# Wednesday February 12th

## Correction to Weil's Height Machine

Let $K$ be a NPFF (from now onwards: just a global field) and $X/K$ a nice variety.
Let $D\in \div(X)$ be ample and consider the points of bounded height $\theset{ p\in V(K) \suchthat h(p) \leq R  }$; by the Northcott property this is finite for every $R \in \RR$.

Counterexample:
Let $X = E$ an elliptic curve and $D = [0]$.
Then $\ell(D) = \dim{f\in K(X) \suchthat \div f\geq -D}$, the rational functions with poles no worse than order $D$.
Then $\ell[0] = 1$.

Then there is a corresponding morphism to projective space 
$$
\phi_D: X \to \PP(L(D)) = \PP^{\ell(d) - 1}
.$$

Here 
$$
\phi_D: E \to \PP^{\ell[0] - 1} = \PP^0 = \pt
.$$

Thus if $E(K)$ is infinite, which is possible, then $\forall R\geq 1$ the set $\theset{p\in E(K) \suchthat h(p) \leq R} = E(K)$ is infinite as well.

How to fix this?
Replace "$D$ is ample" with "$D$ is ample and $\ell(D) \geq 2$".


> Note: not listed as a correction in Silverman's book.


Theorem (C.1.9 Of Hindry-Silverman)
:   Let $K$ be a number field (perhaps works with global function fields) and $n\geq 2$, with $A/K$ an abelian variety of dimension $g$.
    Suppose $A[n] = A[n](K)$ (so a large extension of $K$), and let $S$ be a set of finite non-Archimedean places of $K$ containing places dividing $n$ and places of bad reduction.
  
    Suppose all that the ring of $S\dash$integers, $\ZZ_{K, S}$, is a PID.


    Then 
    $$
    \rk A[k] \leq 2g \cdot \rk \ZZ_{K, S}\units
    .$$
    We can then apply the Dirichlet unit theorem, the RHS is $2g(r_1 + r_2 + \# S - 1)$ where $r_1$ is the number of real places and $r_2$ is the number of complex places.

This is an explicit upper bound, but a pretty bad one!

## Néron-Tate Canonical Height

Let $K$ be a global field.
We'll get a good canonical representative of the height functions studied previously, provided we get one additional piece of data.

Theorem (Néron-Tate)
:   Let $V/K$ be a nice variety and $D \in \div V$, and suppose there exists a morphism $\phi: V\to V$ such that there exists an $a\in \QQ^{> 1}$ with the induced map on $\pic V$ satisfies $\phi^* D \sim \alpha D$. 
  
    > So $D$ is an "eigendivisor".

    Then there exists a unique function 

    \begin{align*}
    \hat{h}_{V, \phi, D}: V(K\sep) \to \RR
    \end{align*}

    such that 

    1. (CH1) $\hat h_{V, \phi, D} = h_{V, D} + O(1)$

    2. (CH2) $\hat h_{V, \phi, D} \circ \phi = \alpha \hat h_{V, D}$

    Moreover, for all $p\in K\sep$, there is a stabilization 

    \begin{align*}
    \hat h_{\phi, D}(p) = \lim_n \frac{1}{\alpha^n} h_{V, D}(\phi^n(p))
    .\end{align*}

Proof
: We first need to show that the stated limit exists.
  Applying Weil's Height Machine to $\phi^* D \sim \alpha D$, there exists a $c$ such that

  \begin{align*}
  \abs{ h_{V, D} \circ \phi  - \alpha h_{V, D}} \leq c
  .\end{align*}

  So take a separable point $p\in V(K\sep)$.

Claim
: $\alpha^{-n} h_{V, D}(\phi^n(p))$ is Cauchy in $\RR$ and thus convergent.

Proof
:   **Step 1**:

    Let $m\leq n$, then we want to show

    \begin{align*}
    \abs{ \alpha^{-n} h_{V, D}( \phi^n (p) ) - \alpha^{-m} h_{V, D}( \phi^n (p) ) } 
    &=
    \abs{ \sum_{i=m+1}^n \alpha^{-i} \qty{ h_{V, D} (\phi^i(p)) - \alpha h_{V, D}( \phi^{i-1}(p)  )  }  } \\
    &=
    \sum_{i=m+1}^n \alpha^{-i} \abs{ \qty{ h_{V, D} (\phi^i(p)) - \alpha h_{V, D}( \phi^{i-1}(p)  )  }  } \\
    &\leq 
    \sum_{i=m+1}^n \alpha^{-i} C \\
    &<
    C \sum_{i=m+1}^\infty \alpha^{-i}\\
    &= \frac{\alpha^{-m-1}{1-\alpha}} C \\
    &\converges{m\to\infty}\to 0
    .\end{align*}

    **Step 2**:
    Write

    \begin{align*}
    \hat h_{V, D}(\phi(p)) 
    &=
    \lim_{n\to\infty} \alpha^{-n} h_{V, D} ( \phi^n(\phi(p))  ) \\
    &=
    \alpha \lim_{n\to\infty} \frac{1}{\alpha^{n+1}} h_{V, D} ( \phi^n(\phi(p))  ) \\
    &=
    \alpha \hat h_{V, \phi, D}(p)
    .\end{align*}

    **Step 3**:
    Suppose $\hat h$ and $\hat{h'}$ both satisfy CH1 and CH2.
    Let $g\definedas \hat h - \hat{h'}$.
    By CH1, $\abs g \leq C$ for some $C$.
    Then by CH2, $g\circ \phi = \alpha g$, and for all $n\in \ZZ^+$, $g\circ \phi^n = \alpha^n g$.
    We thus get

    \begin{align*}
    g = \frac{g\circ \phi^n}{\alpha^n} \implies \abs{g} \leq \frac{C}{\alpha^n} \converges{n\to\infty}\to 0
    .\end{align*}


## 10 Cents of Dynamics

Anytime we have an endomorphism, we can iterate it and consider the associated dynamical system.

Let $S$ be a set, $\phi: S\to S$ a function, $\phi^n = \phi \circ \phi \circ \cdots \circ \phi$ with $\phi^0 = \id_S$.
Then $p\in S$ is $\phi\dash$periodic if there exists an $n\in \ZZ^+$ such that $\phi^n(p) = p$.
We say $p$ is $\phi\dash$preperiodic iff the set $\theset{\phi^n(p) \suchthat n\in \NN}$ is finite.
In this case, there exists some $n\in \NN$ such that $\phi^n(p)$ is $\phi\dash$periodic:

![Image](figures/2020-02-12-13:01.png)\

**Proposition:**
Setup as in the last result, $\phi: V\to V$, $D\in \div V$ and $\phi^*D \sim \alpha D$ for $\alpha \in \QQ^{\geq 1}$.
For $\ell(D) \geq 2$ and $D$ ample, we have

a. For all $p\in V(K\sep)$, $\hat h_{V, D, \phi}(p) \geq 0$
  Moreover, $\hat h_{V, D, \phi}(p) = 0$ off $p$ is $\phi\dash$preperiodic.

b. The set $\theset{p\in V(K) \suchthat p \text{ is $\phi\dash$ preperiodic} }$ is finite.

*Proof of (a):*

We know positivity by the Weil Height Machine.
Let $p\in A(K\sep)$, then $K \to K(p)$ for ?.
Suppose $\hat h_{V, D, \phi}(p) = 0$, then

\begin{align*}
h_{V, D}( \phi^n(p)  ) 
&= \hat h_{V, \phi, D}(\phi^n(p)) + O(1) \\
&= \alpha^n \hat h_{V, \phi, D}(p) + O(1) \\
&= O(1)
.\end{align*}


Then the set of forward orbits $\theset{\phi^n(p) \suchthat n\in \NN}$ has bounded height.
By the finiteness property of the Weil Height Machine, this set is finite and $p$ is thus $\phi\dash$preperiodic.

If $p$ is $\phi\dash$preperiodic, then $\theset{h_{V, D}(p)}$ bounded, so

\begin{align*}
\hat h_{V, \phi, D} (p) = \lim_n \frac{1}{\alpha^n} h_{V, D}(\phi^n(p)) = 0
.\end{align*}


*Proof of (b):*

By part (a), if $p\in V(K)$ then $p$ is $\phi\dash$preperiodic, so $\hat h_{V, \phi, D} = 0$ and thus $h_{V, D}(p) = O(1)$.
By the Weil Height Machine, this set of points is finite.

$\qed$


### Important Examples

**Example 1:**
Let $\phi: \PP^n \to \PP^n$ of degree $d\geq 2$ and $H\subset \PP^n$ the hyperplane divisor, $\phi^* H \sim d H$.
So take $\alpha = d$, this is an eigendivisor, now applying this theory, a consequence is that $\theset{p\in V(K) \suchthat p \text{ is } \phi\dash\text{preperiodic}}$ is finite. 
(Theorem by Northcott, properties of heights were established to show this statement!)

**Example 2:**
Let $A/K$ an abelian variety with $D\in \div A$ ample, $\ell(D) \geq 2$ with $[-1]^* D \sim D$.
Take $\phi = [n]$ for some $n\geq 2$, then $\phi^*(D) \sim n^2 D$ where $\alpha = n^2 > 1$.
The $\phi\dash$preperiodic points of $A(K\sep)$ are precisely the torsion points.

> Note that this proves that the set of torsion points on an abelian variety is finite.