# Lecture 1: Overview and what questions we're interested in
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References:
- Kevin's website:
- Youtube Playlist:
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This course: adapted from handwritten notes from a course by Richard Taylor in 1992 at Caltech.
The original course focused on $\GL_2$, we'll discuss $\GL_n$.
About a year before Wiles-Taylor!
Ways to learn:
- Engage with the material
- Type it up.[^like_me]
Better than hours of videos!
- Complete the exercises and fill in the gaps.
See also the problem set on the [course website](https://www.ma.imperial.ac.uk/~buzzard/MSRI/).
- Talk to the experts at the talks
- Complete the project on the website on the abelian \(p\dash \)adic Langlands correspondence.
This mimics the classical correspondence, and is very much in its infancy at the moment.
In all cases, it's useful to work with other people, communicate, interact, etc.
[^like_me]:
DZG: Like me!
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The story begins with Dirichlet characters -- let $N\in \ZZ_{\geq 1}$ and consider a character
\[
\chi: C_n\units \to \CC && \in \Grp
.\]
Attached to $\chi$ is a Galois representation
\[
\rho_\chi: G \da \Gal(\bar{\QQ}/\QQ) \to \GL_1(\CC)
.\]
Note that $G$ is an infinite group.
This arises as the following composition:
\begin{tikzcd}
G &&&& {\GL_1(\CC)} \\
\\
{\Gal(\QQ(\zeta_N)/\QQ)} &&&& {C_N\units}
\arrow["{=}", from=3-1, to=3-5]
\arrow["\chi"', from=3-5, to=1-5]
\arrow[two heads, from=1-1, to=3-1]
\arrow["{\rho_\chi}", from=1-1, to=1-5]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJHIl0sWzQsMCwiXFxHTF8xKFxcQ0MpIl0sWzQsMiwiQ19uXFx1bml0cyJdLFswLDIsIlxcR2FsKFxcUVEoXFx6ZXRhX24pL1xcUVEpIl0sWzMsMiwiPSJdLFsyLDEsIlxcY2hpIiwyXSxbMCwzLCIiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMCwxLCJcXHJob19cXGNoaSJdXQ==)
Here the equality denotes a canonical isomorphism, and is given by the map $n\mapsto (\zeta_N \mapsto \zeta_N^n)$.
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There is a version of this for $\GL_2$.
Let $f$ be a cuspidal modular form which is an eigenform for the Hecke operators $T_p$, which are endomorphisms of the space of modular forms.
So $T_p f = \lambda f$ for some $\lambda \in \CC$, and it turns out that the subfield $\gens{\lambda_p} \CC$ (which could be infinitely generated) is a number field -- it is equipped with an embedding into $\CC$ and has finite degree over $\QQ$.
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:::{.theorem title="Deligne, 60s/70s"}
Let $\ell \in \ZZ$ is prime and $\lambda \divides \ell$ is a prime of $E_f$, i.e. a nonzero prime ideal of $\OO_{E_f}$.
Following an observation due to Serre, Deligne constructs a map
\[
\rho_f: G \to \GL_2(\cl_\alg (E_f \complete{\lambda} ) )
,\]
where the RHS is completing $E_f$ at $\lambda$ and taking an algebraic closure.
This is an $\ell\dash$adic representation, and formally resembles the $\CC\dash$representation above in the sense that $\rho_f$ is "attached" to $f$.
Deligne's construction uses étale cohomology.
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For $f$ a modular form, there is a level $N\geq 1$, a weight $k\geq 1$, and a Dirichlet character $\chi$.
It turns out that $\rho_f$ is unramified away from $N\ell$ -- note that there is no analog of $\ell$ for the $\GL_1$ case.
If $p$ is a prime not dividing $N\ell$, then $\rho_f(\Frob_p)$ has characteristic polynomial
\[
x^2 + \lambda_p x + p^{k-1}\chi(p)
.\]
Since $p\notdivides N\ell$, $p\notdivides N$ and thus $\chi(p) \neq 0$.
Note that this is the trace of the representation $\rho$, and it turns out that the conjugacy classes $\Frob_p$ are dense in $G$.
By **Chebotarev density**, there is at most one semisimple $\rho_f$ with this property, and Deligne's theorem is that there is *at least* one.
These days, $N\dash$dimensional $\ell\dash$adic Galois representations are common precisely because we now know they are the right things to look at.
Historically, number theorists may not have considered modular forms number-theoretic objects -- instead, they were considered objects of harmonic analysis, and number theory likely focused on things like class numbers and Iwasawa's main conjecture.
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A word on Deligne's construction -- how does he find a 2-dimensional $\ell\dash$adic representation of $G$?
He constructs $\rho_f$ using étale cohomology with nontrivial coefficients.
A modern take is that it would come from a motive, and étale cohomology produces motives, although one would normally take trivial coefficients.
There is a slight issue in constructing $\ell\dash$adic sheaves on the modular curve.
Deligne later uses trivial coefficients and takes cohomology on some power of a universal elliptic curve.
Thus Deligne's proof is a partial proof that the $\rho_f$ are motivic.
This all assumes $k\geq 2$, and the $k=1$ case was handled by Deligne-Serre in the 70s.
> #todo What is $k$
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Some questions arising from Deligne's construction:
- What do the representations locally look like at ramified primes $p$, i.e. $p\divides N\ell$?
There is a formal meaning here: the global Galois group contains the local one, and $G$ is embedded canonically and only ambiguous up to conjugacy.
One can restrict the global representation to this local representation
In the unramified case, we know the characteristic polynomial, and this essentially determines the local behavior, although there is a semisimplicity issue involving the Tate conjecture -- specifying the characteristic polynomial of a matrix doesn't uniquely determine it, due to possible multiplicity in eigenvalues.
- Case 1: if $p\divides N$ and $p\neq \ell$, the answer is given by the conjectured **local Langlands correspondence**.
These are theorems for $\GL_2(\QQpadic)$ and $\GL_n(K)$ for any $K\in\Local\Field$.
This correspondence aims to relate (possibly infinite dimensional) representations of groups like $\GL_n(\QQpadic)$ to Weil-Deligne representations, which are similar to Galois representations.
We'll soon explain what this has to do with modular forms.
- Case 2: if $p=\ell$, so we have a \(p\dash \)adic representation of $G$, there should be a **\(p\dash \)adic local Langlands correspondence**.
This is essentially the boundary of what we currently know, and in a sense we don't even know what the right question should be.
This is a theorem for $\GL_2$, but unknown for $\GL_3$ and above.
Ask Rebecca about this!
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An easier variant: instead of asking for $\rho_f$, ask for $\bar\rho_f: \Gal(\QQbar/\QQ)\to \GL_2(\bar\FF_\ell)$, i.e. reduce $\mod \ell$.
Identify $\bar\FF_\ell$ with the residue field of $\bar E_\lambda$ at $\lambda$.
So reduce Deligne's $\ell\dash$adic representation to get a $\mod \ell$ representation.
These reps are easier to understand since there are tricks, e.g. looking for $\bar\rho_f$ in $A[\ell]$ for some $A\in \Ab\Var$.
Note that $\rho_f$ involves étale cohomology.
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Questions Taylor asked at the time:
Are $\rho_\chi, \rho_f$ special cases of a general story?
Note that this relates Dirichlet characters to modular forms.
Answer: yes, kind of.
There is the following theorem from 2013, which was reproved by Scholze using perfectoid magic.
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:::{.theorem title="?"}
Let $E$ be a totally real (so all embeddings $E\embedsvia{\sigma} \CC$ have $\sigma(E) \subseteq \RR$) or CM number field (totally imaginary extension of a totally real number field) and let $\pi$ be a *cuspidal automorphic representation* of $\GL_n(\AA_E)$.
Suppose that $\pi$ is "cohomological", which is a condition on the weights and the PDEs that the automorphic forms come from and is a strong algebraicity condition.
Then there is a representation $\rho_\pi: \Gal(\bar E/E)\to \GL_n(\QQbar_\ell)$ attached to $\pi$ in a canonical way, which is the analog of $\ch\poly \rho_f(\Frob_p)$
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Automorphic forms will be solutions to elliptic PDEs
Unrelated, but see [Frank Calegari's blog](https://www.galoisrepresentations.com/about/).
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Meta-theorem: Galois representations come from étale cohomology groups and their $p\dash$adic deformations, say of an algebraic variety defined over a number field e.g. a Shimura variety.
General idea: given an algebraic or analytic object like $\chi, f, \pi$ (or more generally *motivic* objects), some technical machinery produces representations of Galois groups.
Can we classify the image of this correspondence?
Which Galois representations come from such things?
I.e. given $\rho: G\to \GL_n(k)$ for *some* $k\in \Field$, is $\rho$ isomorphic to some $\rho'$ coming from an algebraic variety?
Dimension 1: let $K/\QQ$ be finite Galois and $\rho: \Gal(K/\QQ)\to \GL_1(\CC)$.
Is $\rho\cong \rho_\chi$ for $\chi$ a Dirichlet character?
We can take an epi-mono factor any group morphism and Galois theory works better with epis, so replace $K$ with a subfield $L \leq K$ to make $\rho$ *injective*.
This is because $\rho: \Gal(K/\QQ) \surjects \Gal(L/\QQ) \injects \GL_1(\CC)$.
So we assume
\[
\rho: \Gal(K/\QQ) \injects \CC\units
,\]
hence $\Gal(K/\QQ)\in \Ab\Grp$.
A reminder of what a $\rho_\chi$ will look like:
\begin{tikzcd}
{C_N \units} && \CC\units \\
\\
{\Gal(\QQ(\zeta_N)/\QQ)}
\arrow[Rightarrow, no head, from=3-1, to=1-1]
\arrow["\chi", from=1-1, to=1-3]
\arrow["{\rho_\chi}"', from=3-1, to=1-3]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJDX04gXFx1bml0cyJdLFsyLDAsIlxcQ0NcXHVuaXRzIl0sWzAsMiwiXFxHYWwoXFxRUShcXHpldGFfTikvXFxRUSkiXSxbMiwwLCIiLDAseyJsZXZlbCI6Miwic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFswLDEsIlxcY2hpIl0sWzIsMSwiXFxyaG9fXFxjaGkiLDJdXQ==)
By the same trick, we can factor $\rho_\chi$ to assume it is injective from some $L \leq \QQ(\zeta_N)$ a subfield of a cyclotomic field.
So the question reduces to the following:
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If $K\in \Number\Field$ is Galois over $\QQ$ with $\Gal(K/\QQ)\in \Ab\Grp$, does there exist an $N\geq 1$ with $K \injects \QQ(\zeta_N)$.
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Answer: yes, but this takes some work and has a name, the Kronecker-Weber theorem (an explicit special case of global CFT).
Note that the converse is clear: subfields of cyclotomic fields will yield Galois groups which are subgroups of a cyclic group and hence abelian.
So for all $\rho: \Gal(\QQbar/\QQ) \to \GL_1(\CC)$ (so the image is finite order), there is a $\chi: C_N\units \to \CC\units$ with $\rho\cong \rho_\chi$.
Thus is the proposed correspondence, the image is everything, and the proof is class field theory.
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What about $\GL_2$?
Let $f$ be a cuspidal modular eigenform as before, there is a Galois representation $\rho_f: \Gal(\QQbar/\QQ) \to \GL_2(\QQbar_\ell)$ such that
- $\rho_f$ is absolutely irreducible
- $\rho_f$ is "odd", i.e. writing $\conj\in \Gal(\QQbar/\QQ)$ for complex conjugation, $\det \rho_f(\conj) = -1$.
- $\rho_f$is unramified away from a finite set of primes and carries some $p\dash$adic Hodge theory and is *potentially semistable*.
Étale cohomology always has the third property, which is sometimes called "being geometric", and the conjecture is that geometric representations come from geometry.
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Fontaine-Mazur asked in the 1990s if all $\rho$ satisfying these properties are of the form $\rho_f$ for some $f$, which became the Fontaine-Mazur conjecture.
This is basically known now, see Emerton and Kisin.
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Similarly, we can ask that if $\rho: \Gal(\bar E/\E) \to \GL_n(\QQbar_\ell)$ satisfies some assumptions, is it the case that $\rho \cong \rho_\pi$ for some $\pi$.
State of the art: BLGGT, proves this in many cases using very technical $p\dash$adic Hodge theory.
See also the 10-author paper.
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