# p-adic Hodge Theory: An Overview

In The Theory of Motives we discussed the notion of a Weil cohomology, and mentioned four “classical” examples, the singular (also known as Betti) cohomology, the de Rham cohomology, the $\ell$-adic cohomology, and the crystalline cohomology.

Cohomology theories may be thought of as a way to study geometric objects using linear algebra, by associating vector spaces (or more generally, modules or abelian groups) to such a geometric object. But the four Weil cohomology theories above actually give more than just a vector space:

• The singular cohomology has an action of complex conjugation.
• The de Rham cohomology has a Hodge filtration.
• The $\ell$-adic cohomology has an action of the Galois group.
• The crystalline cohomology has an action of Frobenius (and a Hodge filtration as well).

There are relations between these different cohomologies. For example, for a smooth projective variety $X$ over the complex numbers $\mathbb{C}$, the singular cohomology of the corresponding complex analytic manifold $X(\mathbb{C})$, with complex coefficients (this can be obtained from singular cohomology with integral coefficients by tensoring with $\mathbb{C}$) and the de Rham cohomology are isomorphic:

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H_{\mathrm{dR}}^{k}(X)$

The roots of this idea go back to de Rham’s work on complex manifolds, where chains in singular homology (which is dual to singular cohomology, see also Homology and Cohomology) can be paired with the differential forms of de Rham cohomology (see also Differential Forms), simply by integrating the differential forms along these chains. By the machinery developed by Alexander Grothendieck, this can be ported over into the world of algebraic geometry.

Again borrowing from the world of complex manifolds, the machinery of Hodge theory gives us the following Hodge decomposition (see also Shimura Varieties):

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{C}}^{j})$

Now again for the case of smooth projective varieties over the complex numbers , $\ell$-adic cohomology also has such an isomorphism with singular cohomology – but this time if it has $\ell$-adic coefficients (i.e. in $\mathbb{Q}_{\ell}$).

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Q}_{\ell}\simeq H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell})$

Such isomorphisms are also known as comparison isomorphisms (or comparison theorems).

More generally, if we have a field $B$ into which we can embed both $\mathbb{Q}_{\ell}$ and $\mathbb{C}$ (for instance $\mathbb{C})$, we obtain the following comparison theorem:

$\displaystyle H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell}) \otimes_{\mathbb{Q_{\ell}}} B\simeq H_{\mathrm{dR}}^{k}(X) \otimes_{\mathbb{C}} B$

Here is a very interesting thing that these comparison theorems can give us. Let $X$ be a modular curve. Then the Hodge decomposition for the first cohomology gives us

$\displaystyle H_{\mathrm{sing}}^{1}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H^{1}(X,\Omega_{X/\mathbb{C}}^{0})\oplus H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$

But the $H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$ is the cusp forms of weight $2$ as per the discussion in Modular Forms (see also Galois Representations Coming From Weight 2 Eigenforms). By the results of Hodge theory, the other summand $H^{1}(X,\Omega_{X/\mathbb{C}}^{0})$ is just the complex conjugate of $H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$. But we now also have a comparison with etale cohomology, which has a Galois representation! For this the modular form must lie in the cohomology with $\mathbb{Q}$ coefficients, which happens if it is a Hecke eigenform whose Hecke eigenvalues are in $\mathbb{Q}$. So one of the great things that these comparison theorems gives us is this way of relating modular forms and Galois representations.

The comparison isomorphisms above work for smooth projective varieties over the complex numbers, but let us now go to the p-adic world, and let us consider smooth projective varieties over the p-adic numbers.

It was observed by John Tate (and later explored by Gerd Faltings) that the p-adic cohomology (i.e. the etale cohomology of a smooth projective variety over $\mathbb{Q}_{p}$, or more generally some other p-adic field, with p-adic coefficients, distinguishing it from $\ell$-adic cohomology where another prime $\ell$ different from $p$ must be brought in) can have a decomposition akin to the Hodge decomposition, after tensoring it with the p-adic complex numbers (this is the completion of the algebraic closure of the p-adic numbers):

$\displaystyle H^{k}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C}_{p}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{Q}}^{j})\otimes_{\mathbb{Q}}\mathbb{C}_{p}(-j)$

The p-adic complex numbers here play the role of the complex numbers in the singular cohomology case above or the $\ell$-adic numbers in the $\ell$-adic case.

The ideas conjectured by Tate, and later completed by Faltings, was but the prototype of what is now known as p-adic Hodge theory. In its modern form, p-adic Hodge theory concerns comparison isomorphisms between different Weil cohomology theories on smooth projective varieties over the p-adic numbers. However, the role played by the complex numbers, $\ell$-adic numbers (for the complex case), and p-adic complex numbers (for the p-adic case) must now be played by much more complicated objects called period rings, which were developed by Jean-Marc Fontaine. We will discuss the construction of the period rings at the end of this post, but first let us see how they work.

Let $X$ be a smooth projective variety over $\mathbb{Q}_{p}$ (or more generally some other p-adic field). Let $H_{\mathrm{dR}}^{i}(X)$ and $H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})$ be its de Rham cohomology and the p-adic etale cohomology of its base change to the algebraic closure $\overline{\mathbb{Q}}_{p}$ respectively. The comparison isomorphism at the center of p-adic Hodge theory is the following:

$\displaystyle H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}=H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}$

The object denoted $B_{\mathrm{dR}}$ here is the aforementioned period ring. It is equipped with both a Galois action and a filtration akin to the Hodge filtration. More than just that isomorphism above, we also have a way of obtaining the de Rham cohomology if we are given the p-adic etale cohomology, simply by taking the part that is invariant under the Galois action:

$\displaystyle \displaystyle H_{\mathrm{dR}}^{i}(X)=(H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

To go the other way, i.e. to recover the p-adic etale cohomology from the de Rham cohomology, we will need a different kind of period ring. This period ring is $B_{\mathrm{cris}}$, which aside from having a Galois action and a filtration also has an action of Frobenius. Aside from providing us the same isomorphism between de Rham and p-adic etale cohomology upon tensoring, it also provides us with a solution to our earlier problem (as long as $X$ has a smooth proper integral model) as follows:

$\displaystyle H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})= \mathrm{Fil}^{0}(H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{cris}})^{\varphi=1}$

This idea can be further abstracted – since etale cohomology provides Galois representations, we can just take some p-adic Galois representation instead, without caring whether it comes from etale cohomology or not, and tensor it with a period ring, then take Galois invariants. For instance let $V$ be some p-adic Galois representation. Then we can take the tensor product

$V_{\mathrm{dR}}=(V\otimes B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

If the dimension of $V_{\mathrm{dR}}$ is equal to the dimension of $V$, then we say that the Galois representation $V$ is de Rham. Similarly we can tensor with $B_{\mathrm{cris}}$:

$V_{\mathrm{cris}}=(V\otimes B_{\mathrm{cris}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

If its $V_{\mathrm{cris}}$ is equal to the dimension of $V$ , we say that $V$ is crystalline.

The idea of these “de Rham” and “crystalline” Galois representations is that if they come from the corresponding cohomologies then they will have these properties. But does the converse hold? If they are “de Rham” and “crystalline” does that mean that they come from the corresponding cohomologies (i.e. they “come from geometry”)? This is roughly the content of the Fontaine-Mazur conjecture.

Now let us say a few things about the construction of these period rings. These constructions make use of the concepts we discussed in Perfectoid Fields. We start with the ring $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})$, which, as we recall from Perfectoid Fields, is the ring of Witt vectors of the tilt of $\mathcal{O}_{\mathbb{C}_{p}}$. By inverting $p$ and taking the completion with respect to the canonical map $\theta: A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}) \to\mathcal{O}_{\mathbb{C}_{p}}$, we obtain a ring which we suggestively denote by $B_{\mathrm{dR}}^{+}$.

There is a special element $t$ of $B_{\mathrm{dR}}^{+}$ which we think of as the logarithm of the element $(1, \zeta^{1/p},\zeta^{1/p},\ldots)$. Upon inverting this element $t$, we obtain the field $B_{\mathrm{dR}}$.

The field $B_{\mathrm{dR}}$ is equipped with a Galois action, carried over from the fields involved in its construction, and a filtration, given by $\mathrm{Fil}^{i}B_{\mathrm{dR}}=t^{i}B_{\mathrm{dR}}$.

To construct $B_{\mathrm{cris}}$, we once again start with $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})$ and invert $p$. However, to have a Frobenius, instead of completing with respect to the kernel of the map $\theta$, we take a generator of this kernel (which we shall denote by $\omega$). Then we denote by $B_{\mathrm{cris}}^{+}$ the ring formed by all the power series of the form $\sum_{n=0}^{\infty} a_{n}\omega^{n}/n!$ where the $a_{n}$‘s are elements of $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$ which converge as $n\to\infty$, under the topology of $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$ (which is not the p-adic topology!). Once again there will be an element $t$ like before; we invert $t$ to obtain $B_{\mathrm{cris}}$.

There is yet another period ring called $B_{\mathrm{st}}$, where the subscript stands for semistable; in addition to a Galois action, filtration, and Frobenius, it has a monodromy operator. Since this is less extensively discussed in introductory literature, we follow this lead and leave this topic, and the many other wonderful topics related to p-adic Hodge theory, to future posts on this blog.

References:

de Rham cohomology on Wikipedia

Hodge theory on Wikipedia

An introduction to the theory of p-adic representations by Laurent Berger

# Rigid Analytic Spaces

This blog post is inspired by and follows closely an amazing talk given by Ashwin Iyengar at the “What is a…seminar?” online seminar. I hope I can do the talk, and this wonderful topic, some justice in this blog post.

One of the most fascinating and powerful things about algebraic geometry is how closely tied it is to complex analysis, despite what the word “algebraic” might lead one to think. To state one of the more simple and common examples, we have that smooth projective curves over the complex numbers $\mathbb{C}$ are the same thing as compact Riemann surfaces. In higher dimensions we also have Chow’s theorem, which tells us that an analytic subspace of complex projective space which is topologically closed is an algebraic subvariety.

This is all encapsulated in what is known as “GAGA“, named after the foundational work “Géometrie Algébrique et Géométrie Analytique” by Jean-Pierre Serre. We refer to the references at the end of this post for the more precise statement, but for now let us think of GAGA as giving us a fully faithful functor from proper algebraic varieties over $\mathbb{C}$ to complex analytic spaces, which gives us an equivalence of categories between their coherent sheaves.

One may now ask if we can do something similar with the p-adic numbers $\mathbb{Q}_{p}$ (or more generally an extension $K$ of $\mathbb{Q}_{p}$ that is complete with respect to a valuation that extends the one on $\mathbb{Q}_{p}$) instead of $\mathbb{C}$. This leads us to the theory of rigid analytic spaces, which was originally developed by John Tate to study a p-adic version of the idea (also called “uniformization”) that elliptic curves over $\mathbb{C}$ can be described as lattices on $\mathbb{C}$.

Let us start defining these rigid analytic spaces. If we simply naively try to mimic the definition of complex analytic manifolds by having these rigid analytic spaces be locally isomorphic to $\mathbb{Q}_{p}^{m}$, with analytic transition maps, we will run into trouble because of the peculiar geometric properties of the p-adic numbers – in particular, as a topological space, the p-adic numbers are totally disconnected!

To fix this, we cannot just use the naive way because the notion of “local” would just be too “small”, in a way. We will take a cue from algebraic geometry so that we can use the notion of a Grothendieck topology to fix what would be issues if we were to just use the topology that comes from the p-adic numbers.

The Tate algebra $\mathbb{Q}_{p}\langle T_{1},\ldots,T_{n}\rangle$ is the algebra formed by power series in $n$ variables that converge on the $n$-dimensional unit polydisc $D^{n}$, which is the set of all n-tuples $(c_{1},\ldots,c_{n})$ of elements of $\mathbb{Q}_{p}$ that have p-adic absolute value less than or equal to $1$ for all $i$ from $1$ to $n$.

There is another way to define the Tate algebra, using the property of power series with coefficients in p-adic numbers that it converges on the unit polydisc $D^{n}$ if and only if its coefficients go to zero (this is not true for real numbers!). More precisely if we have a power series

$\displaystyle f(T_{1},\ldots,T_{n})=\sum_{a} c_{a}T_{1}^{a_{1}}\ldots T_{n}^{a_{n}}$

where $c_{a}\in \mathbb{Q}_{p}$ and $a=a_{1}+\ldots+a_{n}$ runs over all n-tuples of natural numbers, then $f$ converges on the unit polydisc $D^{n}$ if and only if $\lim_{a\to 0}c_{a}=0$.

The Tate algebra has many important properties, for example it is a Banach space (see also Metric, Norm, and Inner Product) with the norm of an element given by taking the biggest p-adic absolute value among its coefficients. Another property that will be very important in this post is that it satisfies a Nullstellensatz – orbits of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ in $D^{n}$ correspond to maximal ideals of the Tate algebra. Explicitly, this correspondence is given by the evaluation map $x\in D^{n}\mapsto \lbrace f\vert f(x)=0\rbrace$.

A quotient of the Tate algebra by some ideal is referred to as an affinoid algebra. The maximal ideals of the underlying set of an affinoid algebra $A$ will be denoted $\mathrm{Max}(A)$, and this should be reminiscent of how we obtain the closed points of a scheme.

Once we have the underlying set $\mathrm{Max}(A)$ , we need two other things to be able to define “affinoid” rigid analytic spaces, which we shall later “glue” to form more general rigid analytic spaces – a topology, and a structure sheaf (again this should be reminiscent of how schemes are defined).

Again taking a cue from how schemes are defined, we define “rational domains”, which are analogous to the distinguished Zariski open sets of (the underlying topological space of) a scheme. Given elements $f_{1},\ldots,f_{r},g$ of the affinoid algebra $A$, the rational domain $\displaystyle A\left(\frac{f}{g}\right)$ is the set of all $x\in\mathrm{Max}(A)$ such that $f_{i}(x)\leq g(x)$ for all $1\leq i\leq r$.

The rational domains generate a topology, however this is still just the p-adic topology, and so it still does not solve the problems that we originally ran into when we tried to define rigid analytic spaces by mimicking the definition of a complex analytic manifold. The trick will be to make use of something that is more general than just a topology – a Grothendieck topology (see also More Category Theory: The Grothendieck Topos).

Let us now define the particular Grothendieck topology that we will use. Unlike other examples of a Grothendieck toplogy, the covers will involve only subsets of the space being covered (it is also referred to as a mild Grothendieck topology). Let $X=\mathrm{Max}(A)$. A subset $U$ of $X$ is called an admissible open if it can be covered by rational domains $\lbrace U_{i}\rbrace_{i\in I}$ such that for any map $Y\to X$ where $Y=\mathrm{Max}(B)$ for some affinoid algebra $B$, the covering of $Y$ given by the inverse images of the $U_{i}$‘s admit a finite subcover.

If $U$ is an admissible open covered by admissible opens $\lbrace U_{i}\rbrace_{i\in I}$, then this covering is called admissible if for any map $Y\to X$ whose image is contained in $U$, the covering of $Y$ given by the inverse images of the $U_{i}$‘s admit a finite subcover. These admissible coverings satisfy the axioms for a Grothendieck topology, which we denote $G_{X}$.

If $A$ is an affinoid algebra, and $f_{1},\ldots,f_{k},g$ are functions, we let $\displaystyle A\left\langle \frac{f}{g}\right\rangle$ denote the ring $A\langle T_{1},\ldots T_{k}\rangle/(gT_{i}-f_{i})$. By associating to a rational domain $\displaystyle A\left(\frac{f}{g}\right)$ this ring $\displaystyle A\left\langle\frac{f}{g}\right\rangle$, we can define a structure sheaf $\cal{O}_{X}$ on this Grothendieck topology.

The data consisting of the set $X=\mathrm{Max}(A)$, the Grothendieck topology $G_{X}$, and the structure sheaf $\mathcal{O}_{X}$ is what makes up an affinoid rigid analytic space. Finally, just as with schemes, we define a more general rigid analytic space as the data consisting of some set $X$, a Grothendieck topology $G_{X}$ and a sheaf $\mathcal{O}_{X}$ such that locally, with respect to the Grothendieck topology $G_{X}$, it is isomorphic to an affinoid rigid analytic space.

Under this definition, we have in fact a version of GAGA that holds for rigid analytic spaces – a fully faithful functor from proper schemes of finite type over $\mathbb{Q}_{p}$ to rigid analytic spaces over $\mathbb{Q}_{p}$ that gives an equivalence of categories between their coherent sheaves.

Finally let us now look at an example. Consider the affinoid rigid analytic space obtained from the affinoid algebra $\mathbb{Q}_{p}\langle T\rangle$. By the Nullstellensatz the underlying set is the unit disc $D$. The “boundary” of this is the rational subdomain (and therefore an admissible open) $\displaystyle D\left(\frac{1}{T}\right)$, and its complement, the “interior” is covered by rational subdomains $\displaystyle D\left(\frac{T^{n}}{p}\right)$. With this covering the interior may also be shown to be an admissible open.

While it appears that, since we found two complementary admissible opens, we can disconnect the unit disc, we cannot actually do this in the Grothendieck topology, because the set consisting of the boundary and the interior is not an admissible open! And so in this way we see that the Grothendieck topology is the difference maker that allows us to overcome the obstacles posed by the peculiar geometry of the p-adic numbers.

Since Tate’s innovation, the idea of a p-adic or nonarchimedean geometry has blossomed with many kinds of “spaces” other than the rigid analytic spaces of Tate, for example adic spaces, a special class of which generalize perfectoid fields (see also Perfectoid Fields) to spaces, or Berkovich spaces, which are honest to goodness topological spaces instead of relying on a Grothendieck topology like rigid analytic spaces do. Such spaces will be discussed on this blog in the future.

References:

Rigid analytic space on Wikipedia

Algebraic geometry and analytic geometry on Wikipedia

Several approaches to non-archimedean geometry by Brian Conrad

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

p-adic families of modular forms [after Coleman, Hida, and Mazur] by Matthew Emerton

# 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 – namely 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

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 – 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.

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

# 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

# Hecke Operators

A Hecke operator is a certain kind of linear transformation on the space of modular forms or cusp forms (see also Modular Forms) of a certain fixed weight $k$. They were originally used (and now named after) Erich Hecke, who used them to study L-functions (see also Zeta Functions and L-Functions) and in particular to determine the conditions for whether an L-series $\sum_{n=1}^{\infty}a_{n}n^{-s}$ has an Euler product. Together with the meromorphic continuation and the functional equation, these are the important properties of the Riemann zeta function, which L-functions are supposed to be generalizations of. Hecke’s study was inspired by the work of Bernhard Riemann on the zeta function.

An example of a Hecke operator is the one commonly denoted $T_{p}$, for $p$ a prime number. To understand it conceptually, we must take the view of modular forms as functions on lattices. This is equivalent to the definition of modular forms as functions on the upper half-plane, if we recall that a lattice $\Lambda$ can also be expressed as $\mathbb{Z}+\tau\mathbb{Z}$ where $\tau$ is a complex number in the upper half-plane (see also The Moduli Space of Elliptic Curves).

In this view a modular form is a function on the space of lattices on $\mathbb{C}$ such that

• $f(\mathbb{Z}+\tau\mathbb{Z})$ is holomorphic as a function on the upper half-plane
• $f(\mathbb{Z}+\tau\mathbb{Z})$ is bounded as $\tau$ goes to $i\infty$
• $f(\mu\Lambda)=\mu^{-k}f(\Lambda)$ for some nonzero complex number $\mu$, and $k$ is the weight of the modular form

Now we define the Hecke operator $T_{p}$ by what it does to a modular form $f(\Lambda)$ of weight $k$ as follows:

$\displaystyle T_{p}f(\Lambda)=p^{k-1}\sum_{\Lambda'\subset \Lambda}f(\Lambda')$

where $\Lambda'$ runs over the sublattices of $\Lambda$ of index $p$. In other words, applying $T_{p}$ to a modular form gives back a modular form whose value on a lattice $\Lambda$ is the sum of the values of the original modular form on the sublattices of $\Lambda$  of index $p$, times some factor that depends on the Hecke operator and the weight of the modular form.

Hecke operators are also often defined via their effect on the Fourier expansion of a modular form. Let $f(\tau)$ be a modular form of weight $k$ whose Fourier expansion is given by $\sum_{n=0}^{\infty}a_{i}q^{i}$, where we have adopted the convention $q=e^{2\pi i \tau}$ which is common in the theory of modular forms (hence this Fourier expansion is also known as a $q$-expansion). Then the effect of a Hecke operator $T_{p}$ is as follows:

$\displaystyle T_{p}f(\tau)=\sum_{n=0}^{\infty}(a_{pn}+p^{k-1}a_{n/p})q^{n}$

where $a_{n/p}=0$ when $p$ does not divide $n$. To see why this follows from our first definition of the Hecke operator, we note that if our lattice is given by $\mathbb{Z}+\tau\mathbb{Z}$, there are $p+1$ sublattices of index $p$: There are $p$ of these sublattices given by $p\mathbb{Z}+(j+\tau)\mathbb{Z}$ for $j$ ranging from $0$ to $p-1$, and another one given by $\mathbb{Z}+(p\tau)\mathbb{Z}$. Let us split up the Hecke operators as follows:

$\displaystyle T_{p}f(\mathbb{Z}+\tau\mathbb{Z})=p^{k-1}\sum_{j=0}^{p-1}f(p\mathbb{Z}+(j+\tau)\mathbb{Z})+p^{k-1}f(\mathbb{Z}+p\tau\mathbb{Z})=\Sigma_{1}+\Sigma_{2}$

where $\Sigma_{1}=p^{k-1}\sum_{j=0}^{p-1}f(p\mathbb{Z}+(j+\tau)\mathbb{Z})$ and $\Sigma_{2}=p^{k-1}f(\mathbb{Z}+p\tau\mathbb{Z})$. Let us focus on the former first. We have

$\displaystyle \Sigma_{1}=p^{k-1}\sum_{j=0}^{p-1}f(p\mathbb{Z}+(j+\tau)\mathbb{Z})$

But applying the third property of modular forms above, namely that $f(\mu\Lambda)=\mu^{-k}f(\Lambda)$ with $\mu=p$, we have

$\displaystyle \Sigma_{1}=p^{-1}\sum_{j=0}^{p-1}f(\mathbb{Z}+((j+\tau)/p)\mathbb{Z})$

Now our argument inside the modular forms being summed are in the usual way we write them, except that instead of $\tau$ we have $((j+\tau)/p)$, so we expand them as a Fourier series

$\displaystyle \Sigma_{1}=p^{-1}\sum_{j=0}^{p-1}\sum_{n=0}^{\infty}a_{n}e^{2\pi i n((j+\tau)/p)}$

We can switch the summations since one of them is finite

$\displaystyle \Sigma_{1}=p^{-1}\sum_{n=0}^{\infty}\sum_{j=0}^{p-1}a_{n}e^{2\pi i n((j+\tau)/p)}$

The inner sum over $j$ is zero unless $p$ divides $n$, in which case the sum is equal to $p$. This gives us

$\displaystyle \Sigma_{1}=p^{-1}\sum_{n=0}^{\infty}a_{pn}q^{n}$

where again $q=e^{2\pi i \tau}$. Now consider $\Sigma_{2}$. We have

$\displaystyle \Sigma_{2}=p^{k-1}f(\mathbb{Z}+p\tau\mathbb{Z})$

Expanding the right hand side into a Fourier series, we have

$\displaystyle \Sigma_{2}=p^{k-1}\sum_{n}a_{n}e^{2\pi i n p\tau}$

Reindexing, we have

$\displaystyle \Sigma_{2}=p^{k-1}\sum_{n}a_{n/p}q^{n}$

and adding together $\Sigma_{1}$ and $\Sigma_{2}$ gives us our result.

The Hecke operators can be defined not only for prime numbers, but for all natural numbers, and any two Hecke operators $T_{m}$ and $T_{n}$ commute with each other. They preserve the weight of a modular form, and take cusp forms to cusp forms (this can be seen via their effect on the Fourier series). We can also define Hecke operators for modular forms with level structure, but it is more complicated and has some subtleties when for the Hecke operator $T_{n}$ we have $n$ sharing a common factor with the level.

If a cusp form $f$ is an eigenvector for a Hecke operator $T_{n}$, and it is normalized, i.e. its Fourier coefficient $a_{1}$ is equal to $1$, then the corresponding eigenvalue of the Hecke operator $T_{n}$ on $f$ is precisely the Fourier coefficient $a_{n}$.

Now the Hecke operators satisfy the following multiplicativity properties:

• $T_{m}T_{n}=T_{mn}$ for $m$ and $n$ mutually prime
• $T_{p^{n}}T_{p}=T_{p^{n+1}}+p^{k-1}T_{p}$ for $p$ prime

Suppose we have an L-series $\sum_{n}a_{n}n^{-s}$. This L-series will have an Euler product if and only if the coefficients $a_{n}$ satisfy the following:

• $a_{m}a_{n}=a_{mn}$ for $m$ and $n$ mutually prime
• $a_{p^{n}}T_{p}=a_{p^{n+1}}+p^{k-1}a_{p}$ for $p$ prime

Given that the Fourier coefficients of a normalized Hecke eigenform (a normalized cusp form that is a simultaneous eigenvector for all the Hecke operators) are the eigenvalues of the Hecke operators, we see that the L-series of a normalized Hecke eigenform has an Euler product.

In addition to the Hecke operators $T_{n}$, there are also other closely related operators such as the diamond operator $\langle n\rangle$ and another operator denoted $U_{p}$. These and more on Hecke operators, such as other ways to define them with double coset operators or Hecke correspondences will hopefully be discussed in future posts.

References:

Hecke Operator on Wikipedia

Modular Forms by Andrew Snowden

Congruences between Modular Forms by Frank Calegari

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

Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman