# Thursday, December 02

## Setup

:::{.conjecture title="Tate conjecture, general"}
Let $X\in \smooth\Proj\Var\slice k$ for $k$ a finitely generated field.
Conjecturally the cycle class map is surjective:
\[
\CH^i(X)_{\QQladic} \surjects H^{2i}(X_{\bar k}; \QQ_\ell(i) )^{G_k}
.\]

Replacing rational equivalence by *homological equivalence* is conjectured to yield a bijection, although this is closer to tautological -- one quotients by the kernel of this map.
A conjecture with more content: replacing rational equivalence with *numerical equivalence* yields a bijection, where one essentially quotients by the intersection pairing.
:::

:::{.remark}
Recall that we're proving $\# X(k) < \infty$.
Note
\[
X(k) \mapsvia{kP} \mcm_{g'}(\OO_{k, S}) \mapsvia{\Jac} \mca_{g'}(\OO_{k, S}) \mapsvia{V_\rho} \Rep_{2g'}^*(G_{k, S}; \QQladic)
.\]
Last time we constructed $\mathrm{KP}$ and showed that $\Rep_{2g'}$ above is finite.
The strategy: proving some finiteness statement, prove a special case of the Tate conjecture (generally widely open), then get a stronger finiteness result.
:::

:::{.theorem title="Faltings, Tate conjecture for AVs over a number field"}
For $A, B \in \Ab\Var\slice k$, $k\in \Number\Field$,

1. The Tate modules $V_\ell(A), V_\ell(B)$ are semisimple $G_k\dash$modules.

2. There is a bijection
  \[
  \Hom(A, B)\tensor \ZZladic \iso \Hom_{G_k}(T_\ell(A), T_\ell(B))
  ,\]
  where the LHS are isogenies.
:::

:::{.remark}
This won't be true for $p\dash$adic fields, but there is a version that works for finitely-generated fields, e.g. function fields of varieties defined over number fields.
:::

:::{.theorem title="Faltings, a finiteness result"}
Isogeny classes over $k$ are finite.
:::

:::{.theorem title="Tate?"}
Faltings' finiteness result implies the Tate conjecture
:::

:::{.remark}
The proof idea: use a **height** defined by Faltings:
\[
\mca_g(k) \to \RR
\]
satisfying

- For all $N$, $\size \ts{[A] \in \mca_g(k) \st h([A] ) < n }$ is finite.
- If $A \sim B$ are isogenous, then $\abs{ h(A) - h(B) } < C = C(A)$ a constant.

Note that these imply the finiteness result since $\ts{h(B) \st B\sim A} < \infty$.
There is a general height machine theory here, which extends this theory to (nice, e.g. DM) stacks.
Recent work is going into extending height theories to more complicated stacks, e.g. algebraic spaces.
:::

## Proof: Finiteness implies the Tate conjecture

:::{.claim}
$\Hom(A, B)$ is torsionfree.
:::

:::{.proof title="?"}
This is because $\Hom(A, B) \injects \Hom(T_\ell(A), T_\ell(B))$ since the $\ell^\infty\dash$torsion points are dense, and the RHS is torsionfree.
:::

:::{.claim}
There is a functor 
\[
T_\ell: \Ab\Var\slice k \tensor \ZZladic \to \Rep_{G_k}
\]

which is 

- Faithful (and we'll want to show it's full)
- $\Hom(A, B)$ is finitely generated.

:::

:::{.proof title="?"}
The proof is essentially checking over $\CC$.
:::

:::{.claim}
The following cokernel is torsionfree:
\[
\coker\qty{
\Hom(A, B)\tensor \ZZladic \too \Hom(T_\ell(A), T_\ell(B))
} 
.\]
:::

:::{.proof title="?"}
Again, check over $\CC$!
:::

:::{.claim}
The truth of the Tate conjecture is preserved under base change: given $K/k$ a finite extension, the Tate conjecture for $K$ implies the Tate conjecture for $k$.
:::

:::{.proof title="?"}
Without loss of generality, let $K'/k$ be finite Galois.

:::{.claim}
For semisimplicity, the claim is that if $\Gamma' \leq \Gamma$ is a subgroup of finite index and $\rho: \Gamma\to \GL_n(\QQladic)$, then if $\ro{\rho}{\Gamma'}$ semisimple implies $\rho$ is semisimple.
:::

:::{.proof title="?"}
Proof: get a splitting $\rho \mapsvia{s} \rho'$ and use an averaging argument on $\Gamma/\Gamma'$ (a la Maschke's theorem).
This works since $\QQladic$ is positive characteristic.
:::

:::{.claim}
For the homs, the claim is that $\Hom(A_{K'}, B_{K'}) \tensor \ZZladic \to \Hom(T_\ell(A), T_\ell(B))$ is bijective.
:::

:::{.proof title="?"}
Noting that we already have injectivity over any field.
Let $\sigma \in \Hom_{G_k}(T_\ell A, T_\ell B)$, then there exists $f: A\to B$ over $k$ such that $T_\ell(f) = \sigma$.
Since $f$ is $G_k\dash$equivariant, we get density by Galois descent.
:::

Reduce to $A=B$ by using $A\times B$.
Reduce to proving the following:

:::{.claim}
If $W \subseteq V_\ell(A)$ is a Galois stable subspace, there exists a $u\in \Endo(A)\tensor \QQladic$ with $u(V_l(A)) = W$.
:::

:::{.proof title="?"}
For semisimplicity, use that subrepresentations are continuous?
Then use a double coset trick.
:::

Reduce to showing this for $W$ a maximal isotropic subspace for the Weil pairing (a symplectic form).
This is Zahrin's trick: replace $A$ with $A\cartpower{4}$, gives a way to "complete" any subspace $W \subseteq T_\ell(A)$ to a maximal isotropic.
The proof uses the Lagrange 4-squares theorem, how neat!

Write
\[
G_n \da {W \intersect T_\ell(A) + \ell^n T_\ell(A) \over \ell^n T_\ell(A) } \subseteq A[\ell^n], && B_n \da A/G_n
.\]
Then if $W$ is maximal isotropic, $B_n$ is a PPAV and thus so is $A$.
The actual theorem Tate proves:

:::{.theorem title="?"}
It suffices to show that $\ts{B_n}_{n\geq 1}$ is finite.
:::

:::{.proof title="Idea"}
Write $A \mapsvia{\psi_n} A/G_n = B_n$ and $\psi_n\dual: B_n\to A$, and from the construction of $G_n$ and unwinding definitions yields
\[
\psi_n(T_\ell(B_n)) = W \intersect T_\ell(A) \cdot \ell^n T_\ell(A)
.\]
So $\im(\psi_n)$ is "converging" to $W \intersect T_\ell(A)$.
Use that eventually the $B_n$ stabilize to compose $\gamma_n: A \mapsvia{\psi_n}  B_n \mapsvia{\psi_n\dual}  A$ and replace this with $\gamma_n:A\to B\to A$.
Use that $\Endo(A)\tensor \ZZladic$ is a compact space to get a convergent subsequence.
:::

:::

:::{.remark}
The upshot: there exist finitely many PPAVs isogenous to $A$.
:::

## Faltings Heights

:::{.remark}
Write $\tilde \mca \to \mca \da \da \mca_{\mathrm{univ}, g}\to \mca_g$ for the universal family, and define a line bundle $\mcl \da \Extpower^{\text{top}} s^* \Omega^1_{\tilde \mca / \mca}$.
A fact is that $\mcl$ is amply, since there exists a compactification of $\mca_g$ to which $\mcl$ is ample.
This is a hard theorem and involves Siegel modular forms.
:::

:::{.remark}
Define a height function associated to $\mcl$:
\[
\tilde h: \mca_g(\QQ) \mapsvia{\mcl} \PP^N(\QQ) &\to \RR \\
\tv{ {x_0 \over x_i}:\cdots : {x_n\over x_i} } &\mapsto \max(\abs{x_i}, \abs{x_i'} )
.\]
This is the height machine, but it's fairly incomputable.
:::

:::{.remark}
A definition by Faltings: for $A$ semisimple, write
\[
H(A) = \prod_{k\embeds \CC} \int_{A(\CC)} \eta \wedge \bar\wedge && \eta \in H^0(\bar A\slice{ \OO_K}; \Omega_A^q ) 
,\]
taking a Néron model $\bar A$ for $A$.
Then define
\[
h(A) = {1\over [k: \QQ]} \log(H(A))
.\]
:::

:::{.theorem title="Faltings"}
There exist constant $c_1, c_2$ such that 
\[
\abs{h(A) - \tilde c_1 h(A) } < c_2
\]
where the $c_i$ do not depend on $A$.
:::

:::{.proposition title="?"}
For $f: A\to B$ an isogeny induces $\bar A\to \bar B$ on Néron models, and
\[
h(B) - h(A) = {1\over 2} \log \deg(f)\cdot {1\over [K:\QQ] } \log \length s^* \Omega^1\slice{?}
.\]
:::