# Chabauty-Coleman (Thursday, November 11)

:::{.theorem title="Coleman-Chabauty"}
For $X\in \Sch\slice \QQ$ a proper curve with $g(X) \geq 2$ and $\rank_\ZZ \Jac(X)(\QQ) < g$ where $X$ has good reduction at $p>2g$, $\size X(\QQ) \leq \size X(\FF_p) + 2g-2$
:::

:::{.remark}
Idea: choose $x\in X(\QQ)$, define a map
\[
\AJ_x: X &\mapsvia{\text{Abel-Jacobi}} \Jac(X) \\
y &\mapsto [\OO(y-x) ]
.\]
Then $X^n\to \Jac(X)$ by $\vector y \mapsto \sum \AJ_x(y_i)$ is surjective for $n\geq g$.
Note that 
\[
\Hom(T, \Jac(X)) = \ts{ \mcl \in \Pic(X\times T) \st \mcl \text{ has degree 0 on each fiber of } X\times T\to T}/\Pic(T)
\]
This equals $\ts{\mcl\in \Pic X\times X \st \mcl \text{ has fiberwise degree } 0}/\Pic(X)$.
Note that the LHS is $\OO(\ts{x}\times X\to \Delta)$.
:::

:::{.claim}
For $X^n \mapsvia{\sum \AJ_x} \Jac(X)$ is surjective for $n\geq g$.
:::

:::{.proof title="?"}
ETS that the following map is surjective:
\[
X^n &\to \Pic^n(X) \\
\vector y &\mapsto \OO\qty{ \sum y_i }
.\]
Given $[\mcl]\in \Pic^n(X)$, for $\mcl = \OO(D)$ for $D$ effective iff $H^0(\mcl) = 0$, so ETS $H^0(\mcl\neq 0)$ for $\deg \mcl \geq g$.
By Riemann-Roch:
\[
\dim H^0(\mcl) - \dim H^1(\mcl) = \deg \mcl + 1 - g
.\]

The RHS is 1, so $\dim H^0(\mcl) \geq 1$.
:::

:::{.slogan}
$X$ generates $\Jac(X)$, and $\Jac(X)$ is the Albanese of $X$.
:::

:::{.remark}
Let $\Gamma = \Jac X(\QQ) \subseteq \Jac X(\QQpadic)$.
We have a factorization:

\begin{tikzcd}
	{X(\QQ)} && \Gamma \\
	\\
	{X(\QQpadic)} && \bar\Gamma && {\Jac X(\QQpadic)}
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow[curve={height=18pt}, from=3-1, to=3-5]
	\arrow[from=1-3, to=3-5]
	\arrow["\exists"', dashed, from=1-1, to=3-3]
	\arrow[from=3-3, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJYKFxcUVEpIl0sWzAsMiwiWChcXFFRcGFkaWMpIl0sWzIsMCwiXFxHYW1tYSJdLFsyLDIsIlxcYmFyXFxHYW1tYSJdLFs0LDIsIlxcSmFjIFgoXFxRUXBhZGljKSJdLFswLDFdLFswLDJdLFsxLDQsIiIsMCx7ImN1cnZlIjozfV0sWzIsNF0sWzAsMywiXFxleGlzdHMiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMyw0XV0=)

Here $\bar \Gamma$ has $\dim n < g$ and $\Jac X(\QQpadic)$ has dimension ??.
A fact from \(p\dash \)adic Lie groups: they're all direct sums of balls!

Now construction functions on $\Jac X (\QQpadic)$ vanishing on $\bar\Gamma$.
Then $\ro{f_i}{X(\QQ_p)}$ vanish on  $X(\QQpadic) \intersect \bar\Gamma \contains X(\QQ)$.

This will show $X(\QQ)$ is finite.
We'll show that not all of the $f_i$ are identically zero: ETS that $X(\QQpadic) \not\subseteq \bar\Gamma$, which will be true since $X$ generates its Jacobian.
Take whichever $f_i \not\equiv 0$, which is an \(p\dash \)adic analytic function, then $f_i$ will have finitely many zeros.
:::

:::{.remark}
There is a hypothesis that $\rank_\ZZ \Jac X(\QQ) < g$ can be replaced by $\dim \bar\Gamma < g$.
:::

## Proofs

:::{.lemma title="?"}
Let $\Gamma \leq A(\QQpadic)$ be a finitely-generated abelian subgroup for $A\in \Ab\Var$.
Then $\dim_{\QQpadic} \bar\Gamma \leq \rank_\ZZ \Gamma$.
:::

:::{.fact}
Any \(p\dash \)adic manifold $X$ is locally isomorphic to $\ZZpadic\cartpower{\dim X}$.
:::

:::{.proof title="?"}
For $\Gamma' \subseteq \ZZpadic\cartpower{n}$ a finitely-generated abelian group, $\dim \bar{ \Gamma}' \leq \rank_\ZZ \Gamma'$.
Pick generators $g_1, \cdots, g_n$ generators, then write a linear map
\[
\gamma: \ZZpadic^r &\to \ZZpadic^n \\
e_i &\mapsto g_i
.\]
The image is closed of dimension $n = \rank \gamma$.
We'll now reduce the original lemma to proving this statement.
This is because any *abelian* \(p\dash \)adic Lie group is of the form $\ZZpadic\cartpower{n} \times \text{something finite}$ for some $n$.
If $A$ has good reduction at $p$, then $A(\QQpadic) \cong \ZZpadic \times A(\FF_p)$.
Recall that there are maps:
\[
\exp: T_0 A(\ZZpadic) &\to A(\QQpadic) \\
\log: \mathrm{sp}\inv(0) &\to A(\ZZpadic)
,\]
where $\mathrm{sp}$ is the specialization map, and this induces an isomorphism of abelian groups.

The goal is to now construct a \(p\dash \)adic analytic function on $A(\QQpadic)$ vanishing on $\bar\Gamma$.
This construction: due to Coleman.
Fix $\omega \in H^0(A_{\QQpadic}, \Omega^1_{A/\QQpadic})$, then there exists a \(p\dash \)adic analytic function
\[
A(\QQpadic) &\to \QQpadic \\
Q &\mapsto \int_0^Q \omega
.\]
This map is determined by

- $Q\to \int_0^Q \omega$ is a morphism of topological groups,
- FTC: locally near 0, if $f$ is a \(p\dash \)adic analytic function with $\omega = df$, then $\int_0^Q \omega = f(Q) - f(0)$.

:::

:::{.remark}
Take $\bar\Gamma \subseteq \QQpadic$, then for $U \da \bar\Gamma \in B_0$ a ball around the origin, consider $\log(U) \subseteq T_Q(A) = \QQpadic\cartpower{n}$.
Let $v_1,\cdots, v_s \subseteq \T_0\dual A$ be dual vectors vanishing on $\log \bar \Gamma$.
Then

- $\T_0\dual A \cong H^0(A, \Omega^1_{A})$
- $f_i(Q) = \int_0^Q v_i$, viewing $v_i$ as a 1-form.
:::