The title of this post is a bit of a misnomer. We won’t be talking about representations of elliptic curves, but about representations attached to (associated with?) elliptic curves. In particular, the goal of this post is to prove that given an elliptic curve $E/\Q$ defined over the rationals, and a prime $\l$, the $\l$adic representation
is irreducible where $G_{\Q}=\Gal(\Qbar/\Q)$ is the absolute Galois group of $\Q$ and $T_\l(E)=\invlim E[\l^n]$ is $E$’s $\l$adic Tate module. If you don’t know what some of these words mean, don’t worry; I’ll explain. ^{1}
It’ll be quite a while before we get into defining and proving irreducibility of these representations since doing so requires a lot of ideas I have not introduced on this blog before. To begin, we’ll introduce some of the basics of the general theory of algebraic curves before focussing specifically on elliptic curves. Since I’ve said the word $\spec$ here before, I could set things up in those terms ^{2}, but I won’t; instead, I’ll take an approach that’s more concrete. ^{3} With that said, let’s get started$\dots$
Note: I’m writing this post like a madman (in a rushed manner manner split over several days without much of a game plan ahead of time), so it likely contains more mistakes than usual (e.g. there may be some circular arguments here). Some time after finishing it and putting it online, I’ll take another look at it and try to resolve all these issues. Until then, recovering a coherent post from what’s below is left as an exercise to the reader ^{4}.
Algebraic Varieties
This section is mostly definitions ^{5}, and so can be skipped and referred back to whenever you see something but don’t know what it means.
Fix a field $k$ with algebraic closure $\bar k$. Let $\bar k[X]=\bar k[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables. Let $\A^n=\A^n(\bar k)=\bracks{P=(p_1,\dots,p_n):p_i\in\bar k}$ denote affine $n$space (over $\bar k$).
Note that Hilbert’s basis theorem tells us that $\bar k[X]$ is noetherian, so any algebraic set is given by the vanishing of only finitely many polynomials.
That was a lot of definitions, and there are only more to come. If you haven’t seen material like this before, this may be a good place to mention that my previous post on algebraic geometry might be helpful for understanding why some of the definitions here are reasonable. For example,
We now turn to projective algebraic sets and projective varities. Projective $n$space (over $k$), denoted $\P^n=\P^n(\bar k)$, is the set of lines through the origin in $\A^{n+1}$. Succienctly, $\P^n=(\A^{n+1}\sm{0})/\units{\bar k}$. More explicitly,
Note that $G_{\bar k/k}$ acts on $\P^n$ be acting on each coordinate individually. Since individual coordinates in projective space are not welldefined, evaluating a polynomial at a projective point doesn’t make sense. However, for homogeneous polynomials, we can still check if one vanishes at a point.
Given a homogenous polynomial $f$, the answer to the question, “does $f(P)=0$?” does not depend on how we write down $P$, so we can define a vanishing set $V(f)\subset\P^n$. Similarly, we get some vanishing set $V(I)\subset\P^n$ for any homogenous ideal $I$.
Note that there are many ways of embedding $\A^n\subset\P^n$. For example, each set
is easily seen to be a copy of $\A^n$, e.g. via the natural bijection $\phi_i:\A^n\to U_i$ given by
Hence, given any projective algebraic set $C$ with homogeneous ideal $I(C)$ (and a choice of $i$), $\inv\phi_i(V\cap U_i)$, which we call $C\cap\A^n$, is an affine algebraic set with ideal
This shows that any projective algebraic set (resp. variety) is covered by a bunch of affine algebraic sets (resp. varieties) $C\cap U_0,\dots,C\cap U_n$. The process of replacing a homogeneous $f(X_0,\dots,X_n)$ with $f(Y_1,\dots,Y_{i1},1,Y_i,\dots,Y_n)$ is called dehomogenization with respect to $X_i$, and can be reversed by taking $f(Y)\in\bar k[Y]=\bar k[Y_1,\dots,Y_n]$ to the homogenization of $f$ with respect to $X_i$.
This let’s us take an affine algebraic set $V\subset\A^n$ to its projective closure which is the projective algebraic set whose homogenous ideal is generated by $\bracks{\ast f(X):f\in I(V)}$. When we do this to an affine variety $V$, we get out a projective variety. Because we can move back and forth between the affine and projective worlds like this, we often make definitions on projective varieties by referring to the analagous thing on one of their affine covers. For example,
At this point, we’ve moreorless laid the groundwork for what a (projective) variety is. So, the next thing would be to describe maps between them. One quirky thing about varieties is that their maps are not required to be defined everywhere. ^{6}
 each $gf_i$ is regular at $P$, and
 there is some $i$ for which $(gf_i)(P)\neq0$.
Phew. Alright, I think we can move on to the next section now. I hope you have all of this memorized.
Algebraic Curves
An algebraic curve is a projective variety of dimension 1. For curves, $\bar k[C]$ is nicer than just an arbitrary integral domain.
Now, here’s some nice information about regular functions that we’ll just take for granted.
WIth this, let’s move onto differentials.
Differentials
Because calculus is quite useful for doing analysis/geometry over $\R$ and $\C$, we’d like something similar for studying curves over any field $k$.
 $\d(x+y)=\dx+\dy$
 $\d(xy)=x\dy+y\dx$
 $\d a=0$
Now, it’ll be useful to know that $\Omega_C$ is a $1$dimensional $\bar k(C)$vector space for any curve $C$; I will not prove this, but it’s good to know. As a consequence of this (+ something else I didn’t bother mentioning), any $\omega\in\Omega_C$ can be written as $\omega=g\dt$ for a unique $g\in\bar k(C)$ where $t\in\bar k(C)$ is some fixed uniformizer. We’ll denote this $g$ by $\omega/\dt$. Note that the quantity $\ord_P(\omega/\dt)$ does not depend on the choice of uniformizer $t$, and so we simply denote it by $\ord_P(\omega)$. Furthermore, $\ord_P(\omega)\neq0$ for only finitely many $P\in C$ (when $\omega\neq0$). Call $\omega$ holomorphic (I don’t know what the standard term is) is $\ord_P(\omega)\ge0$ for all $P\in C$.
Divisors & RiemannRoch
This is the point in this blog post where I may actually start bothering to prove things I claim. ^{7} We’ll state and prove one of the main tools in the study of algebraic curves: the RiemannRoch theorem. Before we can state it though, we first need to introduce some terminology:
Part of the utility of divisors is that they are a convenient way of describing the location and number of zeros/poles of functions. To make this more precise, given two divisors $D_1,D_2$, we say that $D_1\ge D_2$ if $\ord_P(D_1)\ge\ord_P(D_2)$ for all $P\in C$. If $D\ge0$, then we say that $D$ is effective. Note that for a function $f\in\units{\bar k(C)}$, point $P\in C$, and number $n\in\Z_{>0}$, we have $\div(f) + n[P]\ge0$ iff $f$ has a pole of order at most $n$ at $P$, and we similarly have $\div(f)  n[P]\ge0$ iff $f$ has a zero or order at least $n$ at $P$.
A natural question to ask, and one mostly answered by RiemannRoch, is, “Can we calculate $\l(D)$ for an arbitrary divisor?” Our main tool for getting a handle of $\l(D)$ ^{8} is a certain 7term exact sequence.
Before we can prove this lemma, we have to be able to say what all these maps are. While most of them are fairly straightforward (at least when $D=0$), the map $\bar k\to\dual{\Omega(PD)}$ is more involved. To define it, first fix a uniformizer $t\in\bar k(C)$ at $P$ (i.e. $\ord_P(t)=1$). Now, given any differential form $\omega\in\Omega_C$, we can write $\omega=g\dt$ for some unique $g\in\bar k(C)$. Forming the completion $\wh{\bar k[C]_ P}=\invlim_n\bar k[C]_ P/\mfm_P^n\bar k[C]_ P$, and letting $\bar k(C)_ P:=\Frac\parens{\wh{\bar k[C]_ P}}$, one notes that $\bar k(C)_ P$ is isomorphic to the ring of formal power series in $t$ with coefficients in $\bar k$ ^{9}. This let’s us write a Taylor (or Laurent) expansion for $g$:
and there exists some $m>0$ s.t. $n<m\implies a_n(g)=0$. With this in mind, define
modelled after residues in complex analysis ^{10}. Note that $\ord_P(g)$ is the smallest $n$ such that $a_n(g)\neq0$, so $\res_P(\omega)=0$ for any holomorphic form $\omega$. In our to be able to effectively employ residues, one needs to know the following. ^{11}
Now, we’ll prove a special case of the previous lemma, and leave the construction of this exact sequence for an arbitrary divisor $D$ as an exercise.
 The first map $\mc L(DP)\to\mc L(D)$ is the natural inclusion map.
 The second map $\mc L(D)\to\bar k$ is "evaluation" at $P$, i.e. $f\mapsto (t^{\ord_P(D)}f)(P)$
 The third map $\bar k\to\Omega(PD)^\vee$ sends $c\in\bar k$ to $$\omega\longmapsto c\cdot\res_P(t^{\ord_P(D)}\omega)$$
 The fourth map $\dual{\Omega(PD)}\to\dual{\Omega(D)}$ is the natural restriction map.
 Exactness at $\mc L(DP)$ is obvious because this is literally an inclusion map.
 Exactness at $\mc L(D)$ follows from the fact that $\mc L(DP)$ is the space of maps $f\in\mc L(D)$ s.t. $\ord_P(f)+\ord_P(D)\ge1$.
 Exactness at $\bar k$ is the statement that there exists $\omega\in\Omega(PD)$ such that $\res_P(t^{\ord_P(D)}\omega)\neq0\iff(t^{\ord_P(D)}f)(P)=0$ for all $f\in\mc L(D)$.
Elliptic Curves
Group Law
Isogenies
Torsion Points
$\l$adic Representations

There is a lot to explain, so I anticipate this becoming one of my longer (and also more fun) posts to date…. In the end, this became my second post that I feel is overly ambitious and so hard to follow. If you’ve read some of this, please tell me how hard it is to understand so I can know for sure if I really am trying to do too much all at once. ↩

I’ve also said the word sheaf before, so I could really try to be fancy ↩

My main reference for this post is Silverman who doesn’t use $\spec$, and rather not have to do the extra work associated with translating terminology and making sure I don’t say anything false. ↩

I’m sure things aren’t as bad as I make them sound hear, but gotta set the bar low so it’s easier to exceed expectations. ↩

This section is basically just the first chapter of Silverman because I’m unoriginal. ↩

Think of meromorphic maps from complex analysis ↩

Could you imagine how long this post would be if I did this the whole time? ↩

Including proving RiemannRoch ↩

At the very least, showing that the underlying additive group of this ring is formall power series can be done relatively easily in much the same manner as in my padic post ↩

This value does not depend on the choice of uniformizer $t$, but to avoid having to show this, we fixed a canonical choice in the beginning (if you’re bothered by this, show independence as an exercise). ↩

Working over $\C$, one can show this easily using Stoke’s theorem. However, we’re doing algebraic geometry, not complex geometry, so we need to be more creative ↩