# Friday February 21st

Question: how do we define $h_{V, D}$?

Answer: write $D = D_1 - D_2$ which are (very) ample divisors and basepoint free.
We then obtain embeddings

\begin{align*}
\varphi_1: V &\injects \PP_K^{n_1} \\
\varphi_2: V &\injects \PP_K^{n_2}
.\end{align*}

So write
$$
h_{V, D}(p) = h(\varphi_1(p)) - h(\varphi_2(p)) + O(1)
$$

Example
: For $E/K$ an elliptic curve,

    - $2[0]$ is an ample divisor
    - $3[0]$ is a very ample divisor.

Let $K$ be a local field (i.e. $\CC, \RR,$ a $p\dash$adic field, or $\FF_q((t))$ formal Laurent series) and $A/K$ be an abelian variety; we want to understand $A(K)$.
We know this has the structure of compact abelian $K\dash$analytic Lie group.

- Question 1: What does Lie theory say?

- Question 2: What extra information comes from $A/K$ being a $g\dash$dimensional abelian variety?

If $K = \CC$, then $A(K) \cong (\RR/\ZZ)^{2g}$.
If $K = \RR$, then $A(K) \cong (\RR/\ZZ)^g \oplus \prod_{i=1}^d \ZZ/2\ZZ$ where $0\leq d \leq g$.

Fix $d$, then

- Let $E_1/\RR$ with $\Delta > 0$ (and thus 3 real roots), then $E_1(\RR)[2] = \qty{\ZZ/2\ZZ}^2$.

- Let $E_2/\RR$ with $\Delta < 0$ (and 1 real root), then $E_2(\RR)[2] = \ZZ/2\ZZ$.

By taking products of $E_1$ and $E_2$, i.e. $A =(E_1)^{d} \cross (E_2)^{g-d}$.


> Todo: find reference in Silverman?

Fact
: $A(K)$ is totally disconnected and homeomorphic to a Cantor set.

Fact (From Lie Theory, Serre p.116)
:   If $G$ is an abelian compact $K\dash$analytic Lie group, then there exists a filtration by open finite index subgroups

    $$
    G = G^0 \supset G^1 \supset \cdots \supset G^n \supset \cdots
    $$

    such that

    1. The successive quotients are finite, and each $G^i$ is *standard*, i.e. obtained by evaluating a formal group law on $\qty{\mfm^i}^g$.

    2. $\intersect_i G^i = (0)$.

    3. $G^i / G^{i+1}$ has exponent $p$, i.e. it is a finite dimensional $\ZZ/p\ZZ\dash$vector space.

    4. $G'[\tors] = G'[p^\infty]$, all of the prime-to-$p$ torsion is $p\dash$primary.

> Todo: define $p\dash$primary torsion, $\QQ_p$.

What structure theorem does this give?


Theorem (C-Lacy)
:   Let $G$ be a compact, second countable, $K\dash$analytic abelian Lie group of dimension $g\geq 1$.
    Then

    a. If $\ch K = 0$ and $d = [K: \QQ_p]$, then
    $$
    G \cong_{\text{TopGrp}} \ZZ_p^{dg} \oplus G[\tors]
    $$
    where $G[\tors]$ is finite.

    b. If $\ch K = p$, i.e. $K = \FF_q((t))$, and if it is true that $G[\tors]$ is finite iff $G[p]$ finite, then
    $$
    G \cong_{\text{TopGrp}} \prod_{i=1}^\infty \ZZ_p \oplus G[\tors]
    $$

For any $g\geq 1$, $(T, +)$ a finite discrete abelian group $(R, +) \cong (\ZZ_p^d, +)$ and $R^g \oplus T$ is a compact abelian $K\dash$analytic Lie group isomorphic to $\ZZ_p^{dg} \oplus T$ (?).

> Question: what does this mean for $G = S^1$? Ask Pete!

Theorem (Cartan)
: Let $K$ be a local field, $\QQ\injects K$ dense (so $K = \RR, \QQ_p$).
  Then if $G_1, G_2$ are $K\dash$analytic, and $\varphi \in \hom_{\text{TopGrp}}(G_1, G_2)$, then $\varphi \in \hom_{k\dash\text{analytic}}(G_1, G_2)$.

Example
: For $R = \FF_q[[t]]$, $(R, +)^g [p] = (R, +)^g$.

Example
: Take $G = \GG_a^g(K)$ the additive group or $A/K$ a $g\dash$dimensional abelian variety (i.e. $G = A(K)$) then $G[p] \subsetneq (\ZZ/p\ZZ)^{2g}$ and is finite.


## Proof of Cartan's Theorem


### Step 1

We want to show that $G[p] < \infty$, then $G[\tors] < \infty$.
We'll use the filtration in Serre's result; then for $i \gg 1$, we'll have $G^i[p] = 0$.
Thus for $i \gg 1$, we'll have $G^i[p^\infty] = 0$; but this is the only torsion that can appear.
We'll then obtain an injection $G[\tors] \injects G/G^i < \infty$.

Lemma
: Let $H$ be an abelian torsionfree pro$\dash p$ group (e.g. $\prod \ZZ_p$).
  Then $H \cong \prod_{i\in I} \ZZ_p$, and if $H$ is second-countable, then $I$ is countable.

Proof
:   Omitted.
    Idea: use Pontrayagin duality $G\dual \definedas \hom_{\text{Top}}(G, \RR/\ZZ)$ for locally compact abelian groups.

    \

    Use the fact that this is naturally an exact contravariant functor, this lets you trade in profinite groups for discrete torsion abelian groups.

> Note: look up pro-p groups. Is the Pontrayagin dual a cohomotopy group?