=

$, 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. \newpage \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{Beilinson} A.~A. 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