# Wednesday January 29th

## Remarks on Weak Mordell-Weil

Definition (Small Groups)
: A profinite group (e.g. group occurring as an absolute Galois group) is **small** if for all $n \in \ZZ^+$, the set of open subgroups of $g$ of index $n$ is finite.

Proposition
: For any $g = g_K = \Aut(K\sep/K)$, $g$ is small iff $K$ has only finitely many separable extensions of any fixed degree $d$.

Definition (Big Fields)
: A field $K$ is *big* iff its Galois group $g_K$ is small.

**Warning**: In general not related to cardinality.

If $K$ is big and contains $n$th roots of unity $\mu_n \in K$, then for all $n\geq 2$, the maximal abelian extension of exponent dividing $n$ is finite (by Kummer theory)
Therefore, if $A/K$ is any commutative algebraic group (plus conditions?), then $A(K)/ nA(K)$ is finite.

Fields that are big:

- Finite fields
- Algebraically closed fields
- $\RR$ and any real closed field $\iff$
- $p\dash$adic fields

But $\FF_q((A))$ is *not* big.

> The Weak Mordell Weil proof goes through if you look at integral points instead of just rational points.
> Dirichlet unit theorem gives WMW for the multiplicative group.

## Height Functions

Let $(K, \Sigma_K, A)$ be a product formula field where $K$ is a number field and $K = k(C)$ for $C/K$ is a nice curve for $k$ any field.

> Don't need to assume separability, but we will!

A height function is a function $H: \PP^n(K) \to \RR$.
We assume the product formula field $K$ has the *Northcott property*, i.e. $\theset{ p\in \PP^n(K) \suchthat H(P) \leq R  }$ is finite for ever $n$ and every $R$.

Remark
: A number field $K$ has the Northcott property, and $k(C)$ has this property iff $k$ is finite.

1. This gives an estimate for change in $H$ under finite morphisms $f: \PP^n \to \PP^m$.
2. For $E/K$ with $y^2 = P_S(x)$, then $H: E(K) \to \RR$ given by $p\mapsto H([X(p): 1])$.

View $X: E \to \PP^1$ is a 2:1 map given by taking $x$ coordinates.

Sticking point:
The "near-quadraticity" of $H$, i.e. the second property of height functions.
We won't have Weierstrass equations for abelian varieties, so we don't want to do this.
Instead, we'll develop 

1. Weil's height machine
2. Néron-Tate canonical heights on abelian varieties $A/K$.

Note that if $(x_0, \cdots, x_n) \in \PP^n(\QQ)$, where $\QQ$ is the fraction field of $\ZZ$, a UFD, so this is a UFD.
Appropriately clearing denominators, we can get $\gcd(x_0, \cdots, x_n)$.
So $H(x_0, \cdots, x_n) = \max_{0\leq i \leq n} \abs{x_i}$, and (check!) the Northcott property holds.

Let $(K, \Sigma_K, A)$ where $\Sigma_K$ is a set of inequivalent places $\abs{\wait}_v$.

1. PFF1: The Archimedean places $\Sigma_K^{\text{Arch}}$ have valuations $v: K \to \RR \union \infty$ where $\abs{\wait}_v = c_v > 1$.

2. PFF2: For all $x\in k\units$, $\abs{x}_v = 1$ for all $v$.

3. PFF3: Height is invariant under scaling, i.e. for all $x\in k\units$, $\prod_{v\in\Sigma_K} \abs{x}_v = 1$.

Example
: 	Take $K = \QQ$, then 

   - $\abs{x}_p = p^{-\ord_p(x)}$
   - $\abs{x}_\infty = \abs{x}$ the standard real absolute value
   - $\abs{x}_\infty = \qty{\prod_p \abs{x}_p  }\inv$ for all $x\in \ZZ^\bullet$.

Example
: $k$ any field, $R = k[t]$, and $K = k(t)$.
	Then it's not quite true that $\Sigma_K = \theset{v_p, ~p \text{ monic irreducible polynomials}}$.
	Question: what is $C_{v_p}$.

	For $c>1$, we can write 
	$$
	\abs{\wait}_p \definedas c^{-v_p}
	.$$
	Write $x = \eps p_1^{a_1} \cdots p_{r}^{a_r}$ with the $p_i$ monic irreducibles and $\eps \in k\units$.
	Then 
	$$
	\prod_p \abs{x}_p = \prod_p c^{-v_p(x)} = c^{-\sum^r a_i}
	.$$
	
	But this isn't one!
	We don't have the "counterbalance" here, so we may want to add an infinite place to compensate.

Remark
: If $k$ is algebraically closed, then $p_i = t - x_i$ and $\sum a_i = \deg v$.

Adding a place:
Define $v_\infty(f/g) = v_\infty(f) - v_\infty(g)$, and 
$$
\abs{\wait}_{v_\infty} = c^{-v_\infty(f/g)} = c^{\deg f - \deg g}
.$$
This yields

\begin{align*}
\abs{x}_\infty \prod_p \abs{x}_p 
&= \prod_p C^{-v_p(x)} C^{\deg x} \\
&= c^{-\sum^r a_i} C^{\deg x} \\
&= C^{-\deg x}C^{\deg x}
.\end{align*}

In general, 
$$
\deg\qty{\prod_{i=1}^r p_1^{a_1} \cdots p_r^{a_r}} = \sum_{i=1}^r a_i \deg(p_i)
.$$

Then 
$$
k_v \definedas \deg p_v \cdot c \quad\text{and}\quad \abs{x}_p \definedas C^{-\deg(p) v_p(x)}
.$$