Serre’s uniformity conjecture, concerns mod \(p\) representations of Elliptic curves over the rationals. Let \(E/\mathbb{Q}\) be an elliptic curve without CM, let \(p>37\). Then
\[\overline{\rho}_{E,p}(G_\mathbb{Q}) = \mathrm{GL}_2(\mathbb{F}_p)\]
i.e. the mod \(p\) representation is surjective.
Theorem The subgroup \(H\le \mathrm{GL}_2(\mathbb{F}_p)\) not containing \(\mathrm{SL}_2(\mathbb{F}_p)\) is one of
\(H\subset B_0(p)=\left\{\begin{pmatrix} *&*\\0&*\end{pmatrix}\right\}\)
Normalizer of split Cartan
\[H\subset N_s^+(p)=\left\{\begin{pmatrix} \alpha& 0\\0 &\beta\end{pmatrix}, \begin{pmatrix} 0& \alpha\\ \beta& 0\end{pmatrix}\right\}\]
\(H\subset N_{ns}(p)\) (normalizer of non-split Cartan), a subgroup of \(\mathrm{GL}_2(\mathbb{F}_p)\) which is..
image of \(H\) in \(\mathrm{PGL}_2(\mathbb{F}_p)\) is isomorphic to \(A_4\), \(S_4\) or \(A_5\), these are called the exceptional subgroups of \(\mathrm{GL}_2(\mathbb{F}_p)\).
Theorem (Serre). Let \(p\ge 11\) and \(E/\mathbb{Q}\) be an elliptic curve. Then the mod \(p\) image of representation \(\overline{\rho}_{E,p}(G_\mathbb{Q})\) is not exceptional.
Given \(K\) a field, \(N\ge 1\) and \(H\subset \mathrm{GL}_2(\mathbb{Z}/N)\), the thing we want to understand is the set of Elliptic curves over \(K\) whose mod \(N\) image is contained inside \(H\) (up to conjugation, because the image only make sense up to conjugation).
\[\left\{ E/K \mid \overline{\rho}_{E,N}(G_K)\subset_{conj} H\right\}\]
(With certain assumptions on \(H\)) such elliptic curves give rise to non-cuspidal \(K\) points on the modular curve \(X_H\).
The upshot is understanding \(X_H(K)\Rightarrow\) understand the set of elliptic curves with mod \(N\) image (conjugate) in \(H\).
Let \(X\) be an algebraic curve (smooth projective), defined over a number field \(K\). Let \(g\) be the genus of \(X\). Then
\(g=0 \Rightarrow X(K)=\varnothing\) or \(X(K)\cong \mathbb{P}^1(K)\)
\(g=1\), it could have no points, or the points form a finitely generated abelian group \(X(K)\) is an elliptic curve.
\(g\ge 2\), Faltings’ theorem says \(X(K)\) is finite.
(\(H=\mathrm{GL}_2(\mathbb{Z}/N)\)). Here the condition \(\subset H\) is vacuous, so this is all elliptic curves over \(K\).
\[\mathbb{H}=\{z\in \mathbb{C}\mid \Im(z)>0\}\]
also the extended upper half plane
\[\mathbb{H}^*=\mathbb{H}\cup \mathbb{P}^1(\mathbb{Q}).\]
There exists an action of \(\mathrm{SL}_2(\mathbb{Z})\) on \(\mathbb{H}^*\) given by fractional linear transformations
\[\mathrm{SL}_2\times \mathbb{H}^*\to \mathbb{H}^*\quad \left(\begin{pmatrix} a&b\\c&d \end{pmatrix}, z\right)\mapsto \frac{az+b}{cz+d}\]
from the theory of elliptic functions,
Associated to a \(\tau\in \mathbb{H}\) there is an elliptic curve \(E_\tau/\mathbb{C}\) such that
\[E_\tau(\mathbb{C})\cong \frac{\mathbb{C}}{\mathbb{Z}+\mathbb{Z}\tau}\]
Any complex elliptic curve \(E/\mathbb{C}\) there is a \(\tau\in \mathbb{H}\) such that \(E\cong E_\tau\).
\(E_{\tau_1}\cong E_{\tau_2}\) if and only if \(\tau_1=\gamma\tau_2\) for some \(\gamma\in \mathrm{SL}_2(\mathbb{Z})\).
Therefore we have a bijection
\[ \mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}\leftrightarrow \{ \text{iso class of complex elliptic curves}\}\] \[ [\tau]\mapsto [E_\tau]\]
The Riemann surface \(\mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}\) is non-compact whose compactification is \(\mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}^*\). We know that any compact Riemann surface is just a complex algebraic curve, and in fact \(\mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}^*\) has genus \(0\), i.e.
\[ \mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}^*\cong \mathbb{P}^1(\mathbb{C})\]
In this context we denote that \(\mathbb{P}^1\) by \(X(1)\).
\[ j:\mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}\to X(1)(\mathbb{C}) \quad [\tau]\mapsto j(\tau)\]
Let \(Y(1) = X(1)\backslash \{\infty\}\), there is a one to one correspondence between
\[\{\text{iso class }E/\mathbb{C}\}\leftrightarrow Y(1)(\mathbb{C})\quad [E]\mapsto j(E)\]
Now let \(K\) be any field, we still have a bijection
\[\{\text{iso class }E/\overline{K}\}\leftrightarrow Y(1)(\overline{K})\]
points on \(Y(1)(K)\) correspond to elliptic curves \(E/\overline{K}\) stable under the action of Galois group
\[E^\sigma \cong_{\overline{K}} E\]
(this isomorphism feels like higher category or derived category here). This is classes of elliptic curves defined over \(K\) but isomorphic over \(\overline{K}\).
Suppose \(E/\overline{K}\) and \(E^\sigma\cong_{\overline{K}} E\) for all \(\sigma\), then \(j(E)^\sigma = j(E^\sigma)=j(E)\), so \(j(E)\in Y(1)(K)\).
Isomorphism class of elliptic curves with a \(N\) torsion point \((E,P)\), corresponds to points in the modular curve \(X_1(N) = \Gamma_1(N)\backslash \mathbb{H}^*\) ( a compact Riemann surface.)
\[Y_1(N)=X_1(N)\backslash \{\text{cusps}\}.\]