# Friday February 7th

## Weil Height Machine

What is the analog of the $x\dash$coordinate of an elliptic curve for a general abelian variety?
Answer: an ample base point divisor.

Let $K$ be an NPFF (Northcott Product Formula Field), $V/k$ a projective variety and $\phi: V \to \PP^n$, then define

\begin{align*}
h_\phi: V(K\sep) &\to [0, \infty) \\
p &\mapsto \log H(\phi(p))
.\end{align*}

*Key property:*
Take hyperplanes $\psi: V \to \PP^m \supset H'$ and $\phi: V\to \PP^n \supset H$, noting that the Picard group of $\PP^n$ is $\ZZ$.

Suppose $\phi^* \sim \psi^* H'$, then $h_\phi = h_\psi + O(1)$.
Generally, we'll want to identify functions whose difference is a bounded function.

> Highly recommended: Diophantine Geometry, Handry-Silvermann GTM

Theorem (Weil Height)
:  Let $K$ be a NPFF (e.g. a global field) and $V/K$ a smooth projective variety.
  There exists a map $h_{V, \wait}: \div V \to \RR^{V(K\sep)}$ from divisors on $V$ to ? which is unique modulo bounded functions such that 

  a. For $H \subset \PP^n$ a hyperplane, the divisor $h_{\PP_1^n H} = h + O(1)$ (for the previously defined $h$).

  b. Pullbacks: For $\phi: V\to W$ a smooth morphism and $D\in \div W$, we have $h_{V, \phi^* D} = h_{W, D}\circ \phi + O(1)$.

  c. Linearity: For all divisors $D_1, D_2 \in \div V$, $h_{V, D_1 + D_2} = h_{V, D_1} + h_{V, D_2} = O(1)$.

  d. If $D_1 \sim D_2$, then $h_{V, D_1} = h_{V, D_2} + O(1)$.

  e. Positivity: Let $D\subset \div V$ be effective and $B$ the base locus (common intersection of linearly equivalent effective divisors), then$h_{V, D}(p) \geq O(1)$ for all $P \in (V\setminus B)(K\sep)$.

  f. Let $D, E. \in \div V$ with $D$ ample and $E$ algebraically equivalent to $0$. 
  Then 
  $$
  \lim_{P \in V(K\sep), h_{V,D}(p) \to \infty} h_{V, E}(p) / h_{V,D}(p) = 0
  .$$

  > Ample: Chern class of line bundle is zero.

  g. Finiteness: $D \in \div V$ is ample. For all map into $L/K$ and $r\in \RR$, the set $\theset{p\in V(L) \suchthat h_{V, 0}(p) \leq R  }$ is finite.

Note every ample divisor has the Northcott property.
Every abelian variety has an ample *symmetric* divisor.

Let $V = A/K$ be an abelian variety and $D\in \div A$ a divisor, see section A.7 in Silverman.

*Some properties of such divisors*:

1. (A.7.25) For all $n\in \ZZ$, 
  $$
  [n]^* D \sim \frac{n^2 + n}{2}D + \frac{n^2 - n}{2} [-1]^*D
  $$ 
  (standard fact).
  
  If $[D]$ is symmetric, i.e. 
  $[-1]D \sim D$, then 
  $$
  [n]^* D \sim n^2 D
  $$
  
  If $[D]$ anti-symmetric, i.e. 
  $[-1]^* D \sim -D$, then 
  $$
  [n]^*D \sim nD \text{ .i.e. } [-1]^* D \sim -D
  ,$$ 
  then 
  $$
  [n]^*D \sim nD \text{ i.e. } [-1]^* D \sim -D
  ,$$ 
  then $[n]^*D \sim nD$.

2. (A.7.210?) Let $D$ be effective, then $2D$ is basepoint free and TFAE:

  - $D$ is ample.
  - $\psi_\ell: A \to A\dual$ where $x \mapsto \tau_x^*D - D$ (where $\tau_x$ is translation by $x$) has finite kernel.
  - $A \to \PP(L(2D))$ is finite.

3. **(A.7.33)** Define 4 maps, $\sigma, \delta, \pi_1, \pi_2: A\cross A \to A$,
\begin{align*}
\sigma(P, Q) = P + Q \quad \delta(P, Q) = P - Q \quad \pi_1(P, Q) = P \quad \pi_2(P, Q) = Q
,\end{align*}

  Then
  
  a. For $D\in \div A$, $[D]$ is symmetric iff 
  $$
  \sigma^* D + \delta^* D \sim 2\pi_1^* D + 2\pi_2^* D
  .$$
  
  b. [D] is antisymmetric iff 
  $$
  \sigma^* \sim \pi_1^* D + \pi_2^* D \iff [D] \in \pic^0 A
  .$$

  > Proof: see Silverman, uses "Theorem of the Square" and the "Seesaw Principle".

Corollary
: For $A/K$ an abelian variety over $K$ a NPFF and $D\in \div A$, then

  a. $p\in A(K\sep)$ and 
  $$
  h_{A, 0}([n]p) = \frac{n^2 + n}{2} h_{A, D}(p) + \frac{n^2 - n}{2} h_{A, D}(-p) + O(1)
  .$$
    If $D$ is symmetric, then 
    $$
    h_{A, D}([n]p) = n^2 h_{A, D}(p) + O(1)
    .$$
    If $D$ is antisymmetric, then 
    $$
    h_{A, D}([n]p) = nh_{A, D}(p) + O(1)
    .$$
    
  > Note: This follows from properties of the Weil height machine.

  b. If $[D]$ is symmetric, than we get the **quadraticity** relation: for all $P, Q \in A(K\sep)$, 
  $$
  h_{A, D}(P+Q) + h_{A, D}(P-Q) = 2h_{A, D}(p) + 2h_{A, D}(Q) + O(1)
  .$$

Proof
: We have 
  $$
  h_{A\cross A, D}(\sigma(P, Q)) + h_{A\cross A, D}(\delta(P,Q)) = 2h_{A\cross A, D}(\pi_1^* D) + 2h_{A\cross A, D}(\pi_2^* D) + O(1)
  .$$

  For $\phi: V \to W$ and $D\in \div W$, we have 
  $$
  h_{C, \phi^*(D)}(p) = h_{W, D}(\phi(p)) + O(1)
  .$$
  So 
  $$
  h_{A\cross A, D}(\sigma(P, Q)) = h_{A, D}(\sigma(P, Q)) = h_{A, D}(P + Q)
  ,$$ 
  and applying it to the other terms works the same way.

Abelian variety has attached *polarization*, and are projective group varieties.
Every projective variety has an ample divisor, by definition?

## Sketch of Proof of Mordell-Weil for $A/K$ a Global Field

1. Weak MW

2. Choose an ample symmetric $D \in \div A$, by definition $A$ carries an ample divisor $D_0$; then $D \definedas D_0 + [-1]^* D_0$ is ample and symmetric.

3. Work with $h_{A, D}$ instead of $h_X$ for elliptic curves

Then the proof goes through as before.

Note $h$ is trying to be a quadratic form (we'll review such forms on groups) except for the $O(1)$.
Maybe we can modify $h$ by a bounded function to make it quadratic -- this is the Neron-Tate canonical height.

> Regulator: defined in terms of the Neron-Tate canonical height on the Mordell-Weil group of an elliptic curve.