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\title{Height Pairings}

\date{}

\author{Jesse Silliman}

\begin{document}
\maketitle

\section{Introduction}
\textbf{Notation}
\begin{itemize}
\item $K$ = number field with integers $\O$, places $v$, completions $K_v \supset \O_v$, residue field $k_v$ of size $q_v$,
\item $\eta = \Spec(\bar{K})$,
\item $X$ = smooth, projective variety over $K$ of dimension $n$,
\item $X_{\eta} = X \times_{K} \bar{K}$, $X_v = X \times_{K} K_v$, 
\item $Z^i(X)$ = cycles of codimension $i$, defined over $K$
\item $\CH^i(X) = Z^i(X)/(\text{rational equivalence})$
\item $H^i = H^i_{\acute{e}t}$
\item Let $\CH^i(X)^0 = \ker(cl \from \CH^i(X) \to H^{i}_{et}(X_{\eta}, \Z_{l}(i)))$, (to be safe, let's have $l$ not lie under a place of bad reduction) (TODO: What is known about indep. of $l$?)
\item $Z^i(X)^0 = $ preimage of $\CH^i(X)^0$ in $Z^i(X)$.
\end{itemize}

\textbf{Goal}:
We want to construct a pairing \[ \pair{\cdot}{\cdot} \from \CH^i(X)^0 \times \CH^{n-i+1}(X)^0 \to \R, \] generalizing the Neron-Tate pairing on abelian varieties.

Note that our cycles are of a dimension where their expected intersection has dimension $-1$.

\begin{eg}[\cite{Neron}, \cite{Gross}]\label{NeronTheorem}
Let $C/K$ be a smooth projective curve, with $\infty \in C(K)$ giving $i\from C \into \Pic^0(C)$. Let $\pair{\cdot}{\cdot}_{NT} \from \Pic^0(C)(K) \times \Pic^0(C)(K) \to \R$ be the Neron-Tate height pairing, identifying $\dual{\Pic^0(A)} \isom \Pic^0(A)$ via the theta divisor. Then, once we have defined $\pair{\cdot}{\cdot}$, we will have
\[\pair{P - \infty}{Q - \infty} = \pair{i(P)}{i(Q)}_{NT}.\]
\end{eg}

\begin{eg}
More generally, for an abelian variety $A/K$ of dimension $n$, the maps $\CH^1(A)^0 \isom \dual{A}(K)$, $\CH^n(A)^0 \to A(K)$ identify the pairings $\pair{\cdot}{\cdot}$ with $\pair{\cdot}{\cdot}_{NT} \from A(K) \times \dual{A}(K) \to \R$.

Recall that Neron-Tate pairings came from the canonical height function associated to the Poincare bundle. The Poincare bundle is not ample, although it is when restricted to the ``diagonal" of $A \times \dual{A}$ via a polarization $A \to \dual{A}$. In particular, we get an induced pairing $A(K) \times A(K) \to \R$ which is symmetric and non-degenerate modulo torsion. 
\end{eg}

To achieve this goal, Beilinson constructs partial local pairings. In other words, for representative cycles $C_1$, $C_2$ with disjoint support, we want to define, for every place $v$, a number $\pair{C_1}{C_2}_v \in \R$. Then we may define the global pairing as $\pair{C_1}{C_2} = \sum_v \pair{C_1}{C_2}_v$. 

We would need this formula to descend to Chow. 

\textbf{n=1}
For divisors on curves, we could insist that $\pair{D}{div(f)} = \log |f(D)|$, the sum of the value of $f$ at the points of $D$, with sign and multiplicity. If this were true for every local pairing $v$, this would descend to $\Pic^0$ by the product formula.

\textbf{n $>$ 1}
More generally, we could hope that our local pairings were well-behaved with respect to correspondences in $X \times \P^1$, to reduce the descent to Chow to dimension 1.

We can do this as follows:

\begin{itemize}
\item For $v$ a place of good reduction (there exists a smooth model over $\O_v$), we may extend cycles $C_1, C_2$ to a good model by Zariski closure, obtaining $\tilde{C_1}, \tilde{C_2}$. Then define the local pairing in terms of the intersection pairing: \[\pair{C_1}{C_2}_v = - (\tilde{C_1}\cdot \tilde{C_2})\log q_v.\] 

\item For $v$ a place of bad reduction, we might attempt similarly to choose a regular proper model, extend by Zariski closure. This would be incorrect, even for curves. 

Better: extend to cycle whose intersection with the fiber is cohomologous to zero. (TODO: what does this really mean?) Unfortunately, this is not known to be possible. Conjectural solutions!

\item For $v$ an infinite place, there is a construction in terms of Hodge theory. We will present this construction for curves.

\end{itemize}

\begin{conj}[Beilinson \cite{Beilinson}]\label{BadPlace}
For each place $v$, and smooth projective variety $X$ over $K$, there exists a local pairing
\[ \pair{\cdot}{\cdot}_v \from Z^i(X_v)^0 \times Z^{n-i+1}(X_v)^0 \partialto \R, \] 
defined on cycles of disjoint support, such that
\begin{enumerate}
\item if $X$ has good reduction at $v$, it is the pairing above,
\item it is functorial in correspondences $f \subset X \times Y$, in that $\pair{f_*a}{b}_v = \pair{a}{f^*b}_v$.
\end{enumerate} \end{conj}


%The content of the conjecture is thus that local pairings exist and are valued in $\R$. We will see that the conjectural constructions are valued naturally in $\Q_l$, conjecturally in $\Q$, independent of the choice of auxiliary $l$. There is no characterization, just conjectural constructions. TODO: These local pairings are not determined if they exist?

\begin{lem}[\cite{Beilinson}] If local pairings valued in $\R$ exist, satisfying \ref{BadPlace}, then there exists a unique pairing 
\[ \pair{\cdot}{\cdot} \from \CH^i(X)^0 \times \CH^{n-i+1}(X)^0 \to \R, \]
defined, on representative cycles with disjoint support, to be $\sum \pair{\cdot}{\cdot}_v$. 
\end{lem}

\begin{rmk}
This is a frustrating definition: we would like it to be positive-definite, but to compute $\pair{x}{x}$ requires a knowledge of moving! Naive heights for points on varieties don't require moving! At least for divisors alg. equiv. to zero and zero-cycles, Neron has related it to naive heights on Picard/Albanese varieties, similarly to \ref{NeronTheorem}.
\end{rmk}

\begin{rmk}
We needed to have a projective variety so that we can move things: roughly, one can move ample things since they are hyperplane sections. See \cite[Tag 0B0D]{stacks-project}, for a precise statement of Chow's Moving Lemma (TODO: Does this moving lemma work for us?).
\end{rmk}


%TODO: Figure out how Beilinson eliminates $\log q_v$ from his local pairings, or put them back in.



Without assuming other conjectures, we can use one of the following groups instead to define local pairings unconditionally:
\begin{enumerate}
\item $\ker(cl \from Z^i(X_v) \to H^{i}(X_{K_v^{ur}}, \Z_{l}(i)))$ (cycles vanishing in \emph{absolute} etale cohomology over the maximal unramified extension of $K_v$)
\item Image of the fiberwise cohom. to zero cycles in $\CH^i(X_{\O_v})$ (cycles which extend to a local regular model) inside of $\CH$.
\end{enumerate}

The construction under group 1) is not known to lie in $\Q \cdot \log q_v$, only in $\Q \cdot \log q_v$, nor to be independent of $l$. Despite this lack of practical value, we will sketch it later.

When are these groups equal to $\CH^i(X)^0$?
\begin{enumerate}
\item when $X$ is a curve, $CH^i(X)^0$ = group 1 = group 2)
\item when $X$ satisfies the weight-monodromy conjectures at all primes of bad reduction, for the relevant cohomology groups, $CH^i(X)^0$ = group 1.
\end{enumerate}

%\begin{rmk}Apparently Kunneman has constructed heights for abelian varieties: HEIGHT PAIRINGS FOR ALGEBRAIC CYCLES ON ABELIAN VARIETIES\end{rmk}

\begin{warn}There is a natural map $\CH^n(X)^0 \to Alb(X)(K)$, but it is not known to be an isomorphism. However, Beilinson shows that, for $X/\Q$, $\CH^n(X)^0 \tensor \Q \isom Alb(X)(\Q) \tensor \Q$ is implied by his conjectures. Even over $\C$, the Albanese is merely the universal abelian variety quotient of $CH^n(X)$.
\end{warn}
%Esnault, Srinivas, Viehweg: The universal regular quotient of the Chow group of points on projective varieties
%Matsusaka, On the algebraic construction of the Picard variety II
%The Albanese has no precise interpretation in terms of zero cycles: see \url{https://en.wikipedia.org/wiki/Albanese_variety}, Roitman's Theorem.



\textbf{Properties}
\begin{itemize}
\item Correspondences: $f \subset X \times Y$, $\pair{f_*a}{b} = \pair{a}{f^*b}$
\item Pairing for cycles algebraically equivalent to zero can be reduced to Neron-Tate pairings %( (hom~0)/(alg~0) can be countably infinite, I am told, but I don't know any elements myself)
\end{itemize}

%TODO: Why is it not functorial under proper, flat maps? Don't those behave well for intersection theory? Perhaps not at infinite places/bad reduction?

%TODO: Maybe sketch the reduction to Neron-Tate pairing: why do alg equiv to zero form an abelian subvariety of intermediate Jacobian?

%\begin{rmk}
%Over a number field, the Neron-Tate definition of pairings extends to all cycles, by considering intermediate Jacobians, which are complex tori with a natural Poincare bundle induced by Poincare duality on the original variety. This is Bloch's approach - note that it fails over function fields! We will follow Beilinson instead. 
%\end{rmk}

%QUESTION: Why aren't intermediate Jacobians for $H^n, n = 2k + 1$, polarized by Poincare duality? Eg: curves on 3-folds, surfaces in 5-folds,... Is it a failure of the form to be definite? The Weil-Lazerri intermediate jacobian is an alternate complex structure which makes it a polarized algebraic variety, but it no longer has any arithmetic significance...

Of course, the point of all this is to formulate
\begin{conj}[Beilinson \cite{Beilinson}]
\hfill
\begin{enumerate}
\item (Swinnerton-Dyer) The groups $CH^i(X)^0$ are finitely generated, with rank = $ord_{s=0}L(H^{2i-1}(X)(i), s)$.
\item The height pairing on $\CH \tensor \R$ is non-degenerate. 
\item Its determinant, times the determinant of the (real) period matrix for $H^{2i-1}(X)$, equals the leading coefficient of the L-function $L(H^{2i-1}(X)(i), s)$ at $s = 0$.
\end{enumerate}
\end{conj}

But we won't talk more about this.

\begin{question}
How does Beilinson decide to consider all cycles homologous, not just algebraically equivalent to, zero?
\end{question}


\section{Local Pairings on Curves}
\subsection{Neron Local Heights \cite{Gross}}
\textbf{Notation}
\begin{itemize}
\item $X =$ smooth, projective curve over a local field $K_v$.
\item $\Div(X/K_v) = $ divisors on $X$ rational over $K_v$,
\item $Z^0(X/K_v) = $ free abelian group on $X(K_v)$,
\item $|\cdot|_v$ the normalized valuation,
\item for $f$ is a function on $X$, with divisor $div(f)$ relatively prime to $a = \sum m_x(x) \in Z^0(X/K_v)$, we define $f(a) = \prod f(x)^{m_x}$.
\end{itemize}

\begin{rmk}
We may assume our curve has a rational point, by taking a finite field extension. We will see how the pairing then descends back to $K_v$.
\end{rmk}

\begin{prop}[Neron \cite{Gross}]
There is a unique function $\pair{a}{b}_v$ on relatively prime divisors $a \in Z^0(X/K_v)$, $b \in \Div^0(X/K_v)$ with values in $\R$ satisfying:
\begin{enumerate}
\item (Linearity) $\pair{a}{b}_v + \pair{a}{c}_v = \pair{a}{b + c}_v$.
\item (Symmetry) $\pair{a}{b}_v = \pair{b}{a}_v$ whenever $b \in Z^0(X/K_v)$
\item (Principal Divisors) $\pair{a}{div(f)}_v = \log|f(a)|_v$ 
\item (Continuity) Fix $b$ and a point $x_0 \in X(K_v) - |b|$. Then the function $X(K_v) - |b| \to \R$, defined by $x \mapsto \pair{(x) - (x_0)}{b}_v$, is continuous.
\end{enumerate}
\end{prop}

The proof of uniqueness is easy - the difference of two such functions gives, by fixing the left variable, a continuous homomorphism from $J(K_v)$ to $\R$, which must be constant (no compact subgroups of $\R$).

An easy lemma, using uniqueness:
\begin{lem}
For a finite extension $H_v/K_v$, we have 
\[ \pair{\cdot}{\cdot}_{H_v} = [H_v:K_v] \pair{\cdot}{\cdot}_{K_v}. \]
\end{lem}

After defining this partial pairing on $Z^0(X/K_v) \times \Div^0(X/K_V)$, we may extend it to a partial pairing on $\Div^0(X/K_v) \times \Div^0(X/K_v)$ by using the previous lemma.

%We will see the functoriality with respect to correspondences $\pair{f^*(a)}{b}_v = \pair{a}{f_*(b)}_v$ directly from the construction.
%TODO: is this actually true for the intersection stuff? need to have functoriality of regular models, or to use linking number definitions...


\subsection{Finite Places}
Let $K$ be a p-adic field, $X$ a smooth proper curve over $K$. We consider a (proper) regular model $X_{O}$ of $X$, with special fiber $X_k$. We will still sometimes denote the place as $v$.

%TODO: Perhaps replace with semistable model, or at least one whose fibers have no multiplicity, so we can extend line bundles uniquely via $Pic^0 $ = identity of Neron model (otherwise have to worry about non-separatedness of relative Picard.) Can we get away with this? Anyways, we are not quite extending using the Neron model, since we are not equating principal divisors from generic fiber. So we should really just define intersection carefully in all regular models, see Chinburg and BGR.

%We will define an intersection pairing on all divisors on $X_{\O}$ with disjoint support on $X_K$. Note that we will be able to intersect horizontal divisors with other horizontal divisors, since we will not require the pairing to factor through principal equivalence.

\begin{defn}
Let $Z_1, Z_2$ be effective divisors on $X_{O}$, considered as closed subschemes defined by ideal sheaves $I_1, I_2$. When $Z_1$ and $Z_2$ are relatively prime, the interesection $Z_1 \cap Z_2$ is dimension zero, necessarily supported in the special fiber $X_k$. This is equivalent to $I_1 \cap I_2$ being a $k$-module of finite length. In this case, we define 
\[ Z_1\cdot Z_2 = \len_k(I_1 \cap I_2). \]
\end{defn}

This pairing is in fact biadditive, and extends bilinearly from effective divisors to all divisors.
%TODO: Is this biadditive/extend to not necc. effective divisors?

\begin{defn}
We say that an effective divisor is \emph{vertical} if its underlying reduced subscheme is contained in $X_k$. Let $V(X_{\O}) \subset \Div(X_{\O})$ denote the subgroup generated by effective vertical divisors. 
\end{defn}

We then have an exact sequence:
\[ 0 \to V(X_{\O}) \to \Div(X_{\O}) \to \Div(X) \to 0. \]

If we restrict this partial pairing
\[ \Div(X_{\O}) \times \Div(X_{\O}) \partialto \Z \]
to $V(X_{\O}) \times \Div(X_{\O})$,
it turns out that it extends to a full pairing
\[ V(X_{\O}) \times \Div(X_{\O}) \to \Z. \]

We will not define this pairing, but the following result lists some of its properties.
\begin{prop}[\cite{Romagny}] Let $E$, $F$ be divisors on $X_{O}$ with $E$ vertical. Then one has:
\begin{enumerate}
\item if $F$ is a vertical divisor then $E \cdot F = F \cdot E$,
\item if $E$ is prime (subscheme is reduced and irreducible) then $E \cdot F = \deg(\O(F) \tensor \O_E)$,
\item if $F$ is principal then $E \cdot F = 0$
\end{enumerate}
\end{prop}

Now, consider the special fiber $X_k$. Its irreducible components $E_1,\ldots,E_r$ need not be reduced or geometrically connected as $k$-schemes. We write $E_i$ for the corresponding divisor, which is a multiple of the divisor of the underlying reduced subscheme $E_i^{red}$. Then $\Z$-linear combinations of $E_i^{red}$ generate the vertical divisors $V(X_{\O})$.

When we restrict the intersection pairing to vertical divisors, we have the following properties:
\begin{thm}[\cite{Romagny}]\label{Intersection}
\hfill
\begin{enumerate}
\item $X_k \cdot F = 0$ for all vertical divisors $F$,
\item $E_i \cdot E_j \geq 0$ if $i \neq j$ and $E_i^2 < 0$,
\item the bilinear form given by the intersection product of $Div(X/K) \tensor_{\Z} \R$ is negative semi-definite, with isotropic cone equal to the line generated by $X_k$.
\end{enumerate}
\end{thm}
%TODO: Romagny's proof of 2) uses that the special fiber is connected - is there a proof which doesn't use ZMT?
%TODO: Check if his fibers are geometric or not, probably not.

\begin{lem}
A degree zero divisor $D \in \Div^0(X/K)$ extends uniquely to an element $\tilde{D}$ of $Div(X_{O}) \tensor \Q$ such that $\tilde{D} \cdot E_i = 0$ for all irreducible components $E_i$ of the special fiber.
\end{lem}
\begin{proof}
\[ 0 \to \ker(\Sigma) \to \oplus_i \Q\cdot E_i \nmto{\Sigma} \Q \to 0 \]
The intersection form is nondegenerate when restricted to $\ker(\Sigma)$.

Let $D'$ the extension of $D$ by Zariski closure, and consider $v = (D' \cdot E_1, \ldots, D' \cdot E_r)$

We have that
\[ \Sigma(v) = \sum_i D' \cdot E_i = D' \cdot X_{k}. \]

Now, $D' \cdot X_k$ may be identified with the degree of the line bundle $\O(D')$ restricted to $X_k$. Since $\O(D')$ is a locally free sheaf on a proper flat family, this is the same as the degree of $\O(D')_{K} = \O(D)$, which is zero by assumption.

%\begin{claim}$\Sigma(v) = 0$
%\end{claim}
%\begin{proof}[Proof of claim]
%First, we show that $\Sigma(v) = D' \cdot X_k$. This is the only place we use that the model $X_{\O}$ is regular. By assumption, $D'$ is a linear combination of divisors $P'$, where $P'$ is the Zariski closure of $P \in X_{K}(K)$. Regularity then implies that $P'$ does not meet the any singular points of the special fiber. In particular, $P' \cdot E_i \neq 0$ for precisely one component $E_i$. Therefore we have 
%\[ P' \cdot X_k = \sum_i P' \cdot E_i. \]
%\end{proof}
%(all points in supp $D$ reduce to somewhere by proper, intersect with multiplicity $1$ by regular).
Thus there is a divisor $C$ supported only on the irreducible components with the same intersection, and we take $D' = D - C$. 
\end{proof}

%\begin{rmk}Note that numerical and homological equivalence are equivalent on regular arithmetic surfaces: this is the Hodge Index Theorem.\end{rmk}

\begin{defn}Given $D_1$, $D_2 \in \Div^0(X_{K})$, we define the local pairing \[\pair{D_1}{D_2}_v = -(D_1\cdot D_2) \log q_v \in \Q \cdot \log q_v, \] where $q_v = \#(k)$.
\end{defn}

\textbf{Linearity} Obvious

\textbf{Principal Divisors}  In case of good reduction: 

Let $D_1 = div(f), D_2 = \sum P_i - \sum Q_j \in Z^0(X/K)$. Normalize $f$ so that $|f|_{\pi} = 1$ (valuation along special fiber), so that $\bar{f} = f \mod \pi$ is a rational function on $X_{k_v}$, not identically zero. 

Then $-(D_1\cdot D_2) \log q_v = \sum_{P_i \cap X_{k_v}} -ord_{P_i}(\bar{f}) \log q_v - \sum_{Q_j \cap X_{k_v}} -ord_{Q_j}(\bar{f}) \log q_v = \log |f(D_2)|_v$.   

In bad reduction, note that $f$ may have zeros or poles on entire irreducible components. These are precisely the ones we need to add/subtract to $D_1$ for $D_1$ to intersect each component in degree zero. This is why we cannot naively extend divisors by Zariski-closure, in the case of bad reduction! The rest should be similar...

\textbf{Continuity} 
We check in the case $D = div(f)$. Fix $x_0 \in X(K) - |D|$. We must show that $x \mapsto \pair{(x)-(x_0)}{D}$ is a continuous function on $X(K) - |D|$. By the previous property, we have  $\pair{(x)-(x_0)}{D} = \log |f(x)|_v - \log |f(x_0)|_v$. Since $\log: \R_{> 0} \to \R$  is continuous and $|\cdot|_v : K \to \R_{> 0}$ is tautologically continuous, it remains to see that $f \from X(K) - |D| \to K$ is continuous, and this is passage to $K$-points
on the morphism $f:X - |D| \rightarrow \mathbf{A}^1_K$, so it remains to check that
passage to $K$-points carries $K$-morphisms to continuous maps.
But by the method of topologizing the $K$-points (via gluing on Zariski-open affines),
this problem is of local nature and hence reduces to the affine case where
it is clear (as polynomials are continuous).
%TODO: Find a reference for this ``classical fact''

%Since $X_{\O}$ is proper, a point $y \in X(K)$ extends to a point $\tilde{y} \in X_{\O}(\O)$. Since $X_{\O}$ is regular, the completed local ring at $\tilde{y}$ is (non-canonically) $\O[[t]]$ (check that residue field of coefficient ring agrees with $k(\tilde{y}) = k$ and ramification of coefficients over $\O$ agrees with geometric multiplicity of the irred. component $\tilde{y}$ intersects, which must be $1$ by regularity). Now, multiplying $f$ by $\pi^n$ for $n$ large enough, so that its valuation along the irreducible component meeting $\tilde{y}$ is $|f|_{E_i} \leq 1$, we obtain $f \in \O[[t]]$. 

% This power series converges on the set $\{z \mid |y-z|_v < 1\}$, defining a continuous function.

% TODO: Why does it make sense to treat the residue tube as a *disk* in the parameter t? Why does the power series agree with the algebraic function? Disk = take etale map to $\A^1$, do p-adic inverse function theorem. Then if you shrink the disk a little, the function has shrinking coefficients, so we can go mod $p^n$ to make it agree with a polynomial in the algebraic uniformizer $t$.
%Fix $D \in \Div^0(X/K)$, $x_0 \in X(K) - |D|$. By first extending the field $K$ so that $D \in Z^0(X/K)$, we may reduce to the checking the case $D = P - Q$. We extend $D$ to a divisor $\tilde{D} = D + E$ on $X_{\O}$. We consider the map
%$x \mapsto ((x) - (x_0))\cdot \tilde{D}$.   


\textbf{Symmetry} Obvious

\begin{rmk}More generally, for non-principal divisors, we may extend a section of a line bundle once we choose a way to extend the line bundle (choosing a model is a choice of extension of $O_X$), and interpret the pairing as $log|s(D_2)|$. Extending the line bundle is roughly the same as putting a p-adic metric on it. The condition of degree zero may be thought of as a way to get a ``canonical metric'', as in the archimedean setting.
\end{rmk}

\subsection{Infinite Places}
\subsubsection{Green's Functions via thinly veiled Hodge Theory}
It will suffice to construct the local pairing when $K_v = \C$, using $[\C : \R] \pair{a}{b}_{\R} = \pair{a}{b}_{\C}$ to define the pairing over $\R$.

We will define this local pairing for all Riemann surfaces $X$. A reference for this construction, in more generality, is in \cite{Lang}.

\begin{claim} The real integral
\[ \re\int \from H^0(X, \Omega^1) \times H_1(X, \R) \to \R \]
is a perfect pairing.
\end{claim}
\begin{proof}
As vector spaces over $\R$, $H^0(X, \Omega^1)$ and  $H_1(X, \R)$ have the same dimension ($2g$). We know that $H_1(X,\Z) \subset \Hom(H^0(X, \Omega^1),\C)$ is a lattice: the Jacobian is compact. So $\int \from H_1(X,\R) \isom \Hom_{\C}(H^0(X, \Omega^1),\C)$. Then use that the map $\re \from \Hom_{\C}(H^0(X, \Omega^1),\C) \to \Hom_{\R}(H^0(X, \Omega^1),\R
)$ is an isomorphism.
\end{proof}

\begin{prop}\label{forms}
Let $D \in \Div^0(X)$. There exists a unique $\omega_D \in H^0(X-|D|, \Omega^1)$ such that
\begin{enumerate}
\item $\omega_D$ has log poles along $D$
\item $\Res(\omega_D) = D$.
\item $\re\int \omega_D \from H_1(X - |D|,\Z) \to \R$ vanishes.
\end{enumerate}
\end{prop}

\begin{rmk}
For $D = P - Q$, this can be rephrased as a splitting of the residue sequence
\[ 0 \to H^1(X, \R(1)) \to H^1(X-|D|, \R(1)) \nmto{\Res} (\R \oplus \R)/\R \to 0\]
in the category of mixed hodge structures over $\R$. 
%See also ``canonical splitting over $\R$ of MHS with only 2 weights''
\end{rmk}
\begin{proof}
We first check uniqueness. Let $\omega_D, \omega_D'$ be two forms satisfying 1)-3). The difference $\omega_D - \omega_D'$ lies in $H^0(X, \Omega^1)$ by 1) and 2), and has $\re\int(\omega_D - \omega_D') = 0$ by 3). But the claim then implies $\omega_D - \omega_D' = 0$.  

It will suffice to consider the divisor $D = P - Q$, since all degree zero divisors are sums of these. 

We use Riemann-Roch:
\[h^0(K + (P+Q)) - h^0(-(P+Q)) = deg(P+Q+K) - (g-1) = 2 + (g-1) = g+1\]

We see that $H^0(X-|D|, \Omega^1(log))/H^0(X, \Omega^1)$ is 1-dimensional. Take any $\omega \in H^0(X-|D|, \Omega^1(log))$ with $\Res(\omega) = D$.

The map $\re\int\omega_1 \from H_1(X-|D|,\Z) \to \R$ vanishes on the small loops around $|D|$ ($\int dz/z = 2i\pi$), hence factors through $H_1(X, \Z)$.

As a functional $H_1(X, \Z) \to \R$, the claim identifies $\re\int\omega$ with $\re\int \eta$ for $\eta \in H^0(X, \Omega^1)$. We can take $\omega_D = \omega - \eta$. Now, $\re\int \omega_D \from \from H_1(X - |D|,\Z) \to \R$ vanishes.
\end{proof}

Now, the vanishing of $\re\int \omega_D \from H_1(X - |D|,\Z) \to \R$ implies that as a function on $X$, $x \mapsto \re\int_{x_0}^x\omega_D$ is well-defined up to a constant. In other words, we may evaluate it on degree-zero divisors.

\begin{defn}
Given two degree-zero divisors $D_1, D_2$ on $X$, we define \[\pair{D_1}{D_2}_{\infty} = 2 \cdot \re\int_{D_2}\omega_{D_1}.\]
\end{defn}

Let's check that this is a Neron local height:

\textbf{Linearity}: Obvious

\textbf{Continuity}: Obvious

\textbf{Principal Divisors}: $D = div(f), \omega_D = d\log(f), 2 \cdot \re \int \omega_D = \log |f|^2$


\textbf{Symmetry}: 
%The following proof is cobbled together from Colmez and Coleman via Lang - note that we avoid using currents or the equality $\del \bar{\del} log|z| = 2\pi i \delta_0$ (although we might be proving the latter, in a way...)
\begin{prop}[Green's Theorem \cite{Colmez}\cite{Lang}]
Let $a_i, b_i$, $i = 1,\ldots,g$, be a symplectic basis for $X$, thought of as the boundary of a standard fundamental domain. Let $\omega_1, \omega_2$ be differential forms satisfying 1) and 2) of \ref{forms} (no condition on the periods). 

Then integrals $\int_{D_i} \omega_j$ are well-defined mod $\Z(1)$, and we have the following formula:
\[ \int_{D_1} \omega_2 - \int_{D_2} \omega_1  = \frac{1}{2\pi i}\sum_{i=1}^{g} \left( \int_{a_i} \omega_1 \cdot \int_{b_i} \omega_2 - \int_{a_i} \omega_2 \cdot \int_{b_i} \omega_1 \right) \mod \Z(1) \]

In particular, if $\omega_1, \omega_2$ have purely imaginary periods, then 
\[ \int_{D_1} \omega_2 = \int_{D_2} \omega_1  \mod \R(1)\]
\end{prop}
%TODO: Does our proof show the $\Z(1)$ claim or just $\mod \R(1)$?
\begin{proof}
Our proof will only prove $mod \R(1)$, and this is all that is needed for our purposes.

First, two lemmas:

\begin{lem}[\cite{Colmez}]\label{integration1}
For any $C^{\infty}$ differential forms $\omega_1, \omega_2$ on X:

\[ \int_{X} \omega_1 \wedge \omega_2 = \sum_{i=1}^{g} \left( \int_{a_i} \omega_1 \cdot \int_{b_i} \omega_2 - \int_{a_i} \omega_2 \cdot \int_{b_i} \omega_1 \right) \]
\end{lem}
We won't show this: it is straightforward to prove.

\begin{lem}\label{integration2}
For forms $\omega_1, \omega_2$ satisfying 1) and 2) of \ref{forms}, \[ \int_{D_1} \omega_2 - \int_{D_2} \omega_1 = \frac{1}{2\pi i}\int_{X - |D_1| \cup |D_2|} \omega_1 \wedge \omega_2 \mod \Z(1) \]
\end{lem} 


Assuming these lemmas, we may modify $\omega_1, \omega_2$ so that they are $C^{\infty}$ across $|D_1| \cup |D_2|$, and apply \ref{integration1}.

\begin{proof}[Proof of \ref{integration2}]
Define $\re(\omega) = \frac{1}{2}(\omega + \bar{\omega})$, $\im(\omega) = \frac{1}{2}(\omega - \bar{\omega})$. Observe that $\int_{\gamma} \bar{\omega} = \bar{\int_\gamma \omega}$ (take holomorphic/antiholomorphic primitives locally, and then it is obvious).

Let $g_{D_i} = \re \int \omega_{i} = \int \re(\omega_i)$



\begin{align}
2\pi i (g_{D_1}(D_2) - g_{D_2}(D_1)) &= \int_{X - |D_1| \cup |D_2|} d(g_{D_1} \omega_2 - g_{D_2} \omega_1)  \text{(Residue)}\\
&= \int_{X - |D_1| \cup |D_2|} \re(\omega_1)\wedge \omega_2 + \omega_1 \wedge \re(\omega_2) \\
\im \int_{X - |D_1| \cup |D_2|} \re(\omega_1)\wedge \omega_2 + \omega_1 \wedge \re(\omega_2) &=  \int_{X - |D_1| \cup |D_2|} \re(\omega_1)\wedge \im(\omega_2) + \im(\omega_1) \wedge \re(\omega_2) \\ &= \int_{X - |D_1| \cup |D_2|} \im (\omega_1 \wedge \omega_2) \\
g_{D_1}(D_2) - g_{D_2}(D_1) &= \re \left(\frac{1}{2\pi i} \int_{X - |D_1| \cup |D_2|} \omega_1 \wedge \omega_2\right) \\
\end{align}
\end{proof}


\end{proof}

\begin{rmk}
It is not hard, using the symmetry of the local pairing, to prove that Weil reciprocity holds after taking absolute values: in other words, if $f, g$ have disjoint divisors, then $|f(div(g))| = |g(div(f))|$.

There is an approach by Deligne to local heights which takes a pair of metrized line bundles $L, M$ on X (for example, degree zero line bundles have canonical Hermitian metrics, roughly by constructing Green's functions as above) and produces a hermitian vector space $\pair{L}{M}$. However, this approach requires the full Weil reciprocity as input. 
\end{rmk}
%TODO: Check that we really have $\mod \Z(1)$, as opposed to the $\mod \Z$ that Colmez claims...


%\subsubsection{Arakelov Theory}
%Consider the line bundle $\O(D)$. The fact that its Chern class (= degree) is zero implies that it admits a form $\omega$ such that the integral $\int \omega$ has periods in $\Z(1)$ (this is the content of the exponential exact sequence). Then the function $s = \exp(\int\omega)$ is a section of $\O(D)$, with $div(s) = D$.

%This is roughly the content of the exact sequence
%\[ 0 \to H^0(X, \Omega^1) \to H^1(X, \C^*) \to H^1(X,\O_X^*)^{0} \to 0, \] which may be obtained by exponentiating the Hodge exact sequence 
%\[ 0 \to H^0(X, \Omega^1) \to H^1(X, \C) \to H^1(X,\O_X) \to 0. \]

%TODO: Say more, that is not enough. Do I need to mention DSK, DTK?

%Our integral $\re\int\omega_D$ provides an interpretation to the symbol $\log|s|$, and can be used to define a Hermitian metric on $\O(D)$. 

\subsubsection{Examples}

If we can write down real-valued harmonic functions $|f|$ on $X$ with $div(|f|) = D$, then $\log(|f|) - \re\int\omega_D$ will be a constant, by the maximum principle for harmonic functions.

\begin{enumerate}
\item $\P^1$: $\log|z|$ for $D = (0) - (\infty)$.
%\item $E = \C^*/q^\Z$: Letting $t$ be the parameter on $\C^*$, $\log|\theta(t)|$ is a function for $\O(\infty)$, where $\theta(t) = (1-t) \prod_{n > 0}(1-q^n t)(1-q^n t^{-1})$. This doesn't fit in our framework, since $D = \infty$ is not degree zero, and $\theta(t)$ fails to descend to a section of $\O(\infty)$. However, it can be used to create functions on degree zero line bundles. For example, let $D = (E[n]) - n^2(\infty)$. Then $\log|\frac{\theta(t^n)}{\theta(t)^{n^2}}|$ is a legitimate function arising from our framework.
\item $\C/\Lambda$: A natural multi-valued function with a pole of order 1 at $\infty$ is the Weierstrass $\sigma$ function:
\[ \log \sigma = \int \zeta(z) dz = \int (\int - \wp(z) dz) dz.\]
We want to modify this so that it is single-valued, i.e. we want to modify $\sigma$ so that it transforms under $\Lambda$ by norm 1 elements.

Our new function (the Klein function) is
\[ k(z) = \Delta(\Lambda)^{1/12}e^{-z \eta(z)/2} \sigma(z),\]
where $\eta(z)$ is the $\R$-linear extension of $\eta \from \Lambda \to \C$ by $\eta(\lambda) = \zeta(z+\lambda) - \zeta(z)$.

\begin{claim} $-\log |k(z)| = \frac{1}{2}\re(z \eta(z)) - \log|\sigma(z)| - \frac{1}{12}\log(\Delta(\Lambda))$ will be a real-analytic function on $E$, $\sim \log|z|$ near $z=0$.
\end{claim}

%CUT THIS OUT/FIX IT UP:
%We have ``clearly'' just subtracted off the real part of the periods of the multivalued differential form $\omega = \zeta(z) dz$, up to a constant.

%$\log \sigma$ has small periods in $\Z(1)$ by Legendre (use the lemma in Green's thm above), so it suffices to fix big periods by a real function with no poles.

%\begin{claim}
%$\re \log \sigma(z)$ and $\frac{1}{2} \re (z \eta(z))$ transform the same under translation by $\Lambda$.
%\end{claim}
%\begin{proof}
%Idea: $\wp(z)dz$ and $\frac{\pi}{\im(\tau)} d\bar{z}$ agree on big loops. agree mod $\Z(1)$ upon iteration? I.e. claim that real periods of $\int \zeta(z) dz$ agree with those of $\frac{\pi}{\im(\tau)} \int \bar{z} dz$...  Then the comparison to $\int dz \cdot \int d\bar{z}$ is easy (at least for rectangular lattice).
%\end{proof}

%The function $\sigma$ transforms as \[ \sigma(z + \omega) = \pm  e^{\eta(\omega)(z + \frac{1}{2} \omega)}\sigma(z).\]  

%$\frac{1}{2}(z+\omega) \eta(z+\omega) - \frac{1}{2}z \eta(z) = \frac{1}{2} (\omega \eta(\omega)) + \frac{1}{2}(z \eta(\omega) + \omega \eta(z))$.

%TODO: Explain why $- \log |k(z)|$ relates to the Neron local height $\pair{(P) - (\infty)}{(P) - (\infty)}_{\C}$. Silverman uses almost-quadratic local functions! We don't want that!

When we have a degree zero divisor $D$, we can use translates of this function to compute $\pair{D}{\cdot}_{\C}$. For example, if $D = \sum m_i z_i$, then $\re\int \omega_D = \log|\prod_{i} k(z - z_i)^{m_i} |$.

%Although we won't prove this, it is PROBABLY true that $\pair{(P) - (\infty)}{(P) - (\infty)}_{\C}$ and $-2\pair{(P)}{(\infty)} = \log|k(z(P))|$ differ by at worst a constant.

%\item The same thing should work for all abelian varieties, giving canonical Green's functions for all divisors on curves (not just degree zero) via theta functions on Jacobians. Take Chern class $H$ for polarizing line bundle $\O(D)$, calculate $H(u,u)$ from this, use to kill the real periods of the complex-analytic, multivalued standard ``section''. Note that $E(u_1, u_2)$ calculates the small periods, in $\Z(1)$.  Exercise: Interpret this an improved version of the Appell-Humbert theorem as stated in Mumford, where the middle group is now line bundles with Hermitian metrics, whose Chern connection is no longer flat, merely ``standard''.

\end{enumerate}

%\subsection{Divisors with Non-Disjoint Support, Non-zero Degree}
%Maybe should split degree and disjointness - disjoint works over all places, degree works over good places.

%Motivation: Want to understand what $\pair{a}{a}_v$ means.

%Idea: Extend to divisors of non-disjoint support, degree $\neq 0$ divisors, so that $\pair{(P) - (\infty)}{(P) - (\infty)}_v = (\pair{(P)}{(P)}_v + \pair{(\infty)}{(\infty)}_v) - 2\pair{(P)}{(\infty)}_v$ makes sense.

%\textbf{Non-Disjoint Support}
%Choose local parameter $t_x$ for every $x \in X(K_v)$.
%Redefine Neron's characterization where we allow $\pair{D}{div(f)} = \log|f[D]|$, with $f[D]$ defined using the choice $t_x$ to evaluate $f[D]$ as the leading term of $f$ at the points in the support of $D$.
%Show that, having done this, we still get same global pairing, by product formula.

%Moral: By making non-canonical choices, we can extend to degree zero divisors without disjoint support.

%\textbf{Non-Zero Degree}
%For all finite places of good reduction, there is an obvious extension.
%For infinite places, there is Arakelov theory (canonical metrics on line bundles).
%For finite places of bad reduction: ...

%Why should these two things play nicely? They 


%Show that with a good choice of $t_x$, we have $\pair{(x)}{(x)}_v = 0$ (will be true for almost all $v$)


%TODO: Do I believe this claim at a place of bad reduction? It implies $\pair{(P) - (\infty)}{(P) - (\infty)}_v = - 2\pair{(P)}{(\infty)}_v$. But this is not an issue, since this RHS is not defined!

%\begin{claim}[Gross]
%If $dt_x$ is a generator for $\Omega^1_{X_{O_v}}$, then $\pair{(x)}{(x)}_v = 0$.
%\end{claim}
%\begin{proof}
%Consider $div(f) = n\cdot a - b$, where $f = t_a^n \mod t_a^{n+1}$. 
%$\pair{a}{n\cdot a} = \log|f[a]| + \pair{a}{b} = \pair{a}{b}$.
%$\pair{a}{a} = \frac{1}{n} \pair{a}{b}$

%We have a bunch of freedom to choose $f$: sections of $\O(n\cdot a)$, for example, will be the functions with poles at $a$, only zeros elsewhere (can assume $b$ is effective!)
%Can we be sure there are no *basepoints*: zero at $a$ does not imply zero at $x$ for any $x$? I am pretty sure this is true.
%Can we make sure it has no basepoints as we extend over the special fiber? vanishing at $a$ doesn't imply that you pass through any particular point of the special fiber...

%I think the claim is that, for $n$ large enough $\O(n \cdot a)$ gives an embedding of $X$ into projective space so that the natural model arising is nice: the map $X_{O_v} \to \P^n_{O_v}$ is an embedding on special fibers.

%In other words, we could show this by showing that *generically ample* line bundle *naturally* extend to ample line bundle on the total (naturally = new divisor has no vertical components?).

%Alessandro says: take your ample line bundle, push it forward, use projection formula + cohomological criterion.

%So what line bundle do we get when we push-forward? Surely is the one with either the whole vertical component or no vertical component (as zeros or poles?). If we had whole vertical component, it would be ample? Yes: $\O(vert)|vert = (\Omega^1_vert)^*$ is ample on vert. But also no vertical would be ample - at least if we had a point landing in every component... we could worry that we are blowing down components not containing the point $a$. So no matter what, $j_*L$ must have divisors for the irred components our point fails to land in.
%\end{proof}

%TODO: Does the above work for points/divisors on abelian varieties? We should be moving the divisors so that they don't contain the points. This should again be possible using ampleness.
%Extend to all points and divisors with disjoint support (is this actually possible?)
%$<p, D> = <p, E = D + div(f)>$, need to show that we can move the irred comps containing $p$ to other ones. If only one irred. comp of $D$ needs to be moved, can do as above. Can we iterate without screwing up?


\section{Beilinson Heights}

\textbf{Pragmatic Motivation}: We cannot do intersection pairing unconditionally in the case of bad reduction. We introduce one possible abstract pairing which, when it works, does not require choosing a model, and agrees with intersection pairings when those work too. Unfortunately, it is not known to be independent of $l$ (but that hasn't stopped us before!)

\textbf{Theological Motivation 1} (Unconditional Archimedean Heights) The same definitions produce the local pairing at infinite places, when applied to Deligne cohomology (extensions of mixed hodge structures) instead of absolute etale cohomology (extensions of l-adic representations). See \cite{Beilinson}.

\textbf{Theological Motivation 2} (Weight-Monodromy implies unconditional p-adic Heights) Very similar definitions, when applied to p-adic etale cohomology ($p = l$), produce local heights valued in $\Q_p$, which give a cohomological interpretation of various ``p-adic Green's functions'', and allow for the statement of p-adic BSD conjectures. See \cite{Nekovar}.

\subsection{Etale Abel-Jacobi Maps} 
%TODO: Is this section necessary? Not really...
%TODO: I could use mixed extensions, and avoid non-geometric etale cohomology.
Goal: Algebraic cycles create extension of Galois representations.


Let $X$ be a smooth, proper variety over a field $K$, with $\eta$ the geometric generic point. Let $G_K$ be the absolute galois group of $K$. Consider the cycle class map
\[ cl \from \CH^i(X) \to H^{2i}(X_\eta, \Z_l(i)). \]

Let $\CH^i_Z(X)$ denote the cycles supported on a fixed codimension $i$ subvariety $Z$. Then we have also
\[ cl \from \CH^i_Z(X) \to H^{2i}_{Z}(X_\eta, \Z_l(i)).\]

TODO: reference for etale cycle class maps for singular subvarieties of smooth varieties?

The image is the local cohomology with support on $Z$. This fits into a long exact sequence
\[\ldots \to H^{2i-1}(X_{\eta}, \Z_l(i)) \to H^{2i-1}((X-Z)_{\eta}, \Z_l(i)) \to H^{2i}_{Z}(X_\eta, \Z_l(i)) \to H^{2i}(X_\eta, \Z_l(i)) \to \ldots,\]
which is compatible with the above maps $cl$ in the obvious way.

In particular, for a cycle $W$ supported on $Z$ which is cohomologous to zero, we obtain by pullback
\[\ldots \to H^{2i-1}(X_{\eta}, \Z_l(i)) \to E \to \Z_l \to 0.\]

This defines a map
\[ \CH^i(X)^0 \to \Ext^1_{G_K}(\Z_l, H^{2i-1}(X_\eta, \Z_l(i))) \]

Similarly, we obtain
\[\CH^{n-i+1}(X)^0 \to \Ext^1_{G_K}(\Z_l, H^{2n-2i+1}(X_\eta, \Z_l(n-i+1))).
\]

Now, write $V = H^{2i-1}(X_\eta, \Z_l(i))$, $W = H^{2n-2i+1}(X_\eta, \Z_l(n-i+1))$. 

Poincare duality for etale cohomology tells us that $V^* = H^{2n-2i+1}(X_\eta, \Z_l(n-i)) = W(-1)$. In other words, our two maps become

\begin{align*}
j_i \from \CH^i(X)^0 \to & \Ext^1_{G_K}(\Z_l, V), \\
j_{n-i+1} \from \CH^{n-i+1}(X)^0 \to & \Ext^1_{G_K}(\Z_l, V^*(1)),
\end{align*}


\begin{eg}
When $X$ is a curve, we have
\[0 \to H^1(X_\eta, \Z_l(1)) \to H^1((X-U)_\eta, \Z_l(1)) \to Div^0_Z(X) \to 0.\]
This is an example of a Kummer map: fixing a basepoint $\infty$, we obtain a map 
\[ \kappa: X(K) \to \Ext^1_{G_K}(\Z_l, H^1(X_\eta, \Z_l(1))),\]
which agrees with the classical Abel-Jacobi map followed by the Kummer map for the Jacobian of $X$ (also using poincare duality/duality for Jacobians, i.e. $V = V^*(1)$).
\end{eg}

Now, let us specialize to $K$ a p-adic field of residue characteristic $p \neq l$. 

%TODO: Maybe cut this out, and just quote Nekovar.
%\begin{claim}
%Suppose $X$ has good reduction. Then the map $j_i$ is trivial.
%\end{claim}
%\begin{proof}
%Let $X_{\O}$ be a good model. A cycle $W \in CH^i(X)^0$ extends to $W_{\O}$ in $CH^i(X_{\O})^0$. Checking compatibilities, one deduces that the extension $j_i(W) \in \Ext^1_{G_K}(\Z_l, V)$ is unramified (relative etale cohomology of a pair with good reduction is unramified). Do we then need to argue via Frobenius weights? Surely...
%\end{proof}

\begin{prop}[\cite{NekovarAbelJacobi}]\label{purity}
If $X$ has potentially good reduction or, more generally, if the purity
conjecture for the monodromy filtration on V holds, then $\Ext^1_{G_K}(\Q_l, V) = 0$.
\end{prop}
The proof uses the purity from the Weil conjectures + Tate's Euler characteristic formula.
%TODO: Copy Nekovar's proof. It is only two lines.


\textbf{A wishful digression on mixed extensions}

Let $K$ a global field. Pretend we had an category like $G_K$-reps in which $V$ lived, which did Hodge theory when restricted to the decomp. group at infinity. We will consider $H^1(V) \times H^1(V^*(1)) \to H^2(\Q_l(1))$. If we impose a self-dual Selmer condition (say, $f$), we would get $H^1(V)_f \times H^1(V^*(1))_f \to 0$ (since $Ext^2(\Q_l, \Q_l(1))$ should inject into product of local versions). We suppose that we have algebraic cycles whose cycle-classes are crystalline, giving extensions $E_1 \in H^1(V)_f, E_2 \in H^1(V^*(1))_f$.

We would then attempt to find a canonical reason for this class to vanish, in terms of a canonical element of $H^1(\Q_l(1))$. We could try to do this locally at each place $v$.

The fact the a cup-product of extensions vanishes implies that we can fill in the upper-right of the following matrix, to get a cochain valued in $3 \times 3$ matrices (different coordinates have different coefficients):

$\begin{pmatrix}
1 & \Z_l(1) & * \\
0 & 1 & \Z_l \\
0 & 0 & 1
\end{pmatrix}$,

where the upper-left minor is the extension $\Ext^1_{G_K}(V, \Z_l(1))$ and the lower-right is $\Ext^1_{G_K}(\Z_l, V)$.
More diagramatically, but with less meaning, we could write

$\begin{pmatrix}
\Z_l(1) & E_1 & E_3 \\
0 & V & E_2 \\
0 & 0 &  \Z_l \\
\end{pmatrix}$,

with $E_1$, $E_2$ denoting the 1-extensions as above (we have dualized $E_2$ here). This means only that there is a Galois module $E_3$ which has an injection $E_1 \inj E_3$, a surjection $E_3 \surj E_2$, but has only 1 copy of $V$ in it. Such an object is called a ``mixed extension''. There is a canonical mixed extension associated to $E_1 \cup E_2$ when $E_1$ and $E_2$ come from cycle classes, but we won't need this.

If $H^0(K_v, V) = H^1(K_v, V)_f = 0$ for every $v$ (for example, we could assume \ref{purity}), we would be in business:
Take the "mixed extension" $E_3$. These assumptions let us turn the restriction of $E_3$ to $G_{K_v}$ into an element of $H^1(K_v, \Q_l(1)) \isom \Q_l$ (except when $v \mid l$, where we ignore this issue and switch to a different $l$). This is because our extensions $E_1$ and $E_2$ are trivialized as $G_{K_v}$-modules, even canonically so. This lets us put $E_3$ into a canonical shape:
$\begin{pmatrix}1 & 0 & * \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}$, 

again diagramatically as 

$\begin{pmatrix}
\Z_l(1) & 0 & E_3 \\
0 & V & 0 \\
0 & 0 & \Z_l \\
\end{pmatrix}$. 

At this point, $E_3$ has become an element of $Ext^1_{G_{K_v}}(\Z_l, \Z_l(1)) \isom \widehat{\Q_l} \isom \Z_{l}$, by Kummer theory and the $ord_l$ map. This is a local height (after normalizing by $\log q_v$).

Then just add up these numbers: this is a global height.

\subsection{Linking Numbers}
Now, changing notation, let $K$ be the maximal unramified extension of a p-adic field.

We note that the etale cohomology of $\O_K$ is quite like that of a disk:
\begin{claim}\label{DiskMagic}
\hfill
\begin{enumerate}
\item $\Spec(K)$ has $\Q_l$ cohomological dimension 1.
\item $H^1(\Spec(K), \Q_l(1)) \isom \hat{K} \tensor \Q_l \isom \Q_l$
\item $H^1(\Spec(K), \Q_l(1)) \isom H^2_{s}(\Spec(\O_K), \Q_l(1))$, for $s$ the closed point
\end{enumerate}
\end{claim}
\begin{proof}
We consider the residue sequence
\begin{align} H^1(\Spec(O_K), \Q_l(1)) &\to H^1(\Spec(K), \Q_l(1)) \to H^2_s(\Spec(\O_K), \Q_l(1)) \\
&\to H^2(\Spec(O_K), \Q_l(1)) \to H^2(\Spec(K), \Q_l(1)).
\end{align}
1): pro-l inertia is a quotient of $\widehat{\Z}$, of cohomological dimension 1, and for l-torsion modules $M$, $H^1(\text{pro-p}, M) = 0$ by $l \neq p$.

2):  follows from Kummer theory and that $l \neq p$.

For 3):
$H^1(\Spec(\O_K), \Q_l(1))$ vanishes, since $\Spec(\O_K)$ is strictly henselian.

$H^2(\Spec(\O_K), \Q_l(1))$ vanishes, since $Br(\O_K) = Br(k) = 0$ (Brauer group of Henselian ring = that of its residue field).
\end{proof}

Let $a_1 \in \CH^i(X)^0$, $a_2 \in \CH^{n-i-1}(X)^0$. 
Unfortunately, we must make an assumption.

\begin{ass}\label{bounding-disks}
Assume that $a_1, a_2$ are zero under the ``absolute" cycle class map
\[ cl \from \CH^i(X)^0 \to H^{2i}(X, \Q_l(i)). \]
\end{ass}

\begin{claim}
Assumption \ref{bounding-disks} holds under the conditions of \ref{purity}.
\end{claim}
\begin{proof}
Pretend that $K$ = p-adic field, as opposed to maximal unramified extension. The proof becomes a little more tedious otherwise, but the result is still true, using that our cycles were defined over the p-adic field anyways, and that $\Spec(\F_q)$ has cohomological dimension 1.

The cycle class maps should have the following compatibility:

\begin{tikzcd}
\CH^i(X) \arrow{r}{cl}\arrow{rd}{cl}
&H^{2i}(X, \Q_l(1)) \arrow[two heads]{d}
\\
&H^{2i}(X_{\eta}, \Q_l(1))^{G_K}
\end{tikzcd}

The vanishing $\Ext^1_{G_K}(\Q_l, V) = 0$ implies that $H^{2i}(X, \Q_l(1)) = H^{2i}(X_\eta, \Q_l(i))^{G_K}$, by Leray-Serre spectral sequence and 1) of \ref{DiskMagic}. 

Therefore, since our classes were cohomologous to zero in $H^{2i}(X_{\eta}, \Q_l(1))$, they are also zero in $H^{2i}(X, \Q_l(1))$.
\end{proof}

\begin{claim}
Assumption \ref{bounding-disks} holds when $a_1, a_2$ extend to cycles homologous to zero on a regular model $X_{\O}$. 
\end{claim}
\begin{proof}
\hfill
\begin{tikzcd}
\CH^i(X_{\O}) \arrow{r}{cl}\arrow{d}
&H^{2i}(X_{\O}, \Q_l(1)) \arrow{d}
\\
\CH^i(X) \arrow{r}{cl} &H^{2i}(X, \Q_l(1))
\end{tikzcd}
\end{proof}

\begin{claim}
A cycle is homologous to zero on a regular model if and only if its intersection with $X_k$ is homologous to zero on $X_k$.
\end{claim}
\begin{proof}[Sketch]
The natural restriction map $H^{2i}(X_{\O}) \to H^{2i}(X_k)$ is given by cup-product with the fundamental class of $X_k$ in $X_{\O}$, hence agrees with intersection on cycle classes in $H^{2i}(X_{\O})$. But the map $H^{2i}(X_{\O}) \to H^{2i}(X_k)$ is an isomorphism, by proper base change.
\end{proof}

%\begin{rmk}
%More importantly, can we use only *fiberwise* homologous to zero? Beilinson says yes.

%I.e. geometrically generically + geometrically fiberwise implies generically?

%\begin{tikzcd}
%0 \arrow{r}& H^1(\Spec(K), H^{2i}(X_{\eta})) \arrow{r}& H^{2i}(X) \arrow{r}& H^0(\Spec(K), H^{2i}(X_{\eta})) \arrow{r}& 0 \\
% {}& H^1(\Spec(\O), Rf_*^{2i-1}(X_{\O})) \arrow{r}\arrow{u}\arrow{d}& H^{2i}(X_{\O}) \arrow{r}\arrow{u}\arrow{d}& H^0(\Spec(\O), Rf_*^{2i}(X_{\O})) \arrow{r}\arrow{u}{ker \inj H^{2i}(X_k)?}\arrow{d}& 0 \\
% {}& 0 \arrow{r}& H^{2i}(X_k) \arrow{r}& H^{2i}(X_k) \arrow{r}& 0 
%\end{tikzcd}

%Also, note Standard Conjecture D says numerical and homological equivalence are the same.
%\end{rmk}


From the analogs of the long exact sequences above, we have that
$cl(a_1) \in H^{2i}_{|a_i|}(X, \Q_l(i))$ is the image of some $\alpha_1 \in H^{2i-1}(X - |a_1|, \Q_l(i))$. Similarly, $cl(a_2)$ is the image of some $\alpha_2$.

%Galois-cohomologically, we are taking $\alpha_1 \in H^{2i-1}((X - U)_{\eta}, \Q_l(i))^{G_K}$, $\alpha_2$ similar, trivializing the extensions $j(a_1), j(a_2)$.

%TODO: If I explained mixed extensions, we could avoid using non-geometric etale cohomology, perhaps, but that sounds like it would be annoying. Maybe that only works naively when we get to ignore the real places, i.e. in Nekovar's p-adic heights, so that we can get a global mixed extension.

%TODO: it is plausible that Beilinson would be okay with an element of $H^1(G_K, H^{2i-2}((X - U)_{\eta}, \Q_l(i))$, but I don't care..



%TODO: We surely don't need all of these claims.
%\begin{claim}
%\hfill
%\begin{enumerate}
%\item $H^1(\Spec(\O_K), \Q_l(1)) = 0$,
%\item $H^2(\Spec(\O_K), \Q_l(1)) = 0$,
%\item $H^1(\Spec(K), \Q_l(1)) \isom H^2_{s}(\Spec(\O_K), \Q_l(1))$,
%\item $H^1(\Spec(K), \Q_l(1)) \isom \hat{K} \tensor \Q_l \isom \Q_l$,
%\item $H^2(\Spec(K), \Q_l(1)) = 0$.
%\end{enumerate}
%\end{claim}
%\begin{proof}
%We consider the residue sequence
%\begin{align} H^1(\Spec(O_K), \Q_l(1)) &\to H^1(\Spec(K), \Q_l(1)) \to H^2_s(\Spec(\O_K), \Q_l(1)) \\
%&\to H^2(\Spec(O_K), \Q_l(1)) \to H^2(\Spec(K), \Q_l(1)).
%\end{align}
% We see immediately that 1) + 2) implies 3).
% 4) follows from Kummer theory and that $l \neq p$.
%5) The Brauer group of maximal unramified extension of a local field vanishes.
%2) The final map is an inclusion, by the Auslander-Brumer theorem/a result of Grothendieck (see Jeremy's Brauer notes online).

%1) ?
%Maybe we can eliminate 1, use the injectivity of the cycle class map?

%\end{proof}

\begin{defn}
The local linking number $\pair{a_1}{a_2}_v$ is defined as follows: 

We have $\alpha_1 \cup cl(a_2) \in H^{2n+1}_{|a_2|}(X - |a_1|, \Q_l(n+1))$. The linking number is its image under \[H^{2n+1}_{|a_2|}(X - |a_1|, \Q_l(n+1)) \to H^{2n+1}(X, \Q_l(n+1)) \nmto{Tr} H^1(\Spec(K), \Q_l(1)) \isom \Q_l \cdot \log q_v,\] where the first map is via excision and the last is by the identification above. Note that we normalize by the size $q_v$ of the residue field of the local field we originally cared about.
\end{defn}

For excision in etale cohomology, see \cite{Milne}. 
TODO: Find a good reference for the trace/Poincare duality in absolute etale cohomology. Less canonically, can use trace map plus being the only component of a Leray-Serre spectral sequence...

\begin{rmk}
We could have phrased this via ``mixed extensions'', which would remain in the language of extensions of galois representations as in the previous section, at the price of being confusing. It would involve the Galois structure of $H^{2n}_{|a_2|}((X - |a_1|)_{\eta}, \Q_l(n+1))$ being standardized by the trivializations of \ref{purity}. 
\end{rmk}

%Draw a linking picture - Brian, this is just a mnemonic!

\subsection{Linking = Intersection}
See \cite{NekovarHeegner} (2.16) for more details, especially pertaining to sign conventions.

The following diagram commutes, where the maps $\delta$ come from LES of relative cohomology, the upward maps are restriction, and when coefficients are not written they should be $\Q_{l}(i)$ or $\Q_l(n-1+1)$:

\begin{tikzcd}
H^{2i}_{|a_1|}(X) & \times H^{2n-2i+1}(X-|a_2|) \arrow{r}{\cup}&  H^{2n+1}_{|a_1|}(X-|a_2|) \arrow{r}{Tr}& H^1(\Spec(K), \Q_l(1)) \arrow{dd}{\delta} \\
H^{2i}_{|a_1|}(X_{\O}) \arrow{u}\arrow{d}{\isom}& \times H^{2n-2i+1}(X_{\O}-|a_2|) \arrow{u}\arrow{d}{\delta}\arrow{r}{\cup}& H^{2n+1}_{|a_1|}(X_{\O}-|a_2|) \arrow{u}\arrow{d}{\delta} \\
H^{2i}_{|a_1|}(X_{\O}) & \times H^{2n-2i+2}_{|a_2|}(X_{\O}) \arrow{r}{\cup}& H^{2n+2}_{|X_k|}(X_{\O}) \arrow{r}{Tr}& H^2_{|s|}(\Spec(\O), \Q_l(1))
\end{tikzcd}

Then, if we start with classes $(\alpha, \beta)$ in the middle row, mapping down and to the right recovers the intersection product, and mapping up and to the right recovers the linking number (before it is normalized by $\log q_v$).

Thus we see that the linking number and the intersection number agree when one of our cycles $a$ extends to a cycle $\tilde{a}$ on $X_{\O}$ which is cohomologous to zero, so that its cycle class $cl(\tilde{a}) \in H^{2n-2i+2}_{|\tilde{a}|}(X_{\O})$ is in the image of some $\beta \in H^{2n-2i+1}(X_{\O} - |\tilde{a}|)$.


%\section{Neron-Tate Heights}
%Local Neron functions on abelian varieties exist, for divisors of all degrees against degree zero zero cycles, and agree with Beilinson basically by definition on degree zero stuff.
%These Local Neron functions induce Neron functions on curves (really all varieties), which again basically by definition agree with Beilinson on degree zero stuff, as well as agreeing with what we wrote above.

%This reduces us to showing that the sum of local neron functions agrees with Neron-Tate heights on abelian varieties.

%We check, not in degree zero case, but in ample case. Fixing ample $\L$, it is basically clear that the sum of local neron functions is a ``naive height'', measuring distance from point to divisor at infinity in a projective embedding. Thus it lies in the bounded equivalence class of $h_{\L}$ the Neron-Tate height.










%Poincare bundle is ample when restricted to diagonal via polarization. See Brian's notes on polarizations of abelian varieties. $H \from V \times V \to \R$ Hermitian pairing such that $E = \im(H) \from U \times U \to \Z(1)$. Show how this notion of polarization relates to Poincare bundle... $H_1(A, \Z) \times H_1(A, \Z) \to \Z(1)$, $A \times A \to B\G_m$. The Poincare bundle gives local pairings $A(K_v) \times A(K_v) \to \R$: we can change the structure group from $\C^*$ to $\R$ by taking $\log|\cdot|$ of transition functions, I think Bloch or Mazur-Tate shows how we can then get a pairing, using the compactness of $A(K_v)$.

%\subsection{Neron-Tate Height = Beilinson Height}
%We must show that the Beilinson height is given at almost every place by the ``naive height'' with respect to the projective embedding of a polarization $\L$ on $A$.
%This shows that the sum of the local heights lies in the same bounded-class as the naive local height.

%In other words, we need to see that the local self-intersection $\pair{(P) - (\infty)}{(P) - (\infty)}_v$ measure the distance from $P$ to $\infty$ at almost all $v$.

%We can prove this for Jacobians, using the proof above (good reduction = intersection of point with infinity). With a bit more care we could probably show for all abelian varieties at once... This shows that the sum of our local pairings lives in the bounded equivalence class of the naive pairing from a polarization $\L$ on $A$ (when the local model by completing in projective space is a regular model with good special fiber). 

%$\pair{x}{x} = \pair{x - 0}{t^*_{-x}((\infty)) - (\infty)} = 2\pair{0}{(\infty)} - 2 \pair{x}{\infty}$ if we assume $[-1]^*((\infty)) = (\infty)$ (follows from being symmetric divisor? or only modulo rational equivalence?). Unfortunately, it is not clear that this is well-defined, since we cannot use the "tangent vector trick"... 

%Perhaps it is well-defined at all good places?


%Can we then use that all abelian varieties are quotients of Jacobians?

%Claim: Sum of Local Beilinson Heights is (almost) functorial
%Do I believe this??? Usually only have functoriality for correspondences?

%Claim: Neron-Tate Canonical Height is functorial. 
%For the latter claim, see (B.5.6, \cite{HindrySilverman}).

%$f \from A \surj B$
%(point x of $A$, point y of $\dual B$)
%$= (f_*(x), y) = (x, f^*(y))$
%(cannot push forward divisors along positive-dim quotients or pull-back points)

%(seems like naive height of point could drop under quotients $A \mapsto B$, increase under pullback)


%Alternatively, we can look in Moret-Bailly: Compactifications, hauteur et finitude, in Asterisque 127, which proves, roughly, that $h(x, \L) = deg((x, \L)^* \P)$ where $\P$ is the Poincare bundle over the Neron model of $A \times \dual A$ (have to multiply to get into connected component at bad places).

%\subsection{Nondegeneracy of Neron-Tate Height}
%Do I really need to show this? Is in Mumford, was in Brian's class.

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\end{document}