# The Mordell Conjecture (Tuesday, November 23)

:::{.theorem title="Faltings"}
For $K\in\Number\Field$ and $X\slice k$ a smooth curve of genus $g\geq 2$.
Then
\[
\size X(k) < \infty
.\]
:::

:::{.example title="?"}
Consider
\[
x^n+y^n=1
.\]
Note that finiteness of rational points here is a weak form of Fermat: scaling out the denominators yields a rational solution to FLT.
This has finitely many rational solutions for $n\geq 4$, which ensures $g\geq 2$.

Another example is $C \subseteq \PP^2$ a smooth curve, then we have finiteness when $\deg C \geq 4$.
:::

:::{.remark}
Strategy of proof: finite map with finite fibers, a standard way to prove a set is finite.
Note that Faltings proved a number of other famous conjectures along the way to proving this:
:::

:::{.theorem title="Shafarevich for curves"}
Let $K\in \Number\Field$ and $S \subseteq \places{k}$ a finite set of places.
The set of proper curves over $k$ of genus $g$ with good reduction outside of $S$ is finite after modding out by isomorphism.
:::

:::{.remark}
Rephrasing: $\mg$ Deligne-Mumford stack, i.e. the functor sending a ring $R$ to the groupoid of smooth proper curves over $R$, has finitely many $\OO_{k, S}$ points:
\[
\size \mg(\OO_{k, S}) < \infty
.\]
Note the stark contrast for $\QQ$, where $\mcm_2(\QQ)$ has infinitely many points: take
\[
E \da \ts{y^2 = x(x-1) f(x)} && \deg f = 3 \text{ separable}
\]
where $0, 1$ are not roots of $f$.
This produces an infinite family of genus 2 curves.
:::

:::{.remark}
The Shafarevich conjecture for curves implies Mordell: fix $S$ such that $X$ has a smooth proper model $\mcx \slice{\OO_{k, S}}$ with $X(k) = \mcx(\OO_{k, S})$.
Then we win if we can find a finite morphism $\mcx \embeds \mcm_{g'}{}\slice{\OO_{k, S}}$ for some $g'$.
Note that if these were affine, this would be impossible.
The existence of such maps is the **Kodaira-Parshin trick**:

- Over $\CC$, existence was given by Kodaira,
- Over $k$, existence due to Parshin.

Faltings' (and others') proof pass though this trick.
:::

:::{.remark}
Note that a special case of Shafarevich was proved by Faltings, Shafarevich proved it for all curves.
Recent results along these lines: proving $\OO_{k, S}$ points are not Zariski-dense, or are finite, for other interesting moduli spaces.
:::

:::{.theorem title="Shafarevich for abelian varieties"}
For $k\in \Number\Field, S \subseteq \places{k}$ finite, and $g,d > 0$, the set of isomorphism classes of abelian varieties of dimension $g$ with a polarization of degree $d$ over $k$ with good reduction outside of $S$, is finite.
Then
\[
\size \mca_{g, d}(\OO_{k, S}) < \infty
.\]
:::

:::{.remark}
Line bundles are maps to Picard, $d$ is the degree.
For $d=1$, these are principally polarized.
:::

:::{.remark}
It's enough to consider the case $d=1$, which is not obvious -- this is referred to as Zahrin's trick, and shows $A$, $A^8$ always has a polarization of degree 1.
Then Shafarevich for AVs implies Shafarevich for curves, using the Torelli map:
\[
\mcm_g &\to \mca_g \\
[C] &\mapsto [\Jac(C), \Theta]
.\]
This is a finite map!
:::

:::{.definition title="Tate Modules"}
$A\in \Ab\Var\slice k$ and $\ell$ some prime, the $\ell\dash$adic Tate module of $A$ is defined as
\[
T_\ell(A) \da \cocolim_n A[\ell^n](\bar k) \cong \ZZ_\ell\cartpower{2g}
,\]
and $V_\ell(A) = T_\ell(A) \tensor_{\ZZ_\ell} \QQ_\ell$.
:::

:::{.remark}
Recall that finite generation of a field is being finitely generated over a prime field.
:::

:::{.theorem title="Tate Conjecture"}
For $A_1, A_2 \in \Ab\Var\slice k$ with $k$ finitely generated,

1. $V_\ell(A_i)$ are semisimple Galois representations.
2. After tensoring homs with $\ZZ_\ell$, we get an isomorphism of $\ZZ_\ell\dash$modules:
\[
\Hom_k(A_1, A_2)\tensor_\ZZ \ZZ_{\ell} \iso \Hom(T_\ell(A_1), T_\ell(A_2)) \tensor \ZZ_\ell
.\]
:::

:::{.remark}
Part 1 is already very deep, it's a special case of a conjecture we know almost nothing about.
In fact, it's false over $\QQpadic$: take
\[
E: \quad y^2=x(x-1)(x-p) \quad \slice \QQpadic
.\]
This has multiplicative reduction.
Similarly, take
\[
E: \quad y^2=x(x-1)(x-t) \quad \slice{ \CC\functionfield{t} }
,\]
since the monodromy matrix $\matt 1 2 0 1$ is not diagonalizable.
Note that $\bar{\CC\functionfield{t}}$ are the Puiseux series $P$.
Any topological generator of $\Gal(P/\CC\functionfield{t}) \cong \hat{\ZZ}(1)$ acts on $T_\ell(A)$ by a matrix conjugate to this.
:::

:::{.remark}
The general conjecture that we know almost nothing about:
:::

:::{.conjecture}
For $X\slice k$ smooth proper,

1. $G_k\actson H^i(X_{\bar k}; \QQ_\ell)$ semisimply.
2. $\CH^i(X) \tensor_\ZZ \QQ \to H^{2i}(X_{\bar k}; \QQ_\ell(i))^{G_k}$ is surjective.

:::

:::{.remark}
Extremely hard problems! Probably Fields material.
:::

:::{.theorem title="Main difficult ingredient"}
For $A\in \Ab\Var\slice k$, there exist only finitely many isomorphism classes of abelian varieties over $k$ isogenous to $A$.
:::

:::{.proof title="Sketch/idea"}
Take $X\slice k$ smooth proper with $g(X) \geq 2$ and good reduction outside of $S$.
For $\ell$ a prime, there is a map

Define

- $\mcm_{g'}(\OO_{k, S}) = \ts{\text{Curves of genus $g'$ over $k$ with good reduction outside $S$}}/\sim$
  and
\[
\mathrm{KP}: X(k) \to \mcm_{g'}
,\]
  corresponding to the Kodaira-Parshin trick.

- $\mca_{g'}(\OO_{k, S})$ for the PPAVs of dimension $g'$ with good reduction outside of $S$ (mod isomorphism).
- $\Rep_{2g'}(G_{k, S \union\ts{\ell}})$ for the set of semisimple $\QQ_\ell$ Galois representations of rank $2g'$ and *weight* $-1$ unramified outside of $S\union \ts{l}$ such that the characteristic polynomials of $\Frob$ (a conjugacy class) outside of $S\union \ts{\ell}$ have integer coefficients.
  

Here $\ell$ is the ambient prime.
Weight $-1$ means the eigenvalues of $\Frob$ are algebraic numbers $\alpha$ where any embedding $\QQ(\alpha_j) \embeds \CC$ and $\abs{\alpha} = \abs{k(v)}^{1\over 2}$.
$\Frob$ having integer coefficients comes from the Weil conjectures.

Assembling these, we get a chain of maps
\[
X(k) \mapsvia{\mathrm{KP}} \mca_{g'}(\OO_{k, S}) \mapsvia{\Jac} \mca_{g'}(\OO_{k, S}) \mapsvia{V_\ell} \Rep_{2g'}(G_{k, S\union\ts{\ell}}; \QQ_\ell)
,\]
which we claim all have finite fibers, and the last set is finite:

- $\mathrm{KP}$ has finite fibers by construction
- $\Jac$ has finite fibers by the Torelli theorem
- $V_\ell$ has finite fibers, which is more difficult.
  By the Tate conjecture, the fiber over a representation in $V$ are isogenous AVs over $k$, but such sets are finite.
:::

:::{.theorem title="?"}
\[
\size \Rep_N(G_{k, S}; \QQ_\ell) < \infty
.\]
:::

:::{.remark}
We'll discuss this in detail next time, but the idea is that representations are determined by traces of group elements.
For $\Gamma \mapsvia{\rho} \GL(V)$ a representation over $\characteristic k=0$, the semisimplification $\rho^{\semisimplification}$ is determined by $\Tr \circ \rho$.
We can use that Frobenii are dense, and it's enough to determine this representation on a dense set, so we consider $\Tr(\rho(\Frob))$.
This is still an infinite set, so we have to argue that finitely many determine it, but this comes from the bound on eigenvalues.
:::