complex numbers we can always construct a map $R_{\overline{\rho}, \mathrm{conditions} }\to \mathbb{C}$ from the preceding map.

Now a Hecke algebra $\mathbb{T}$ acts on Hecke eigenforms (which say we want to match up with the Galois representations, to show that these Galois representations come from them) and therefore have associated systems of eigenvalues. It is known that any such system of eigenvalues comes from some Hecke eigenform.

We choose only a localization of the Hecke algebra, which we call $\mathbb{T}_{\mathfrak{m}}$ , corresponding to only the modular forms that are expected to give rise to the Galois representations we are considering (the Eichler-Shimura theorem gives relations between the Fourier coefficients of the Hecke eigenform and the form of the characteristic polynomial of the Frobenius under the Galois representation, restricting it). On the other hand, these systems of eigenvalues corresponds to maps $\mathbb{T}_{\mathfrak{m}}\to \mathbb{C}$ .

So if we can show that $R_{\overline{\rho}, \mathrm{conditions} }=\mathbb{T}_{\mathfrak{m}}$ , then these two sets of maps to $\mathbb{C}$ match up, then we can show that these Galois representations come from modular forms. Showing that $R_{\overline{\rho}, \mathrm{conditions} }=\mathbb{T}_{\mathfrak{m}}$ is itself an elaborate process that involves a fascinating strategy pioneered by Richard Taylor and Andrew Wiles known as patching. We will hopefully discuss R=T theorems, and the method of patching, on this blog in more detail in the future.

References:

Deformation on Wikipedia

Modularity Lifting Theorems by Toby Gee

Modularity Lifting (Course Notes) by Patrick Allen

Motives and L-functions by Frank Calegari

Beyond the Taylor-Wiles Method by Jack Thorne

Perturbations, Deformations, and Variations (and “Near-Misses”) in Geometry, Physics, and Number Theory by Barry Mazur

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