The Local Langlands Correspondence for General Linear Groups

The Langlands program is, roughly, a part of math that bridges the representation theory of reductive groups and Galois representations. It consists of many interrelated conjectures, among which are the global and local Langlands correspondences, which may be seen as higher-dimensional analogues of global and local class field theory (see also Some Basics of Class Field Theory) in some way. Just as in class field theory, the words “global” and “local” refer to the fact that these correspondences involve global and local fields, respectively, and their absolute Galois groups.

In this post we will discuss the local correspondence (the global correspondence requires the notion of automorphic forms, which involves a considerable amount of setup, hence is postponed to future posts). We will also only discuss the case of the general linear group \mathrm{GL}_{n}(F), for F a local field, and in particular \mathrm{GL}_{1}(F) and \mathrm{GL}_{2}(F) where we can be more concrete. More general reductive groups will bring in the more complicated notions of L-packets and L-parameters, and therefore also postponed to future posts.

Let us give the statement first, and then we shall unravel what the words in the statement mean.

The local Langlands correspondence for general linear groups states that there is a one-to-one correspondence between irreducible admissible representations of \mathrm{GL}_{n}(F) (over \mathbb{C}) and F-semisimple Weil-Deligne representations of the Weil group W_{F} (also over \mathbb{C}).

Let us start with “irreducible admissible representations of \mathrm{GL}_{n}(F)“. These representations are similar to what we discussed in Representation Theory and Fourier Analysis (in fact as we shall see, many of these representations are also infinite-dimensional, and constructed somewhat similarly). Just to recall, irreducible means that the only subspaces held fixed by \mathrm{GL}_{n}(F) are 0 and the entire subspace.

Admissible means that, if we equip \mathrm{GL}_{n}(F) with the topology that comes from the p-adic topology of the field F, for any open U subgroup of \mathrm{GL}_{n}(F) the fixed vectors form a finite-dimensional subspace.

Now we look at the other side of the correspondence. We already defined what a Weil-Deligne representation is in Weil-Deligne Representations. A Weil-Deligne representation (\rho_{0},N) is F-semisimple if the representation \rho_{0} is the direct sum of irreducible representations.

In the case of \mathrm{GL}_{1}(F), the local Langlands correspondence is a restatement of local class field theory. We have that \mathrm{GL}_1(F)=F^{\times}, and the only irreducible admissible representations of \mathrm{GL}_1(F) are continuous group homomorphisms \chi:F^\times\to\mathbb{C}^{\times}.

On the other side of the correspondence we have the one-dimensional Weil-Deligne representations (\rho_{0},N) of W_F, which must have monodromy operator N=0 and must factor through the abelianization W_F^{\mathrm{ab}}.

Recall from Weil-Deligne Representations that we local class field theory gives us an isomorphism \mathrm{rec}:F^{\times}\xrightarrow{\sim}W_{F}^{\mathrm{ab}}, also known as the Artin reciprocity map. We can now describe the local Langlands correspondence explicitly. It sends \chi to the Weil-Deligne representation (\rho_{0},0), where \rho_{0} is the composition W_{F}\to W_{F}^{\mathrm{ab}}\xrightarrow{\mathrm{rec}^{-1}}F^{\times}\xrightarrow{\chi}\mathbb{C}^{\times}.

Now let us consider the case of \mathrm{GL}_{2}(F). If the residue field of F is not of characteristic 2, then the irreducible admissible representations of \mathrm{GL}_{2}(F) may be enumerated, and they fall into four types: principal series, special, one-dimensional, and supercuspidal.

Let \chi_{1},\chi_{2}:F^{\times}\to\mathbb{C}^{\times} be continuous admissible characters and let I(\chi_{1},\chi_{2}) be the vector space of functions \phi:\mathrm{GL}_{2}(F)\to\mathbb{C} such that

\displaystyle \phi \left(\begin{pmatrix}a&b\\0&d\end{pmatrix}g\right)=\chi_{1}(a)\chi_{2}(d)\Vert a/d\Vert^{1/2}\phi(g)

The group \mathrm{GL}_{2}(F) acts on the functions \phi, just as in Representation Theory and Fourier Analysis. Therefore it gives us a representation of \mathrm{GL}_{2}(F) on the vector space I(\chi_{1},\chi_{2}), which we say is in the principal series.

Now the representation I(\chi_{1},\chi_{2}) might be irreducible, in which case it is one of the things that go into our correspondence, or it might be reducible. This is decided by the Bernstein-Zelevinsky theorem, which says that I(\chi_{1},\chi_{2}) is irreducible precisely if the ration of the characters \chi_{1} and \chi_{2} is not equal to plus or minus 1.

In the case that \chi_{1}/\chi_{2}=1, then we have an exact sequence

\displaystyle 0\to\rho\to I(\chi_{1},\chi_{2})\to S(\chi_{1},\chi_{2})\to 0

where the representations S(\chi_{1},\chi_{2}) and \rho are both irreducible representations of \mathrm{GL}_{2}(F). The representation S(\chi_{1},\chi_{2}) is infinite-dimensional and is known as the special representation. The representation \rho is the one-dimensional representation and is given by \chi_{1}\Vert\cdot\Vert^{1/2}\det.

If \chi_{1}/\chi_{2}=-1 instead, then we have a “dual” exact sequence

\displaystyle 0\to S(\chi_{1},\chi_{2}) \to I(\chi_{1},\chi_{2})\to \rho\to 0

So far the irreducible admissible representations of \mathrm{GL}_{2}(F) that we have seen all arise as subquotients of I(\chi_{1},\chi_{2}). Since characters such as \chi_{1} and \chi_{2} are the irreducible admissible representations of \mathrm{GL}_{1}(F), we may consider the principal series, special, and one-dimensional representations as being built out of these more basic building blocks.

However there exist irreducible admissible representations that do not arise via this process, and they are called supercuspidal representations. For \mathrm{GL}_{2}(F) there is one kind of supercuspidal representation denoted \mathrm{BC}_{E}^{F}(\psi) for E a quadratic extension of F and \psi an admissible character \psi:E\to\mathbb{C}^{\times}.

Now we know what the irreducible admissible representations of \mathrm{GL}_{2}(F) are. The local Langlands correspondence says that they will correspond to F-semisimple Weil-Deligne representations. We can actually describe explicitly which Weil-Deligne representation each irreducible admissible representation of \mathrm{GL}_{2}(F) gets sent to!

Let \chi_{1},\chi_{2}:F^{\times}\to \mathbb{C}^{\times} be the same continuous admissible characters used to construct the irreducible representations as above, and let \rho_{1},\rho_{2} :W_{F}\to \mathbb{C}^{\times} be the corresponding representation of the Weil group given by the local Langlands correspondence for \mathrm{GL}_{1}, as discussed earlier. Then to each irreducible admissible representation of \mathrm{GL}_2(F) we associate a 2-dimensional Weil-Deligne representation as follows:

To the principal series representation I(\chi_{1},\chi_{2}) we associate the Weil-Deligne representation (\rho_{1}\oplus\rho_{2},0).

To the special representation S(\chi_{1},\chi_{1}\times\Vert\cdot\Vert), we associate the Weil-Deligne representation \left(\begin{pmatrix}\Vert\cdot\Vert\rho_{1} & 0\\0 & \rho_{1}\end{pmatrix},\begin{pmatrix} 0 & 1\\0 & 0\end{pmatrix}\right).

To the one-dimensional representation \chi_{1}\circ\det, we associate the Weil-Deligne representation \left(\begin{pmatrix}\rho_{1}\times\Vert\cdot\Vert^{1/2} & 0\\0 & \rho_{1}\times\Vert\cdot\Vert^{-1/2}\end{pmatrix},0\right).

Finally, to the supercuspidal representation \mathrm{BC}_{E}^{F}(\psi) we associate the Weil-Deligne representation (\mathrm{Ind}_{W_{E}}^{W_{F}}\sigma,0), where \sigma is the unique nontrivial element of \mathrm{Gal}(E/F).

We have been able to describe the local Langlands correspondence for \mathrm{GL}_{1}(F) and \mathrm{GL}_{2}(F) explicitly (in the latter case as long as the characteristic of the residue field of F is not 2). The local Langlands correspondence for \mathrm{GL}_{n}(F), for more general n on the other hand, was proven via geometric means Рnamely using the geometry of certain Shimura varieties (see also Shimura Varieties) as well as their local counterpart, the Lubin-Tate tower, which parametrizes deformations of Lubin-Tate formal group laws (see also The Lubin-Tate Formal Group Law) together with level structure.

There has been much recent work regarding the local Langlands program for groups other than \mathrm{GL}_{n}(F). For instance there is work on the local Langlands correspondence for certain symplectic groups making use of “theta lifts”, by Wee Teck Gan and Shuichiro Takeda. Very recently, there has also been work by Laurent Fargues and Peter Scholze that makes use of ideas from the geometric Langlands program. These, and more, will hopefully be discussed more here in the future.

References:

Langlands program on Wikipedia

Local Langlands conjectures on Wikipedia

Local Langlands conjecture on the nLab

MSRI Summer School on Automorphic Forms and the Langlands Program by Kevin Buzzard

Langlands Correspondence and Bezrukavnikov Equivalence by Anna Romanov and Geordie Williamson

The Local Langlands Conjecture for GL(2) by Colin Bushnell and Guy Henniart

Weil-Deligne Representations

Let F be a finite extension of the p-adic numbers \mathbb{Q}_{p}. In Galois Representations we described some continuous Galois representations of \mathrm{Gal}(\overline{F}/F), but all of them were p-adic (or rather \ell-adic, see the discussion in that post for the explanation behind the terminology). What about complex Galois representations? For instance, since the complex \ell-adic numbers (the completion of the algebraic closure of the \ell-adic numbers) are isomorphic to the complex numbers, if we fix such an isomorphism we could just base change to the complex numbers to get a complex Galois representation.

Complex Galois representations, also known as Artin representations, are in fact an interesting object of study in number theory. However, the issue is that if we require these Galois representations to be continuous, like we have required for our \ell-adic representations, we will find that they always have finite image, which also means in essence that we might as well just have been studying representations of finite Galois groups, not the absolute one as we intend to do.

To get a complex representation that will be as interesting as the p-adic ones, we have to make certain modifications. We will look at certain representations of a certain subgroup of the Galois group instead, called the Weil group, and together with some additional information in the form of a “monodromy operator“, we will have a complex representation that will in a way carry the same information as a \ell-adic representation.

Let us first define this Weil group. F be a local field and let \kappa be its residue field. The absolute Galois groups of F and \kappa fit into the following exact sequence

\displaystyle 0\to I\to \mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa)\to 0

where I is the kernel of the surjective map \mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa) and is called the inertia subgroup (this can be considered the “local” and also “absolute” version of the exact sequence discussed near the end of Splitting of Primes in Extensions).

The residue field \kappa is a finite field, say of some cardinality q. Finite fields have the property that they have a unique extension of degree n for every n, and the Galois groups of these extensions are cyclic of order n. As a result, the absolute Galois group \mathrm{Gal}(\overline{\kappa}/\kappa) of the residue field \kappa is isomorphic to the inverse limit \varprojlim_{n} \mathbb{Z}/n\mathbb{Z}, also known as the profinite integers and denoted \widehat{\mathbb{Z}}.

There is a special element of \mathrm{Gal}(\overline{\kappa}/\kappa) called the Frobenius, which corresponds to raising to the power of q. The powers of Frobenius give us a subgroup isomorphic to the integers \mathbb{Z} inside \mathrm{Gal}(\overline{\kappa}/\kappa) (which again is isomorphic to \widehat{\mathbb{Z}}). The inverse image of this subgroup under the surjective morphism \mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa) is what is known as the Weil group of F (denoted W_{F}). Since \widehat{\mathbb{Z}} is the completion of \mathbb{Z}, the Weil group may be thought of as a kind of “decompletion” of the Galois group \mathrm{Gal}(\overline{F}/F).

It follows from local class field theory (see also Some Basics of Class Field Theory) that we have an isomorphism between the abelianization W_{F}^{\mathrm{ab}} of the Weil group and F^{\times}.

A Weil-Deligne representation is a pair (\rho_{0},N) consisting of a representation \rho_{0} of the Weil group W_{F}, together with a nilpotent operator N called the monodromy operator, which has to satisfy the property

\displaystyle \rho_{0}(\sigma)N\rho_{0}(\sigma)^{-1}=\Vert\sigma\Vert N

for all \sigma in W_{F}, where \Vert\sigma\Vert is the valuation of the element of F^{\times} corresponding to \sigma under the isomorphism given by local class field theory as mentioned above.

Grothendieck’s monodromy theorem them says that given a continuous p-adic representation \rho we can always associate to it a unique Weil-Deligne representation (\rho_{0},N) satisfying the property that, if we express an element of the absolute Galois group as \phi^{m}\sigma where \phi is a lift of Frobenius and \sigma belongs to the inertia group, then \rho(\phi^{m}\sigma)=\rho_{0}(\phi^{m}(\sigma))\mathrm{exp}(Nt(\sigma)), where t:\mathrm{Gal}(F^{\mathrm{tame}}/F^{\mathrm{ur}})\to\mathbb{Z}_{\ell}, F^{\mathrm{tame}} being the “tamely ramified” extension of F and F^{\mathrm{ur}} the unramified extension of F. The point is that, we can now associated to a p-adic Galois representation a complex representation in the form of the Weil-Deligne representation, which is the goal we stated in the beginning of this post.

It turns out that certain Weil-Deligne representations (those which are called F-semisimple) are in bijection with irreducible admissible representations of the \mathrm{GL}_{n}(F), thus linking two kinds of representations Рthose of Galois groups like we have discussed here, and those of reductive groups, similar to what was hinted at in Representation Theory and Fourier Analysis. This will be discussed in a future post.

References:

Weil group on Wikipedia

MSRI Summer School: Automorphic Forms and the Langlands Program (Lecture Notes) by Kevin Buzzard

Galois Representations Coming From Weight 2 Eigenforms

In Galois Representations we mentioned briefly that Galois representations can be obtained from modular forms. In this post we elaborate more on this construction, in the case that the modular form is a weight 2 eigenform (a weight 2 cusp form that is a simultaneous eigenfunction for all Hecke operators not dividing the level N). This specific case is also known as the Shimura construction, after Goro Shimura.

Let f be a weight 2 Hecke eigenform, of some level \Gamma_{0}(N) (this also works with other level structures). We want to construct a p-adic Galois representation associated to this Hecke eigenform, such that the two are going to be related in the following manner. For every prime \ell not dividing N and not equal to p, the characteristic polynomial of the image of the Frobenius element associated to \ell under this Galois representation will be of the form

\displaystyle x^{2}-a_{\ell}x+\ell\chi(\ell)

where a_{\ell} is the eigenvalue of the Hecke operator T_{\ell} and \chi is a Dirichlet character associated to another kind of Hecke operator called the diamond operator \langle \ell\rangle. This diamond operator acts on the argument of the modular form by an upper triangular element of \mathrm{SL}_{2}(\mathbb{Z}) whose bottom right entry is \ell mod N. This action is the same as the action of a Dirichlet character \chi:\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}^{\times}. The above polynomial is also known as the Hecke polynomial.

The first thing that we will need is the identification of the weight 2 cusp forms with the holomorphic differentials on the modular curve (as mentioned in Modular Forms in the case of \mathbb{SL}_{2}(\mathbb{Z}), although this is can be done more generally).

The second thing that we will need is the Jacobian. One can think of the Jacobian as the space given by the equivalence classes of all path integrals on a curve (in general we can do this for any algebraic curve, not just modular curves), where two path integrals are to be considered equivalent if they differ by integration along a loop. Since path integration can be considered as a linear functional from holomorphic differentials to the complex numbers, we consider such path integrals as the dual space to the space of holomorphic differentials. However, the loops we wanted to quotient out by can also be expressed as elements of the homology group of the curve (see also Homology and Cohomology)!

Therefore we now define the Jacobian of a curve X as

\displaystyle J(\Gamma)=\Omega^{\vee}/H_{1}(X,\mathbb{Z})

where \Omega denotes the holomorphic differentials on X. The notation \Omega^{\vee} denotes the dual to \Omega, since as we said the path integrals form the dual to the holomorphic differentials. The Jacobian can also described in other ways – for instance it is also the connected component of the Picard group (see also Divisors and the Picard Group), and the connection to the description given here is an important classical theorem called the Abel-Jacobi theorem.

The Jacobian is a higher-dimensional complex torus, and actually more is true – it is also an abelian variety, i.e. a projective variety whose points form a group (and hence a generalization of elliptic curves). Note that every complex torus is an elliptic curve, but this is not true in higher dimensions – only certain special kinds of higher dimensional complex tori (namely those with a polarization) are abelian varieties. In this vein the Jacobian of a curve has yet another description – it is “universal” among abelian varieties in that, if there is a morphism from a curve to any abelian variety, it can be expressed as a morphism from the curve to its Jacobian, followed by a morphism to that other abelian variety.

Now we go back to the case of modular curves. Denoting by S_{2}(\Gamma_{0}(N)) the space of cusp forms of weight two for the level structure \Gamma_{0}(N), which as discussed above is isomorphic to the space of holomorphic differentials on the corresponding modular curve X(\Gamma_{0}(N)), we can now define the Jacobian J(\Gamma_{0}(N)) as

\displaystyle J(\Gamma_{0}(N))=S_{2}(\Gamma_{0}(N))^{\vee}/H_{1}(X,\mathbb{Z})

The third ingredient that we need is a certain ideal of the Hecke algebra (the ring of endomorphisms of S_{2}(\Gamma_{0}(N)) generated by the actions of the Hecke operators and diamond operators) corresponding to the weight 2 Hecke eigenform f (let us denote this ideal by \mathbb{I}_{f}) that we want to obtain our Galois representation from. This ideal \mathbb{I}_{f}) is defined to be the one generated by all elements of the Hecke algebra whose eigenvalue when acting on f is zero.

Since the Hecke operators and diamond operators act on the Jacobian (we can see this this way – since the Jacobian is the quotient of the linear functionals on S_{2}(\Gamma_{0}(N)), the action is obtained by first applying the Hecke operator or diamond operator to the weight 2 eigenform, then applying the linear functional), we can use the ideal \mathbb{I}_{f} to cut down a quotient of the Jacobian which is another abelian variety A_{f}:

\displaystyle A_{f}=J(\Gamma_{0}(N))/\mathbb{I}_{f}J(\Gamma_{0}(N))

Finally, we can take the Tate module of A_{f}, and this will give us precisely the Galois representation that we want. The abelian variety A_{f} will have dimension equal to the degree of the number field generated by the eigenvalues of the Hecke operators.

If the eigenvalues are all rational, then A_{f} will actually be an elliptic curve – in other words, given an eigenform of weight 2 whose Hecke eigenvalues are all rational, we can always use it to construct an elliptic curve! This also gives us a map from the modular curve X(\Gamma_{0}(N)) to this elliptic curve, called a modular parametrization. The resulting elliptic curve will have the property that its L-function, built from point counts when it is reduced modulo primes, is the same as the L-function of the modular form which is built from its Fourier coefficients! This is because the Frobenius and the Fourier coefficients (which are also the eigenvalues of the Hecke operators) are related, as discussed above. The question of whether, given an elliptic curve, it comes from a modular form in this way, is another restatement of the question of modularity. The affirmative answer to this question, at least for certain elliptic curves over \mathbb{Q}, led to the proof of Fermat’s Last Theorem.

This theory, which is only very roughly sketched here, is just a very special case – one can also obtain, for instance, Galois representations from modular forms which are not of weight 2. We leave this for the future.

References:

Jacobian variety on Wikipedia

Abel-Jacobi map on Wikipedia

Modularity theorem on Wikipedia

Course on Mazur’s Theorem Lecture 10: Jacobians by Andrew Snowden

Course on Mazur’s Theorem Lecture 17: Eichler-Shimura by Andrew Snowden

A First Course in Modular Forms by Fred Diamond and Jerry Shurman

More on Galois Deformation Rings

In Galois Deformation Rings we introduced the concept of a Galois deformation ring, and how it is used to prove “R=T” theorems. In this post we will look at a very simple example to help make things more concrete. Then we will explore more about the structure of Galois deformation rings, in particular we want to relate the tangent space of such a Galois deformation ring to the Selmer group in Galois cohomology (which also shows up in a lot of contexts all over arithmetic geometry and number theory).

Let F be a finite extension of \mathbb{Q}, and let k be some finite field, with ring of Witt vectors W(k) (for example if k=\mathbb{F}_{p} then W(k)=\mathbb{Z}_{p}). Let our residual representation \overline{\rho}:\mathrm{Gal}(\overline{F}/F)\to GL_{1}(k) be the trivial representation, i.e. the group acts as the identity. A lift will be a Galois representation \overline{\rho}:\mathrm{Gal}(\overline{F}/F)\to GL_{1}(A), where A is a complete Noetherian algebra over W(k). Then our Galois deformation ring is given by the completed group ring

\displaystyle R _{\overline{\rho}}=W(k)[[\mathrm{Gal}(\overline{F}/F)^{\mathrm{ab,p}}]]

where \mathrm{Gal}(\overline{F}/F)^{\mathrm{ab,p}} means the pro-p completion of the abelianization of the Galois group \mathrm{Gal}(\overline{F}/F). Using local class field theory, we can express this even more explicitly as

\displaystyle R_{\overline{\rho}}=W(k)[\mu_{p^{\infty}}(F)][[X_{1},\ldots,X_{[F:\mathbb{Q}]}]]

Let us now consider a useful fact about the tangent space (see also Tangent Spaces in Algebraic Geometry) of such a deformation ring. Let us first consider the framed deformation ring R _{\overline{\rho}}^{\Box}. It is local, and has a unique maximal ideal \mathfrak{m}. There is only one tangent space, defined to be the dual of \mathfrak{m}/\mathfrak{m^{2}}, but this can also be expressed as the set of its dual number-valued points, i.e. \mathrm{Hom}(R_{\overline{\rho}}^{\Box},k[\epsilon]), which by the definition of the framed deformation functor, is also D_{\overline{\rho}}(k[\epsilon])^{\Box}. Any such deformation must be of the form

\displaystyle \rho(\sigma)=(1+\varepsilon c(\sigma))\overline{\rho}(\sigma)

where c is some n\times n matrix with coefficients in k. If \sigma and \tau are elements of \mathrm{Gal}(\overline{F}/F), if we substitute the above form of \rho into the equation \rho(\sigma\tau)=\rho(\sigma)\rho(\tau) we have

\displaystyle  (1+\varepsilon c(\sigma\tau))\overline{\rho}(\sigma\tau) = (1+\varepsilon c(\sigma))\overline{\rho}(\sigma) (1+\varepsilon c(\tau))\overline{\rho}(\tau)

from which we can see that

\displaystyle  c(\sigma\tau))\overline{\rho}(\sigma\tau) = c(\sigma)\overline{\rho}(\sigma)\overline{\rho}(\tau)+\overline{\rho}(\sigma)c(\tau)\overline{\rho}(\tau)

and, multiplying by \overline{\rho}(\sigma\tau)^{-1}= \overline{\rho}(\tau)^{-1}\overline{\rho}(\sigma)^{-1} on the right,

\displaystyle  c(\sigma\tau))=c(\sigma)(\tau)+c(\tau) \overline{\rho}(\sigma)\overline{\rho}(\sigma)^{-1}

In the language of Galois cohomology, we say that c is a 1-cocycle, if we take the n\times n matrices to be a Galois module coming from the “Lie algebra” of GL_{n}(k). We call this Galois module \mathrm{Ad}\overline{\rho}.

Now consider two different lifts (framed deformations) \rho_{1} and \rho_{2} which give rise to the same deformation of \overline{\rho}. Then there exists some n\times n matrix X such that

\displaystyle \rho_{1}(\sigma)=(1+\varepsilon X)\rho_{2}(\sigma)(1-\varepsilon X)

Plugging in \rho_{1}=(1+\varepsilon c_{1})\overline{\rho} and \rho_{2}=(1+\varepsilon c_{2})\overline{\rho} we obtain

\displaystyle  (1+\varepsilon c_{1})\overline{\rho}=(1+\varepsilon X) (1+\varepsilon c_{2})\overline{\rho}(1-\varepsilon X)

which will imply that

\displaystyle  c_{1}(\sigma)=c_{2}(\sigma)+X-\overline{\rho}(\sigma)X\overline{\rho}(\sigma)^{-1}

In the language of Galois cohomology (see also Etale Cohomology of Fields and Galois Cohomology) we say that c_{1} and c_{2} differ by a coboundary. This means that the tangent space of the Galois deformation ring is given by the first Galois cohomology with coefficients in \mathrm{Ad}\overline{\rho}:

\displaystyle D_{\overline{\rho}}(k[\epsilon])\simeq H^{1}(\mathrm{Gal}(\overline{F}/F),\mathrm{Ad}\overline{\rho})

More generally, when our Galois deformation ring is subject to conditions, it will be given by a subgroup of the first Galois cohomology known as the Selmer group (note that the Selmer group shows up in many places in arithmetic geometry and number theory, for instance, in the proof of the Mordell-Weil theorem where the Galois module used comes from the torsion points of an elliptic curve – in this post we are considering the case where the Galois module is \mathrm{Ad}\overline{\rho}, as stated earlier). The advantage of expressing the tangent space in the language of Galois deformation ring using Galois cohomology is that in Galois cohomology there are certain formulas such as Tate duality and the Euler characteristic formula that we can use to perform computations.

Finally to end this post we remark that under certain conditions (namely that for every open subgroup H of \mathrm{Gal}(\overline{F}/F) the space of continuous homomorphisms from H to \mathbb{F}_{p} has finite dimension) this tangent space is going to be a finite-dimensional vector space over k. Then the Galois deformation ring has the following form

\displaystyle R_{\overline{\rho}}=W(k)[[x_{1},\ldots,x_{g}]]/(f_{1},\ldots,f_{r})

i.e. it is a quotient of a W(k)-power series in g variables, where the number g is given by the dimension of H^{1}(\mathrm{Gal}(\overline{F}/F),\mathrm{Ad}\overline{\rho}) as a k-vector space, while the number of relations r is given by the dimension of H^{2}(\mathrm{Gal}(\overline{F}/F),\mathrm{Ad}\overline{\rho}) as a k-vector space.

Knowing the structure of Galois deformation rings is going to be important in proving R=T theorems, since such proofs often reduce to commutative algebra involving these rings. More details will be discussed in future posts on this blog.

References:

Group cohomology on Wikipedia

Galois cohomology on Wikipedia

Selmer group on Wikipedia

Tate duality 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

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

Galois Representations

The absolute Galois group \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) is one of the most important objects of study in mathematics. However the direct study of this group is very difficult; for instance it is an infinite group, and we know very little about it. To make it easier for us, we will often instead study representations of this group – i.e. group homomorphisms to the group \text{GL}(V) of linear transformations of some vector space V over some field F. When V has finite dimension n, \text{GL}(V) is just \text{GL}_{n}(F), the group of n\times n matrices with entries in F and nonzero determinant. Often we will also want the field F to carry a topology – this will also endow \text{GL}_{n}(F) with a topology. For instance, if F is the p-adic numbers \mathbb{Q}_{p} it has a p-adic topology (see also Valuations and Completions). Since \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) has its own topology, we can then talk about representations which are continuous. In this post we shall consider three examples of these continuous Galois representations.

Our first example of a Galois representation is known as the p-adic cyclotomic character. This is a one-dimensional representation over the p-adic numbers \mathbb{Q}_{p}, i.e. a group homomorphism from \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q} to \text{GL}_{1}(\mathbb{Q}_{p}), which also happens to just be the multiplicative group \mathbb{Q}_{p}^{\times}. Let us explain how to obtain this Galois representation.

Consider a primitive p^{n}-th root of unity \zeta_{p^{n}}. Any element \sigma of \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) acts on \zeta_{p^{n}} and sends it to some p^{n}-th root of unity, which amounts to raising it to some integer power between 1 and p^{n}-1, i.e. an element of (\mathbb{Z}/p^{n}\mathbb{Z})^{\times}. We now define the p-adic cyclotomic character \chi to be the map from \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) to \mathbb{Z}_{p}^{\times} which sends the element \sigma to the element of \mathbb{Z}_{p}^{\times} which after modding out by p^{n} is precisely the integer power to which we raised \zeta_{p^{n}}.

Our second example of a Galois representation is known as the Tate module of an elliptic curve. We recall that we also discussed an example of a Galois representation coming from the p-torsion points of an elliptic curve in Elliptic Curves. The Tate module is a way to package the action of the Galois group not only the p-torsion points but also the p^{n}-torsion for any n, by taking an inverse limit over n. Now the p^{n}-torsion points are isomorphic to (\mathbb{Z}/p^{n}\mathbb{Z})^{2}, so the inverse limit is going to be isomorphic to \mathbb{Z}_{p}^{2}. This is not a vector space, since \mathbb{Z}_{p} is not a field, so we take the tensor product with \mathbb{Q}_{p} to get \mathbb{Q}_{p}^{2}, which is a vector space. Therefore we get a Galois representation, i.e. a homomorphism from \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) to \text{GL}_{2}(\mathbb{Q}_{p}). This construction also works for abelian varieties – higher dimensional analogues of elliptic curves – except that the Tate module is now 2g-dimensional, where g is the dimension of the abelian variety.

Our last example of a Galois representation is given by the \ell-adic cohomology (explanation of this terminology to come later) of a smooth proper algebraic variety X over \mathbb{Q}. This is the inverse limit over n of the etale cohomology (see also Cohomology in Algebraic Geometry) of X with coefficients in the constant sheaf \mathbb{Z}/p^{n}\mathbb{Z}. These etale cohomology groups are somewhat confusingly denoted H^{i}(X,\mathbb{Z}_{p}) – note that they are not the etale cohomology of X with \mathbb{Z}_{p} coefficients! Just as in the case of the Tate module, we take the tensor product with \mathbb{Q}_{p} to produce our Galois representation.

These Galois representations coming from the \ell-adic cohomology somewhat subsume the Tate modules discussed earlier – that is because, if X is an elliptic curve or more generally an abelian variety, we have that the \mathbb{Q}_{p}-linear maps from the Tate module (tensored with \mathbb{Q}_{p}) is isomorphic to the first \ell-adic cohomology H_{1}(X,\mathbb{Z}_{p})\otimes\mathbb{Q}_{p}. We say that the first \ell-adic cohomology is the dual of the Tate module.

Although we discussed representations over \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) in this post, it is also often useful to make our study “local” and focus on a single prime \ell, and study \text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell}) instead. In this case we might as well just have replaced \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) with \text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell}) in the above discussion, and nothing really changes, as long as the primes \ell and p are different primes. In the case that they are the same prime, things become much more complicated (and the theory is far richer)!

Note: Usually, when discussing “local” Galois representations, the notation for the primes p and \ell are switched! In other words, our local Galois representations are group homomorphisms from \text{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}) to \text{GL}_{n}(\mathbb{Q}_{\ell}). This is the reason for the terminology “\ell-adic cohomology”. Since we started out just discussing “global” Galois representations, I switched the notation to use p instead for the only instances were we needed a prime. Hopefully this is not overly confusing. We can also study Galois representations more generally for number fields (“global”) and finite extensions of \mathbb{Q}_{p} (“local”).

Finally, although we stated above that we will only discuss three examples here, let us mention a fourth example: Galois representations can also come from modular forms (see also Modular Forms). To discuss these Galois representations would require us to develop some more machinery first, so we leave this to future posts for now.

References:

Cyclotomic character on Wikipedia

Tate module on Wikipedia

Etale cohomology on Wikipedia

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein