# Tuesday, November 30

## Faltings Theorem

:::{.definition title="?"}
Write $\Rep_N^*(G_{k, S}; \QQladic)$ for semisimple continuous representation $G_{k, S\union\ts{\ell}} \to \GL_n(\OO_\ell)$ of weight -1 such that for all $v\in S\union\ts{\ell}$, the characteristic polynomial of $\Frob_v$ has integer coefficients.
Note that the roots are $\alpha_i \in \bar{\QQ}$ such that $\QQ(\alpha_i) \embeds \CC$ and $\abs{\alpha_i} = q^{i\over 2}$.
:::

:::{.theorem title="Faltings"}
$\size X(k) < \infty$.
:::

:::{.theorem title="?"}
$\Rep_N^*(G_{k, S}, \OO_\ell)$ is finite.
:::

### Proof

:::{.lemma title="A very important fact"}
If $\Gamma$ is a group and $\Gamma \mapsvia{\rho} \GL_n(L)$ is a semisimple representation over $L$ a field of characteristic zero.
Then $\rho$ is determined by $\Tr \circ \rho: \Gamma\to L$, i.e. any two representations for which these composites agree differ by conjugacy.
:::

:::{.fact}
$\rho\in \Rep^*_N(G_{k, S}, \OO_\ell)$ is determined by $\ts{\Tr \circ \rho \circ \Frob_v \st v\in S \union \ts{\ell}}$ since the Frobenii are dense in $G_{k, S\union \ts{\ell}}$ by Chebotarev density.
:::

:::{.fact}
Fix $v\not\in S \union\ts{\ell}$, then there are only finitely many possibilities for $\abs{\Tr \circ \rho \Frob_v} < N\cdot \abs{\kappa(v)}^{1\over 2}$, since traces of Frobenii are sums of eigenvalues.
:::

:::{.remark}
This provides a finite set of Frobenii whose traces determine $\rho$.
Given $\rho_1, \rho_2$, take the group ring of the Galois group and consider the map
\[
\QQ_\ell \adjoin{ G_{k, S\union\ts{\ell}} } \mapsvia{\rho_1 \times \rho_2} \Mat(n\times n; \ZZpadic)\cartpower{2}
,\]
where we used that $\GL_n(\ZZpadic) \leq \GL_n(\QQpadic)$ is a maximal compact subgroup.
The goal is to find a set of places $T$ such that $\ts{ (\rho_1, \rho_2)(\Frob_v) \st v\in T}$ spans the image, which we can do since the RHS is a finite-dimensional $\ZZpadic$ module.
:::

:::{.remark}
Let $\tilde k$ be the compositum of all extensions of $k$ unramified outside of the set of bad places $S\union\ts{\ell}$ of bounded degree $\ell < 2n^2$, the dimension of the RHS above.
This is a finite extension by Hermite-Minkowski.
Let $\tilde k^\cl$ be the Galois closure, and by Chebotarev choose $T$ such that $\ts{\Frob_v}_{v\in T}$ cover all conjugacy classes of $\Gal(\tilde k^\cl/k)$.
:::

:::{.claim}
For $\rho_1, \rho_2 \in \Rep_N^*(G_{k, S}; \OO_\ell)$, if $\Tr(\ro{\Frob_v}{\rho_1}) = \Tr(\ro{\Frob_v}{\rho_2})$ for all $v\in T$, then $\rho_1 \cong \rho_2$.
:::

:::{.proof title="?"}
Consider $\ZZladic [ G_{k, S} ] \mapsvia{(\rho_1, \rho_2)} \Mat_{N\times N}(\ZZladic)\cartpower{2}$, write $M\da \im(\rho_1, \rho_2)$.
ETS that the images of $\Frob_v$ generate this as a $\ZZladic$ module.
By Nakayama, it's enough to check that they generate $M/\ell M$.
Note that $\# (M/\ell M)\units < \ell^{2N^2}$, so write $\tau: G_{k, S\union\ts{\ell}} \to (M/\ell M)\units$, then $\tau$ factors through $\Gal\qty{ \tilde k^\cl/ k}$.
Then $\im\qty{ \ts{\Frob_v \st v\in T} } = \im \tau$.
:::

## The Kodaira-Parshin Trick

:::{.theorem title="Parshin"}
Let $X$ be a curve over $\OO_{k, S}$, $g(X) \geq 2$.
After potentially increasing $k$ and $S$, there exists a finite map of algebraic varieties
\[
X' \to \mcm_{g'}/\OO_{k, S}
\]
for some $g'$ where $X'$ is a finite étale cover of $X$.
:::

:::{.remark}
Note that $\mcm_2$ is affine and can't contain a smooth proper curve.
In general it isn't even proper, so it's difficult to map a proper thing into a non-proper thing.
Note that the RHS is a stack over $\OO_{k, S}$.
Why compactify in general: argue something is an open condition, degenerate to the boundary where the objects are easier to work with and show it holds there and thus on an open containing it.
:::

:::{.remark}
Note that $\Hom(X, \mcm_{g'})$ is the of smooth proper morphisms $Y\to X$ over $\OO_{k, S}$ such that the geometric fibers are curves of genus $g'$.
Take $\mcy \mapsvia{\pi} X$ and $x\in X$ such that $\pi\inv(x)$ is a cover of $X$ ramified only at $x$.

:::{.slogan}
$X$ is a moduli space of branched covers of $X$ branched only at a single point.
:::

Idea: create a moduli space of such covers.
Take $\eta: X\cartpower{2}\sm\Delta \to X$ where the fiber over $X$ is $X\sm\ts{x}$.


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:::

:::{.remark}
Over $\CC$, note that $X$ is covered by $\HH$ so $\pi_2 X =1$ and $\eta$ is a fibration, so taking the LES in homotopy yields a SES
\[
1\to \pi_1(X\smpt) \to \pi_1(X\cartpower{2}\sm\Delta) \to \pi_1(X) \to 1
.\]
Here $\pi_1(X) = \gens{a_1, b_1,\cdots, a_g, b_g \st \prod [a_i b_i] = 1}$.
Pick any $\gamma: \pi_1(X\smpt) \to S_3$ which doesn't send the loop around the puncture to the identity, i.e. $\gamma \qty{\prod [a_i b_i] } \neq \id$.
This now has ramification at a point.

Define $\Gamma \da \ts{g\st\pi_1(X\cartpower{2} \sm\Delta) \st \gamma^g \text{ is conjugate to } g }$, i.e. there exists $h_g \in S_3$ such that for all $x\in \pi_1(X\smpt)$ we have $\gamma(gxg\inv) = h_g \gamma(x) h_g\inv$ for all $x\in \pi_1(X\smpt)$.
:::

:::{.claim}
$[\Gamma: \pi_1(X\cartpower{2}\sm\Delta)] < \infty$ has finite index and contains $\pi_1(X\smpt)$.
:::

:::{.claim}
This yields a group morphism
\[
\Gamma &\to S_g \\
g &\mapsto h_g
.\]
:::

:::{.remark}
Some words on why these are true: stabilizers of groups acting on finite sets have finite index, and the subgroup claim comes from sending $h_g\to \gamma(g)$.
:::

:::{.remark}
The construction:

\begin{tikzcd}
	\mcy \\
	\\
	{X_\Gamma} && {X\cartpower{2}\sm\Delta} \\
	\\
	{X'} && X
	\arrow[from=1-1, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-1, to=5-3]
	\arrow[from=3-3, to=5-3]
	\arrow["{\text{finite fibers}}"', from=5-1, to=5-3]
	\arrow[from=3-1, to=5-1]
	\arrow[curve={height=30pt}, from=1-1, to=5-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXG1jeSJdLFswLDIsIlhfXFxHYW1tYSJdLFsyLDIsIlhcXGNhcnRwb3dlcnsyfVxcc21cXERlbHRhIl0sWzIsNCwiWCJdLFswLDQsIlgnIl0sWzAsMV0sWzEsMl0sWzEsM10sWzIsM10sWzQsMywiXFx0ZXh0e2Zpbml0ZSBmaWJlcnN9IiwyXSxbMSw0XSxbMCw0LCIiLDEseyJjdXJ2ZSI6NX1dXQ==)

The fibers of $\mcy\to X$ are disjoint unions of covers of $X$ ramified at $x$.
Since we now have a family of curves, $\mcy\to X'$ yields a map $X'\to \mcm_{g'}{}\slice \CC$.
:::

:::{.claim}
This map is not constant.
:::

:::{.remark}
Suppose all of the fibers are isomorphic to some fixed $Y$ of a fixed genus, then $\Hom(Y, X)$ would be infinite.
By Hurwitz, there's a bound on the degree of such a map since $g(Y)$ is fixed, so there are only finitely many. $\contradiction$
:::

:::{.remark}
To do everything over $\OO_{k, S}$, we use $\pi_1^\et$ instead of $\pi_1$ and include $2,3\in S$.
Since the map is unramified over $\CC$, it's ramified at only finitely many primes, so just add those to $S$.
:::