# 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 – those 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

# Perfectoid Fields

Consider the field of $p$-adic numbers $\mathbb{Q}_{p}$. An element of $\mathbb{Q}_{p}$ may be written in the form

$\displaystyle \sum_{n=k}^{\infty}a_{n}p^{n}$

with each $a_{n}$ being an element of the finite field $\mathbb{F}_{p}$. Let us compare this with the field of Laurent series $\mathbb{F}_{p}((t))$ in one variable $t$ over $\mathbb{F}_{p}$. An element of $\mathbb{F}_{p}((t))$ may be written in the form

$\displaystyle \sum_{m=l}^{\infty}a_{m}t^{m}$

We see that they look very similar, even though $\mathbb{Q}_{p}$ is characteristic $0$, and $\mathbb{F}_{p}((t))$ is characteristic $p$.

How far can we push this analogy? The fact that one is in characteristic $0$, and the other is characteristic $p$ means we cannot ask for an isomorphism of fields. However, the Fontaine-Wintenberger theorem gives us another connection between $\mathbb{Q}_{p}$ and $\mathbb{F}_{p}((t))$ – if we modify them by adjoining $p$-power roots of $p$ and $t$ respectively. This theorem states that the fields $\cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}})$ and $\cup_{n}\mathbb{F}_{p}((t^{1/p^{n}}))$ have the same absolute Galois group! By the fundamental theorem of Galois theory, this means the category formed by their extensions will be equivalent as well.

We now let $F$ denote the completion of $\cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}})$, and we let $F^{\flat}$ suggestively denote the completion of $\cup_{n}\mathbb{F}_{p}((t^{1/p^{n}}))$. Completing these fields does not change their absolute Galois groups, so the absolute Galois groups of $F$ and $F^{\flat}$ remain isomorphic. We say that the characteristic $p$ field $F^{\flat}$ is the tilt of the characteristic $0$ field $F$, and that $F^{\flat}$ is an untilt of $F$ (note the subtle change in our choice of article – untilts are not unique).

In this post, we will explore these kinds of fields – which are called perfectoid fields – and the process of tilting and untilting that bridges the world of characteristic $0$ and characteristic $p$. After Fontaine and Wintenberger came up with their famous theorem their ideas have since been developed into even more general and even more powerful theories of perfectoid rings and perfectoid spaces – but we will leave these to future posts. For now we concentrate on the case of fields.

First let us look at a much more primitive example of bridging the world of characteristic $0$ and characteristic $p$. Consider $\mathbb{Q}_{p}$ (characteristic $0$). It has a ring of integers $\mathbb{Z}_{p}$, whose residue field is $\mathbb{F}_{p}$ (characteristic $p$). To got the other way, starting from $\mathbb{F}_{p}$ we can take its ring of Witt vectors, which is $\mathbb{Z}_{p}$. Then we take its field of fractions which is $\mathbb{Q}_{p}$.

More generally, there is a correspondence between characteristic $0$ discretely valued complete fields whose uniformizer is $p$ and characteristic $p$ fields which are perfect, i.e. for which the Frobenius morphism is bijective, and the way to go from one category to the other is as in the previous paragraph.

This is a template for “bridging the world of characteristic $0$ and characteristic $p$“. However, we may want more, something like the Fontaine-Wintenberger theorem where the characteristic $0$ object and the characteristic $p$ object have isomorphic absolute Galois groups. We will be tweaking this basic bridge in order to create something like Fontaine-Winterger theorem, and these tweaks will lead us to the notion of a perfectoid field. However, we already have isolated one property that we want from such a “perfectoid” field:

The first property that we want from a perfectoid field is that it has to be nonarchimedean. This allows us to have a “ring of integers” that serves as an intermediary object between the two worlds, as we have seen above.

Now let us concentrate on the Fontaine-Wintenberger theorem. To understand this phenomenon better, we need to make use of a version of the fundamental theorem of Galois theory, which allows us to think in terms of extensions of fields instead of their Galois groups. More properly, we want an equivalence of categories between the “Galois categories” of certain extensions of these “base” fields and this will be the property of these base fields being perfectoid. Now the problem is that the extensions that we are considering may not fit into the primitive correspondence we stated above – for example the corresponding characteristic $p$ object may not be perfect, i.e. the Frobenius morphism may not be surjective.

The fix to this is a kind of “perfection”, which is the tilting functor we mentioned earlier. Let $R$ be a ring. The tilt of $R$, denoted $R^{\flat}$ is defined to be the inverse limit

$\displaystyle R^{\flat}=\varprojlim_{x\mapsto x^{p}}R/pR$

In other words, an element $x$ of $R^{\flat}$ is an infinite sequence of elements $(x_{0},x_{1},x_{2},\ldots)$ of the quotient $R/pR$ such that $x_{1}\cong x_{0}^{p}\mod p$, $x_{2}\cong x_{1}^{p}\mod p$, and so on. We want $R^{\flat}$ to be a ring, so we define it to have componentwise multiplication, i.e.

$\displaystyle (xy)_{i}=x_{i}y_{i}$

However the addition is going to be more complicated. We define it, for each component, as follows:

$\displaystyle (x+y)_{i}=\lim_{n\to\infty}(x_{i+n}+y_{i+n})^{p^n}$

At this point we take the opportunity to define another important concept in the theory of perfectoid fields (and rings). Let $W$ be the Witt vector functor (see also The Field with One Element). Then we give the Witt vectors of the tilt of $R$, $W(R^{\flat})$, a special name. We will refer to this ring as $A_{\mathrm{inf}}(R)$. It will make an appearance again later. For now we note that there is going to be a canonical map $\theta: A_{\mathrm{inf}}(R)\to R$.

As we can see, we have defined the tilt of an arbitrary ring. This is not exclusive to the ones which are “perfectoid” whatever the definition of “perfectoid” may be (we will come to this later of course). Again what makes perfectoid fields (such as our earlier examples) special though, is that if $F$ is a perfectoid field of characteristic $0$, then $F$ and its tilt $F^{\flat}$ will have isomorphic absolute Galois groups. This will actually follow from the following statement (together with some technicalities involving fiber functors and so on):

There is an equivalence of categories between the category of finite etale algebras over a perfectoid field $F$ and the category of finite etale algebras over its tilt $F^{\flat}$.

This in turn will follow from the following two statements:

1. Finite extensions of perfectoid fields are perfectoid.
2. There is an equivalence of categories between the category of perfectoid extensions of a perfectoid field $F$ and the category of perfectoid extensions over its tilt $F^{\flat}$.

This equivalence of categories is given by tilting a perfectoid extension over $F$. This will actually give us a perfectoid extension over $F^{\flat}$. However, we need a functor that goes in the other direction, a “quasi-inverse” that when composed with tilting gives us back our original perfectoid extension over $F$ (or at least something isomorphic to it, this is what the “quasi-” part means). However, we also said in an earlier paragraph that the “untilt” of a characteristic $p$ field may not be unique (two different untilts may also not be isomorphic). How do we approach this problem?

We recall again the ring $A_{\mathrm{inf}}(R)$ defined earlier as the ring of Witt vectors of the tilt of $R$, and we recall that it has a canonical map $\theta:A_{\mathrm{inf}}(R)\to R$. If we know this map, and if we know that it is surjective, then we can recover $R$ simply by quotienting out by the kernel of the map $\theta$!

The problem is that (aside from not knowing whether it is in fact surjective or not) is that we only know this map if we know that $R^{\flat}$ was obtained as the tilt of $R$. If we were simply handed some characteristic $p$ field for instance we would not be able to know this map.

However, note that we are interested in an equivalence of categories between the category of perfectoid extensions over the field $F$ and the corresponding category over its tilt $F^{\flat}$. By specifying these “bases” $F$ and $F^{\flat}$, it is in fact enough to specify unique untilts! In other words, if we have say just some perfectoid field $A$, we cannot determine a unique untilt for it, but if we say in addition that it is a perfectoid extension over $F$, and we are looking for the unique untilt of it over $F^{\flat}$, we can in fact find it, as long as the map $\theta$ is surjective.

So now how do we guarantee that $\theta$ is surjective? This brings us to our second property, which is that the Frobenius morphism from $\overline{R}$ to itself must be surjective. This is actually the origin of the word “perfectoid”; since as above a field for which the Frobenius morphism is bijective is called perfect; hence, requiring it to be surjective is a relaxation of this condition. This condition guarantees that the map $\phi:A_{\mathrm{inf}}(R)\to R$ is going to be surjective.

The final property that we want from a perfectoid field is that its valuation must be non-discretely valued. The reason for this is that we want to consider infinitely ramified extensions of $\mathbb{Q}_{p}$. The two previous conditions that we want can only be found in unramified (discretely valued) or infinitely ramified (non-discretely valued) of $\mathbb{Q}_{p}$. We have already seen above that if we only look at the ones which are unramified then our corresponding characteristic $p$ objects will be limited to perfect $\mathbb{F}_{p}$-algebras, and this is not enough to give us the Fontaine-Wintenberger theorem. Therefore we will want infinitely ramified extension of $\mathbb{Q}_{p}$, and these are non-discretely valued.

These three properties are enough to give us the Fontaine-Wintenberger theorem. To summarize – a perfectoid field is a complete, nonarchimedean field $F$ such that the Frobenius morphism from $\mathcal{O}_{F}/\mathfrak{p}$ to itself is surjective and such that its valuation is non-discretely valued.

We have only attempted to motivate the definition of a perfectoid field in this post, and barely gone into any sort of detail. For that one can only recommend the excellent post by Alex Youcis on his blog The Fontaine-Wintenberger Theorem: Going Full Tilt, which inspired this post, but barely does it any justice.

Aside from the Fontaine-Wintenberger theorem, the concepts we have described here – the idea behind “perfectoid”, the equivalence of categories of perfectoid extensions that gives rise to the Fontaine-Wintenberger theorem, the idea of tilting and untilting which bridges the worlds of characteristic $0$ and characteristic $p$, the ring $A_{\mathrm{inf}}(R)$, and so on, have found much application in many areas of math, from the aforementioned perfectoid rings and perfectoid spaces, to p-adic Hodge theory, and to many others.

References:

Perfectoid Space on Wikipedia

What is…a Perfectoid Space? by Bhargav Bhatt

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

The Fontaine-Wintenberger Theorem: Going Full Tilt by Alex Youcis

# 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. All of this comes from what is known as the Eichler-Shimura relation, which relates the Hecke operators and the Frobenius.

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

# Representation Theory and Fourier Analysis

In Some Basics of Fourier Analysis we introduced some of the basic ideas in Fourier analysis, which is ubiquitous in many parts of both pure and applied math. In this post we look at these same ideas from a different point of view, that of representation theory.

Representation theory is a way of studying group theory by turning it into linear algebra, which in many cases is more familiar to us and easier to study.

A (linear) representation is just a group homomorphism from some group $G$ we’re interested in, to the group of linear transformations of some vector space. If the vector space has some finite dimension $n$, the group of its linear transformations can be expressed as the group of $n \times n$ matrices with nonzero determinant, also known as $\mathrm{GL}_n(k)$ ($k$ here is the field of scalars of our vector space).

In this post, we will focus on infinite-dimensional representation theory. In other words, we will be looking at homomorphisms of a group $G$ to the group of linear transformations of an infinite-dimensional vector space.

“Infinite-dimensional vector spaces” shouldn’t scare us – in fact many of us encounter them in basic math. Functions are examples of such. After all, vectors are merely things we can scale and add to form linear combinations. Functions satisfy that too. That being said, if we are dealing with infinity we will often need to make use of the tools of analysis. Hence functional analysis is often referred to as “infinite-dimensional linear algebra” (see also Metric, Norm, and Inner Product).

Just as a vector $v$ has components $v_i$ indexed by $i$, a function $f$ has values $f(x)$ indexed by $x$. If we are working over uncountable things, instead of summation we may use integration.

We will also focus on unitary representations in this post. This means that the linear transformations are further required to preserve a complex inner product (which takes the form of an integral) on the vector space. To facilitate this, our functions must be square-integrable.

Consider the group of real numbers $\mathbb{R}$ (under addition). We want to use representation theory to study this group. For our purposes we want the square-integrable functions on some quotient of $\mathbb{R}$ as our vector space. It comes with an action of $\mathbb{R}$, by translation. In other words, an element $a$ of $\mathbb{R}$ acts on our function $f(x)$ by sending it to the new function $f(x+a)$.

So what is this quotient of $\mathbb{R}$ that our functions will live on? For now let us choose the integers $\mathbb{Z}$. The quotient $\mathbb{R}/\mathbb{Z}$ is the circle, and functions on it are periodic functions.

To recap: We have a representation of the group $\mathbb{R}$ (the real line under addition) as linear transformations (also called linear operators) of the vector space of square-integrable functions on the circle.

In representation theory, we will often decompose a representation into a direct sum of irreducible representations. Irreducible means it contains no “subrepresentation” on a smaller vector space. The irreducible representations are the “building blocks” of other representations, so it is quite helpful to study them.

How do we decompose our representation into irreducible representations? Consider the representation of $\mathbb{R}$ on the vector space $\mathbb{C}$ (the complex numbers) where a real number $a$ acts by multiplying a complex number $z$ by $e^{2\pi i k a}$, for $k$ an integer. This representation is irreducible.

If this looks familiar, this is just the Fourier series expansion for a periodic function. So a Fourier series expansion is just an expression of the decomposition of the representation of R into irreducible representations!

What if we chose a different vector space instead? It might have been the more straightforward choice to represent $\mathbb{R}$ via functions on $\mathbb{R}$ itself instead of on the circle $\mathbb{R}/\mathbb{Z}$. That may be true, but in this case our decomposition into irreducibles is not countable! The irreducible representations into which this other representation decomposes is the one where a real number $a$ acts on $\mathbb{C}$ by multiplication by $e^{2 \pi i k a}$ where $k$ is now a real number, not necessarily an integer. So it’s not indexed by a countable set.

This should also look familiar to those who know Fourier analysis: This is the Fourier transform of a square-integrable function on $\mathbb{R}$.

So now we can see that concepts in Fourier analysis can also be phrased in terms of representations. Important theorems like the Plancherel theorem, for example, also may be understood as an isomorphism between the representations we gave and other representations on functions of the indices. We also have the Poisson summation in Fourier analysis. In representation theory this is an equality obtained from calculating the trace in two ways, as a sum over representations and as a sum over conjugacy classes.

Now we see how Fourier analysis is related to the infinite-dimensional representation theory of the group $\mathbb{R}$ (one can also see this as the infinite-dimensional representation theory of the circle, i.e. the group $\mathbb{R}/\mathbb{Z}$ – the article “Harmonic Analysis and Group Representations” by James Arthur discusses this point of view). What if we consider other groups instead, like, say, $\mathrm{GL}_n(\mathbb{R})$ or $\mathrm{SL}_n(\mathbb{R})$ (or $\mathbb{R}$ can be replaced by other rings even)?

Things get more complicated, for example the group may not be abelian. Since we used integration so much, we also need an analogue for it. So we need to know much about group theory and analysis and everything in between for this.

These questions have been much explored for the kinds of groups called “reductive”, which are closely related to Lie groups. They include the examples of $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{SL}_n(\mathbb{R})$ earlier, as well as certain other groups we have discussed in previous posts such as the orthogonal and unitary (see also Rotations in Three Dimensions). There is a theory for these groups analogous to what I have discussed in this post, and hopefully this will be discussed more in future blog posts here.

References:

Representation theory on Wikipedia

Representation of a Lie group on Wikipedia

Fourier analysis on Wikipedia

Harmonic analysis on Wikipedia

Plancherel theorem on Wikipedia

Poisson summation formula on Wikipedia

An Introduction to the Trace Formula by James Arthur

Harmonic Analysis and Group Representations by James Arthur

# 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