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\title{The Bloch-Kato Tamagawa Number Conjecture}
\date{}
\author{Jesse Silliman}
\begin{document}
\maketitle
\section{Introduction}
Bloch and Kato originally thought of their conjectures as a version of the Tamagawa number conjecture for algebraic groups, replacing the algebraic group by a pure motive.
\[\text{algebraic groups} <----> \text{abelian varieties} <----> \text{motives} \]
Abelian varieties are the prototypical motive. We recall how the BSD conjecture for abelian varieties equals a Tamagawa number conjecture.
First, we recall the theorem for algebraic groups:
\begin{thm}[?]
For a connected algebraic group $G$ over a number field $K$, we have
\[ \tau(G) = \frac{|Pic^0(G)|}{|\Sh(G)|}. \]
\end{thm}
Here, $\tau(G)$ is roughly the volume of $G(\A_K)/G(K)$ with respect to Haar measures on $G(K_v)$ for all places $v$, where we have to use L-functions to make the product measure converge, and also have to restrict to measuring some ``compact part'' of $G(K_v)$, by taking the kernel of all $|\chi|_v$, $\chi \from G \to \G_m$.
%\begin{eg}
%$\G_m(\Q_p) = \Q_p^* = p^{\Z} \times \Z_p^*$ is not compact. $\G_m(\Z_p) = \Z_p^* = \mu_{p-1} \times \Z_p$ is compact.
%\end{eg}
Now, let's formulate a version of this for abelian varieties. For simplicity, assume that $E$ is an elliptic curve over $\Q$, with $E(\Q)$ finite. There is a Neron model $\mathcal{E}$ for $E$, with Neron form $\omega$. This induces a measure on $E(\Q_p)$: one way to formulate this is that the map $log = \int \omega \from \mathcal{E}(\Z_p) \to Lie(\mathcal{E})_{\Q_p}$ (multiply till you land in ``kernel of reduction'' $\mathcal{E}(p\Z_p)$, then evaluate power series) induces a measure on $\mathcal{E}(\Z_p)$ by declaring that it preserves measure and that $Lie(\mathcal{E})_{\Z_p}$ has volume 1.
A calculation shows that $vol(E(\Q_p)) = \frac{|\tilde{E}^{0}(\F_p)|}{p} \cdot |\Phi_p(\F_p)|$, where $\Phi_p$ is the component group-scheme. Define $c_p = |\Phi_p(\F_p)|$, the Tamagawa factor at p. Also, $vol(E(\R)) = \int_{E(\R)} \omega$ is the real period.
The product $\prod_{p} vol(E(\Q_p))$ does not converge. However, note that $L(E,1) = \prod_{p} det(1-p^{-1}f|((V_lE)^*)^{I_v})^{-1} = \prod_p det(1-f | (V_l(E))^{I_v})^{-1} = \prod_p \frac{p}{|\tilde{E}^0(\F_p)|}$, using the Cartier duality $T_p(E)^*(1) \isom T_p(E)$.
Thus we can define the renormalized adelic volume to be
\[ vol(E(\A)) = L(E,1)^{-1}vol(E(\R))\prod_{p} c_p \]
%TODO: What happens to Cartier duality at bad reduction?!
The Tamagawa number conjecture then becomes
\[ vol\left( \frac{E(\A)}{E(\Q)} \right) = \frac{L(E,1)^{-1}vol(E(\R))\prod_{p} c_p}{|E(\Q)|} \nmequals{?} \frac{|E(\Q)|}{\Sh(E)}. \]
This is evidently equivalent to BSD. More generally, if $E(\Q)$ is not finite, $L(E,1)$ should vanish, and $|E(\Q)|$, $|Pic^0(E)|$ are not finite. However, if we replace $L(E,1)$ by the leading term $L^*(E,1)$ and introduce height pairings to measure, not the covolume of $E(\Q)/tors$ in $E(\A)$, but its ``density'', we again recover BSD.
How do we generalize this to other motives?
\begin{itemize}
\item Global points modulo torsion via K-theory
\item Torsion in global points, $Pic^0$, $\Sh$, via global etale cohomology
\item Local nonarchimedian points via local etale cohomology
\item Local nonarchimedean volumes via Bloch-Kato exponential
\item Local real volumes via period map, real regulators, as in Beilinson conjecture
\item Height pairings to make sense of quotienting adelic points by global points, when not just torsion
\end{itemize}
Once we make precise what all this means, we will have, for a motive $M = h^i(X)(j)$, the exact same conjecture: \[ \frac{vol(M(\A)) Reg(M)}{|M(\Q)_{tors}|} = \frac{L^*(M,0)^{-1}Reg(M)vol(M(\R))\prod_p c_p}{|M(\Q)_{tors}|} \nmequals{?} \frac{|(M^*(1))(\Q)_{tors}|}{|\Sh(M)_{tors}|}. \]
One convenient way to formalize this was found by Fontaine and Perrin-Riou.
First, recall that, in formulating Beilinson's conjecture, we had, for a motive $M$ of weight $w < -1$, an injective real period map
\[ \alpha \from (M_B^{+})_{\R} \to (M_{dR}/F^0M_{dR})_{\R} =: Lie(M)_{\R} \]
and a conjectural isomorphism
\[ H^{i+1}_{M, \Z}(X, \Q(j))_{\R} \isom coker(\alpha) =: H^{i+1}_{D}(X_{\R}, \R(j)). \]
Using these, we obtain, denoting $[\cdot] := det(\cdot)$ for the top exterior power of a vector space, division meaning tensor with dual,
\[ \theta_{\infty} \from \R \isom \left( \frac{[Lie(M)]}{[H^{i+1}_{M,\Z}(X, \Q(j))][M^+_B]} \right) =: \Xi(M) \tensor \R\]
Beilinson's conjecture (not the rank part) is equivalent to the claim that \[ \theta_{\infty}(1/L^*(M)) \in \Xi(M). \]
In other words $L^*(M)$ should measure how far $\theta_{\infty}$ is from respecting the rational structures $\Q \subset \R$, $\Xi(M) \subset \Xi(M) \tensor \R$.
Fontaine and Perrin-Riou generalize this for all $w$, defining $\Q$-vs $H^*_f(M)$, $* = 0,1,2,3$,
and define
\[ \Xi(M) := \frac{[H^{*}_f(M)][Lie(M)]}{[M^+_B]}, \]
as well as an isomorphism
\[ \theta_{\infty} \from \R \to \Xi(M)_{\R}, \]
and still conjecture \[ \theta_{\infty}(1/L^*(M)) \in \Xi(M). \]
We think of this canonical element in $\Xi(M)$ as defining a $\Z$-integral structure. Using etale cohomology, we can define $\Z_p$-integral structures
\[ \theta_p \from \Lambda \into \Xi(M)_{\Q_p}, \]
($\Lambda \isom \Z_p$, but not canonically so)
The Bloch-Kato conjecture is then:
\begin{conj}[\cite{Flach}]The $\Z_p$-integral structures
\[ \Z_p \cdot \theta_{\infty}(1/L^*(M)) \subset \Xi(M)_{\Q_p} \supset \theta_p(\Lambda) \]
agree for all $p$.
\end{conj}
In terms of volumes, this says, roughly, that
\begin{conj}
For all primes $p$,
\[ ord_p\left(\frac{Reg(M)vol(M(\R))}{L^*(M)}\right) = ord_p\left(\frac{|M(\Q)_{tors}||M^*(1)(\Q)_{tors}|}{|\Sh(M)|\prod_{v} c_v}\right)\]
\end{conj}
\begin{rmk}We could make sense of the p-adic valuation of all the invariants on the RHS in terms of p-adic etale cohomology. \end{rmk}
\section{Determinants}
\textbf{Motivation} This is mostly just book-keeping.
Given a (finite dimensional) vector space $V$, define $[V] = \bigwedge^{top} V$. Note that if $V = 0$, then $[V] \isom \Q_p$ canonically.
Given an exact sequence $0 \to A \to B \to C \to 0$, we have $[B] \isom [A][C]$. We will need to keep track of isomorphisms, or else this is useless.
We consider integral structures $T$ on $\Q_p$-vector spaces $V$, by which we mean finitely-generated $\Z_p$ modules $T$ with a canonical isomorphism $T \tensor \Q_p \isom V$. An integral structure $T$ on $V$ determines a $\Z_p$-submodule $[T] \subset [V]$ as follows:
If $T \subset V$ is torsion-free, then $[T] \subset [V]$ is what you expect.
If $V = 0$, $T = 0$, then $[T] \isom \Z_p \subset \Q_p \isom [V]$.
If $V = 0$, $T = \Z/p\Z$, then $[\Z/p\Z] = \frac{1}{p}\Z_p \subset \Q_p \isom [V]$.
i.e. torsion groups have larger volumes than trivial groups.
For a general integral structure $T$, $[T] = [T/tors][T_{tors}] = |T_{tors}|_p [T/tors] \subset [V]$, where $|\cdot|_p$ is the valuation with $|p|_p = \frac{1}{p}$.
Given a a finite complex $C: A_0 \to A_1 \to \ldots \to A_n$ of $\Q_p$-vs, we define
$[C] \isom \frac{[A_0][A_2]\cdots}{[A_1][A_3]\cdots}$.
We can deduce:
$[C] = [H^*(C)]$.
Consider an integral structure $X \subset C$.
%, meaning an integral structure on its terms, we obtain $[X]
The cohomology complex $H^*(X)$ is an integral structure of $H^*(X)$. Thus we obtain $[H^*(X)] \subset [H^*(C)] \isom [C]$.
%This integral structure on $[C]$ can also be described as follows:
Consider $f \in Aut(V)$, such that $f(T) \subset T$ for $T \subset V$ a lattice. Consider the complexes $(T \nmto{f} T) \subset (V \nmto{f} V)$. Now, $[V \to V] = [H^*(V \to V)] = [0 \to 0] \isom \Q_p \supset \Z_p$ has a canonical integral structure. We compare this to the integral structure $[H^*(T \to T)]$:
\[ [H^*(T \to T)] = [coker(f|T)]^{-1} = |det(f)|_p^{-1} \cdot \Z_p \]
Similarly, the complex $(T \nmto{f} f(T)) \subset (V \nmto{f} V)$ has $[H^*(T \to f(T))] \subset [H^*(V \to V)] \isom [0] = \Q_p \subset \Z_p$.
%, but these two integral structures on $H^*(V \to V)$ agree.
\section{Motivic f-cohomology}
\textbf{Motivation} We need rational structures to compare the $p$-adic, $\infty$-adic computations.
For a motive $M = h^i(X,\Q(j))$, with weight $w = i - 2j$, we define
\begin{itemize}
\item $H^0_f(M) = CH^j(X)_{\Q}/hom. equiv.$ if $i = 2j$, $0$ otherwise
\item $H^1_f(M) = \begin{cases} H^{i+1}_{M, \Z}(X, \Q(j)) = Im(K_{2j-i-1}(\mathfrak{X})_{\Q}^{(2j)} \to K_{2j-i-1}(X)_{\Q}), & i \neq 2j - 1, \\ CH^{j}(X)_{hom \sim 0}, & i = 2j-1\end{cases}$,
for $\mathfrak{X}$ a regular proper model of $X$ over $\Z$ (what if this doesn't exist?)
\item $H^2_f(M) = (H^1_f(M^*(1)))^*$
\item $H^3_f(M) = (H^0_f(M^*(1)))^*$
\end{itemize}
Note that $H^0_f = 0$ if $w \neq 0$, $H^3_f = 0$ if $w \neq -2$ (immediately right and left of the point of symmetry $w = -1$).
\begin{conj}[\cite{Flach}]\hfill
\[ord_{s=0}(L(M,s)) = \dim_{\Q} H^1_f(M^*(1)) - \dim H^0_f(M^*(1)) \]
\end{conj}
\begin{rmk}
This is, conjecturally on the isomorphism of the p-adic regulator (see below), the same conjecture as in Tony's talk in terms of Bloch-Kato selmer groups.
\end{rmk}
\begin{rmk} This conjectures possible poles for $w = -2$, possible zeros for $w \geq -1$, and $ord = 0$ for $w < -2$. Note, for example, that $\zeta(r)$ relates to the motives $\Q(r)$ of weight $-2r$.
\end{rmk}
%Somewhat surprisingly, the BK conjecture will not use the integral structure from K-theory.
%The Bloch-Kato conjecture can be stated only with rational K-theory, since torsion should, conjecturally, be visible in global etale cohomology.
%$H^0_f, H^1_f, H^2_f, H^3_f$, using the ad-hoc definitions that only work for proper, smooth varieties $X$, or perhaps Chow motives since we can have $\Q$-coefficients.
%\begin{rmk}
%The conjecture should apply to mixed motives, once we know the correct definitions of these groups?
%\end{rmk}
We use these groups to define the fundamental $\Q$-line
\[ \Xi(M) = \frac{[H^*_f(M)][Lie(M)_{\R{}}]}{[M^+_B]}.\]
\begin{rmk}
The definition of $H^2_f$ is convenient, but it is bad: $Ext^2_f(\Spec(\Q)_{mot}, M) = 0$ according to Beilinson's conjectures (Scholl says this). Further, these groups definitely do not have the correct torsion even if you decide not to $\tensor \Q$: Should have class groups for number fields.
%Perhaps the true $H^2_f$ should be enhanced via Arakelov theory, and the failure to do that, or even be able to, for coefficient reasons, manifests itself in the height pairing/Deligne cohomology stuff.
%Perhaps all coefficients in life should allow a formal $log(q)$ parameter.
\end{rmk}
\section{Real Volumes}
\textbf{Motivation:} Incorporate Beilinson's conjecture, including height pairings.
We have a real period map
\[ \alpha \from (M_B^{+})_{\R} \to (M_{dR}/F^0M_{dR})_{\R} = (Lie(M))_{\R}. \]
A motive is called ``critical'' when $\alpha$ is an isomorphism. For example, motives of weight -1, such as $H^1(E, \Z(1)) = H_1(E,\Z)$, are always critical.
In this case, we obtain an isomorphism
\[ \R \nmisom{[\alpha]} [Lie(M)_{\R}]/[(M_B^+)_{\R}]. \]
Also, when the weight is $-1$, we have the possibility of height pairings:
\begin{conj}When $w = -1$, the height pairing \[ h \from H^1_f(M)_{\R} \times H^1_f(M^*(1))_{\R} \to \R \]
is nondegenerate.
\end{conj}
Assuming the conjecture, we obtain
\[ \R \nmisom{[h]} [H^1_f(M^*(1))^*]/[H^1_f(M)]. \]
In combination, these give an isomorphism
\[ \theta_{\infty} \from \R \isom \Xi(M)_{\R} \]
Now we deal with the noncritical case, and assume the weight is $< -1$. Here $H^2_f(M) = H^3_f(M) = 0$.
%In this case, $ker(\alpha) = 0$, and $coker(\alpha) = Ext^1_{F_{\infty}-MHS}(\R, (M_{dR})_{\R})$.
\begin{conj}
When $w < -1$, the real regulator
\[ H^1_f(M)_{\R} \to \coker(\alpha) \]
is an isomorphism.
\end{conj}
Since $[\coker(\alpha)] = \frac{[Lie(M)_{\R}]}{[(M_B^+)_{\R}]}$, we again obtain
\[ \theta_{\infty} \from \R \isom \Xi(M)_{\R}. \]
\begin{rmk}When the weight is $ > -1$, we need to use factors from the functional equation to define the map $\theta_{\infty}$ in terms of that for its dual motive $M^*(1)$. See Fontaine and Perrin-Riou.\end{rmk}
\begin{rmk}All cases can be combined into the conjectural exactness of the sequence
\[ 0 \to H^0_f(M) \to ker(\alpha) \to H^1_f(M^*(1))^* \nmto{h} H^1_f(M) \to \coker(\alpha) \to H^0_f(M^*(1))^* \to 0, \]
which perhaps suggests that $H^*_f$, $* = 0, 1$, is dual to a cohomology theory which is ``compactly supported at infinity". See Deninger-Nart.\end{rmk}
The map $\theta_{\infty}$ can also be defined for $w > -1$. For all weights $w$ we have the following conjecture:
\begin{conj}[Beilinson]
\[ \theta_{\infty}(1/L^*(M)) \in \Q. \]
\end{conj}
\section{Local f-cohomology and the Bloch-Kato Exponential}
\textbf{Motivation} Local conditions, being unramified, analogous to $H^*_{M,\Z}$.
Fix a prime $p$.
We define complexes
\[
R\Gamma_f(\Q_v, M_p) = \begin{cases}
v = \infty: & R\Gamma(\R, M_p) \\
v \neq p: & M_p^{I_v} \nmto{1-f} M_p^{I_v} \\
v = p: & D_{cris}(M_p) \nmto{(1-f, \pi)} D_{cris}(M_p) \oplus D_{dR}(M_p)/F^0 D_{dR}(M_p)
\end{cases},\]
with $f$ the geometric Frobenius.
Their cohomology groups $H^i_f$ are the same as those in Tony's talk, as we will shortly see.
%We also have integral structures on these (BE CAREFUL at $v = p$) induced from local etale cohomology
%$H^1(\Q_v, T_p)$.
\textbf{Local L-complexes} (This is just notation for later.)
We have the complexes for $v \neq \infty$:
\[ L^v(T_p) = \begin{cases}
v \neq p: & T_p^{I_v} \nmto{1-f} T_p^{I_v} \\
v = p: & D_{cris}(T_p) \nmto{1-f} D_{cris}(T_p)
\end{cases}, \]
We define $L^v(M_p) = L^v(T_p) \tensor \Q_p$.
We also define $[L^S(M_p)] = \tensor_{v \in S - \{\infty \}} [L^v(M_p)]$, with integral structure $[L^S(T_p)] = \tensor_{v \in S - \{\infty \}} [L^v(T_p)]$.
Note that if $L^v(M_p)$ is acyclic, then $[L^v(T_p)] = [det(1-f|M_p^{I_v})]^{-1}$, like a local $L$-factor. This explains the notation.
%THIS IS AN ABUSE OF NOTATION.
%. These complexes are acyclic (why do't I need some weight assumption to say this???), and the cohomology groups of $L_v(T_p)$ are torsion, with $[L_v(T_p)] = [det(1-f)]^{-1}$ (????). Notation designed to suggest Local L-factors (TODO).
%We define
%\[ H^1_f(\Q_v, T_p) := i^{-1}(H^1_f(\Q_v, V_p)) \]
%where $i$ is the map $i \from H^1(\Q_v, T_p) \to H^1(\Q_v, V_p)$. Note that $H^1(\Q_v,T_p)_{tors} \subset H^1_f(\Q_v, T_p)$.
Recall that $f$-cohomology is a ``self-dual Selmer condition'':
\begin{prop}
$H^1_f(\Q_p, M_p)$ is the exact annihilator of $H^1_f(\Q_p, M_p^*(1))$ under the Tate local duality pairing.
\end{prop}
%WE DON'T DEFINE INTEGRAL COMPLEXES SINCE WE CAN'T AT P
%We really ought to define integral structures, but are confused about $v = p$.
%$[H^*_f(\Q_p, V_p)] = [Lie(V_p)]^{-1}[L_p(V_p)]$
%When not Crystalline, the "natural" integral structure on the above complex is certainly wrong: it cannot distinguish between elliptic curve and $\G_m$. (this required a computation with monodromy operators)
%When Crystalline, we can use complex
%\[ F^0D(T_p) \nmto{1-f} D(T_p) \]
%This has determinant $[Lie(T_p)]^{-1}[L_p(T_p)]$ always.
%$[H^*_f(\Q_p, T_p)] = [Lie(T_p)]^{-1}[L_p(T_p)]$ when for ``Fontaine-Lafaille''.
%We are told by BK that the LHS is correct, not RHS, when not Fontaine-Lafaille.
%At least, it is the one that gives self-dual Selmer structure!
%It follows that WE DON'T KNOW HOW TO COMPUTE $H^1_f(\Q_p, T_p)$ VIA COMPLEXES, SO CAN'T USE IT TO DEFINE INTEGRAL SELMER STRUCTURE?!
%Also! This integral structure is not shown to be precisely the image of Kummer in BK.
We want to define the Bloch-Kato exponential
\[ exp_{BK} \from D_{dR}(M_p)/F^0D_{dR}(M_p) \to H^1_f(\Q_p, M_p) \]
%The Bloch-Kato exponential will induce an isomorphism on $D(V_p)/F^0D(V_p)$.
It arises from the ``fundamental exact sequence of p-adic Hodge theory'': \[ 0 \to \Q_p \to B_{cris} \nmto{(1-f, \pi)} B_{cris} \oplus B_{dR}/B_{dR}^+ \to 0. \]
A sequence similar to this was in Tony's talk.
Tensoring this with our representation $M_p$ (which is assumed to be de Rham), and taking the LES of Galois cohomology
\[ 0 \to H^0(M_p) \to D_{cris}(M_p) \to D_{cris}(M_p) \oplus D_{dR}(M_p)/F^0D(M_p) \to ker(H^1(M_p) \to H^1(M_p \tensor B_{cris})) \to 0, \]
Note that this verifies that the definition of $H^1_f$ in Tony's talk agrees with the 1st cohomology of the above complex.
We can also express the BK exponential in terms the $Ext^1$-consequence of the crystalline comparison theorem.
\begin{prop}[\cite{BlochKato}]
For $M_p$ crystalline, we have the following isomorphism:
\[ D(M_p)/(1-f)F^0D(M_p) \isom Ext^1_{f,Fil}(\Q_p, D(M_p)) \isom Ext^1_{K_p}(\Q_p, M_p)_f. \]
In other words, crystalline extensions of galois representations are identified with extensions of (f,Fil)-modules.
\end{prop}
\textbf{An aside on Fontaine-Lafaille Theory}(\cite{BlochKato})
If the lattice $D(T_p) \subset D(M_p)$ is ``Fontaine-Lafaille'' (strongly divisible and with weights in $[0,p-1]$), we have an integral comparison theorem
\[ D(T_p)/(1-f)F^0D(T_p) \isom Ext^1_{f,Fil}(\Z_p, D(T_p)) \isom Ext^1_{K_v}(\Z_p, T_p)_f. \]
In this case, we have the following:
$\begin{tikzcd}
D(T_p)/F^0D(T_p) \arrow{rd}{1-f} \arrow{r}{exp_{BK}} & H^1(K_v, T_p) \arrow{d}{\isom} \\
& D(T_p)/(1-f)F^0D(T_p)
\end{tikzcd}$
This implies that when a lattice is Fontaine-Lafaille, that the local volume agrees with the local L-factor. Morally, this means that we have good reduction, in some strange new sense, since the Tamagawa factor at $p$ is then 1.
For example, Bloch-Kato shows that the lattice $D(\Z_p(r))$ is not Fontaine-Lafaille for $p < r$, contributing an extra factor of $1/(r-1)!$ to the adelic volume as we vary over all such primes.
\textbf{Bloch-Kato Exponential and Kummer Theory}(\cite{BlochKato})
For abelian varieties and tori, the Bloch-Kato exponential agrees with the Kummer map. We first show it for $\G_m$, using the following diagram:
$\begin{tikzcd}
0 \arrow{r}& \Z_p(1) \arrow{r}\arrow{d}{=}& \invlim{p} \O_{\C_p}^* \arrow{r}\arrow{d}{log[\cdot]} & \O_{\C_p}^* \arrow{r}\arrow{d}{log}& 0 \\
0 \arrow{r}& \Q_p(1) \arrow{r}\arrow{d} & B_{cris}^{f=p} \cap B_{dR}^+ \arrow{r}{\theta}\arrow{d} & \C_p \arrow{r}\arrow{d} & 0 \\
0 \arrow{r}& \Q_p(1) \arrow{r}& (B_{cris}^{f=1})(1) \arrow{r}& (B_{dR}/B_{dR}^+)(1) \arrow{r} & 0
\end{tikzcd}$
To get the result for abelian varieties, use that
$Hom_{FormalGroup}(\widehat{A}, \widehat{\G_m})(\O_{\C_p}) \isom T_p(A)^*(1)$ by Cartier duality. For any choice of $\chi \in T_p(A)^*(1)$, we get a map (not galois equivariant) from the sequence \[ 0 \to T_p(A) \to \invlim{p} A(\O_{\C_p}) \to A(\O_{\C_p}) \to 0 \] to the last row of the above diagram, i.e. we get a (galois equivariant) map from this sequence to the last row tensor $V_p(A)(-1)$.
Bloch-Kato claim this proof works, in some sense, for abelian varieties with bad reduction.
%Mention that we need ``Local Volume = inverse of Local L-factor'' at almost all primes. For abelian varieties, we can check this by hand. For other galois representations, need to use Fontaine-Lafaille theory.
%Failure to be Fontaine-Lafaille is a new form of Tamagawa number.
\section{Global f-cohomology}
There is a homological algebra construction, which, given a map of complexes, formally create a complex fitting into a long-exact sequence:
\[ \ldots \to H^i(A) \to H^i(B) \to H^i(Cone(A \to B)) \to \ldots, \]
%in fact a ``triangle''
%\[ \ldots \to A \to B \to Cone(A \to B) \to A[-1] \to \ldots \]
Note that this implies the determinant formula \[ [Cone(A \to B)] = \frac{[B]}{[A]}. \]
Let $S = \{\infty, p, v \text{ s.t. } V^{I_v} \neq V \}$. Let $R\Gamma(\Z[1/S], N)$ be the complex computing global galois cohomology, for $N$ any reasonable Galois module.
Similarly we use $R\Gamma(\Q_v, V_p)$ for local galois cohomology.
We first define the ``quotient'' of local cohomology by local f-cohomology, $R\Gamma_{/f}(\Q_v, M_p)$, as
\[ R\Gamma_{/f}(\Q_v, M_p) = Cone(R\Gamma_f(\Q_v, M_p) \to R\Gamma(\Q_v, M_p)) \]
We define compactly supported cohomology, global f-cohomology, as
\[ R\Gamma_c(\Z[1/S], N) = Cone(R\Gamma(\Z[1/S], N) \to \oplus_{v \in S} R\Gamma(\Q_v, N))[-1] \]
\[ R\Gamma_f(\Z[1/S], M_p) = Cone(R\Gamma(\Z[1/S], M_p) \to \oplus_{v \in S} R\Gamma_{/f}(\Q_v, M_p))[-1] \]
Note that we defined compactly-supported cohomology for any reasonable coefficients but f-cohomology only for the galois representation $V_p$ associated to our motive.
%Should be imagined to be: cohomology with compact support ``in the interial direction'' at each boundary torus except $\infty$.
%Compactly supported cohomology as cone
%Global f-cohomology (i = 0,...,3) as cone
%Where did we get actual maps of complexes: S-global maps to local at S
%Some triangles:
We obtain, beyond the defining triangles, a triangle relating $H^*_f$ and $H^*_c$ (Flach)
\begin{align*}
%H^*_c(\Z[1/S]) \to H^*(\Z[1/S]) \to \oplus_{v \in S} H^*(\Q_v) \\
R\Gamma_c(\Z[1/S]) \to R\Gamma_f(\Q) \to \oplus_{v \in S} R\Gamma_f(\Q_v)
\end{align*}
%H^*_f(\Q) \to H^*(\Z[1/S]) \to \oplus_{v \in S} H^*_{/f}(\Q_v)
%where $H^*_{/f}$ is also defined by a cone-type construction, fitting into \[H_f(\Q_v, V_p) \to H^*(\Q_v, V_p) \to H^*_{/f}(\Q_v, V_p). \]
%(DO WE NEED THE $/f$ SEQUENCE?)
We also have compactly supported cohomology with integral coefficients $R\Gamma_c(\Z[1/S], T_p)$, using that on local etale cohomology $R\Gamma(\Q_v, T_p)$.
%(TODO: Why are any Galois cohomologies defined via a reasonable complex? continuous etale cohomology???)
\begin{prop}\hfill
\begin{enumerate}
\item For $N$ finite, the Euler characteristic of $H^*_c(\Z[1/S], N)$ is $1$.
\item The integral structure \[
[H^*_c(\Z[1/S], T_p)] \subset [H^*_c(\Z[1/S], M_p)]\]
is independent of choice of lattice $T_p \subset V_p$.
\item The integral structure
\[ [L^S(T_p)] \subset [L^S(M_p)] \]
is independent of choice of lattice $T_p \subset M_p$.
\end{enumerate}
\end{prop}
\begin{proof}
i) We use Tate's Euler Characteristic formula.
$\chi(N) = \frac{|H^0(\R, N)|}{|N|}$, for $\chi$ the Euler characteristic $\frac{H^0(\Z[1/S], N) H^0(\Z[1/S], N)}{H^1(\Z[1/S], N)}$
The local Euler characteristic formula, for $v \neq \infty$, says $\chi_v(N) = \frac{|H^0(\Q_v, N)|}{|H^1(\Q_v, N)|} = |N|_v = 1/|N[v^{\infty}]|$.
For $v = \infty$, $\chi_{\infty}(N) = |H^0(\R, N)| \cdot \frac{|H^2(\R, N)|}{|H^1(\R, N)|} = |H^0(\R,N)|$, where the last equality is because the Herbrand quotient is 1 for finite modules.
%. (really easy to check directly here: Herbrand quotient of $T_p$ only depends on eigenvalues of the action)
%This is true precisely if the number of $+1$ eigenvalues equals the number of $-1$ eigenvalues. This is true precisely when $M_B^+$ is half of $M_B$. This is true for $E$ elliptic curve but not for $\G_m$. What gives?
ii) We can assume that $T_p \subset T_p'$. Then
\[ \frac{[H^*_c(T_p')]}{[H^*_c(T_p)]} = [H^*_c(T_p'/T_p)] \isom \Z_p, \]
where the final isomorphism is not becaue $H^*_c(T_p'/T_p)$ is torsion, but because its Euler characteristic is 1. A little thought shows that this means the integral structures agree, not up to finite difference, but exactly, with changes in an individual $H^1_c(T_p)$, say, being cancelled by changes in $H^0_c(T_p)$, $H^2_c, H^3_c$ as well.
iii) When the $L$-complex $L^v(M_p)$ is acyclic, note that $[L^v(T_p)] = [det(1-f| M_p^{I_v})]^{-1}$ does not depend on the lattice at all.
More generally, we can use the exact sequence \[ 0 \to T_p^{f=1} \to T_p \to T_p/T_p^{f=1} \to 0 \] to obtain $[L^v(T_p)] = [L^v(T_p/T_p^{f=1})] \cdot [L^v(T_p^{f=1})]$. By the acyclic case, we have that $[L^v(T_p/T_p^{f=1})] \subset [L^v(M_p/M_p^{f=1})]$ is independent of choice of $T_p$.
Further, the determinants $[L^v(T_p^{1-f})]$ and $[L^v(M_p^{1-f})]$ have canonical elements due to the morphism $1-f$ in the complexes being zero. These canonical elements are the same, hence the integral structure $[L^v(T_p^{1-f})] \subset [L^v(M_p^{1-f})]$ is independent of $T_p$.
\end{proof}
\begin{conj}
The map
\[ H^1_f(M) \to H^1(M_p) \]
lands in the subspace $H^1_f(M_p)$.
\end{conj}
A preprint by Nekovar (\cite{NekovarSyntomic}) claims to prove the above conjecture for $p$ a prime of potentially good reduction.
\begin{conj}
The p-adic regulators
\[ H^i_f(M)_{\Q_p} \to H^i_f(M_p), i = 0,1,2,3 \]
are isomorphisms.
\end{conj}
%(remark on the definition for $i = 2$ as using the duality of f-cohomology and the definition of $H^2_f(M)$)
Recall that \[ \Xi(M) = \frac{[H^*_f(M)][Lie(M)]}{[M_B^+]} \]
Assuming these conjectures, we have the following isomorphism
\[ \theta_{p} \from [H^*_c(\Z[1/S], M_p)][L^S(M_p)] \isom \Xi(M)_{\Q_p}. \]
This uses the isomorphism
\[ [H^*_c(\Z[1/S], M_p)] = \frac{[H^*_f(\Q, M_p)] [L^S(M_p)]^{-1}[H^1_f(\Q_p, M_p)]} {[(M_B^+)_{\Q_p}]} \]
followed by the Bloch-Kato exponential
\[ exp_{BK} \from D(M_p)/F^0D(M_p) \isom H^1_f(\Q_p, M_p) \]
and the de Rham comparison theorem
\[ D(M_p) \isom (M_{dR})_{\Q_p}, F^0D(M_p) \isom (F^0M_{dR})_{\Q_p}. \]
%Integral coefficients - to cone or not to cone?
%This might not actually depend on S, even if we cone? Fantastic!
%Why are we not *allowed* to have smaller S? Complex might not have determinant = $\Q_p$.
%Need to check: Integral structures on $H^1_f, H^2_f$ independent of $S$.
%"Regulator = map from motivic cohomology to absolute cohomology w/ various coefficients"
%Why have we not used absolute etale cohomology here?
%We already pushed our motive down to $\Spec(\Q)$... can this be avoided/explained?
%\section{Conjectures via Volumes}
%How do we actually do concretely this past rank 1? It is sortof hard: need to use compactly supported cohomology $H^2_c$ + image of global lattice inside of $H^1_f(\Q_p, T_p)$, take the image of that... Why have I not yet noticed this in the BSD setting? Does this solve the final convergence factor issue?
%Choices to make:
%Choice of $\omega$ in Lie(T), for volumes: should disappear by product formula
%Height requires basis of global points/torsion for motive/dual motive
%This choice cancels with choice involved in p-volumes?
\section{Statement of Conjecture}
%Why is it well-defined? Justify varying $S$, varying $T_p$.
%Why is this definition of S-compact independent of S? The cohomologies differ by the complex $[T_p --(1-f)--> T_p]$
Recall that Beilinson's conjecture predicts that
$\theta_{\infty} : \R \to \Xi(M)_{\R}$
has $\theta_{\infty}(L(M)^{-1}) \in \Xi(M)_{\Q}$.
\begin{conj}[Bloch-Kato]
For all $p$, the following holds:
Let $S = \{p, \text{ primes of bad reduction } \}$. Then the following $\Z_p$-integral structures agree:
\[ \theta_{\infty}([H^*_c(\Z[1/S],T_p)][L^S(T_p)]) \subset \Xi(M)_{\Q_p} \supset \theta_{\infty}(L(M)^{-1})\cdot\Z_p\]
\end{conj}
Note that both integral structures are isogeny-invariant: the LHS by Euler characteristic and the RHS by definition.
\section{Comparison with BSD}
Let $E/\Q$ be an elliptic curve.
\begin{ass}
$\Sh(E)$ is finite.
\end{ass}
\begin{rmk}
There is no reason to restrict to $E$ an elliptic curve, except to avoiding discussing Neron models. (This is silly, and we should change it, especially since we use Neron forms below)
\end{rmk}
We consider the motive $T = H^1(E, \Z(1)) = H_1(E,\Z)$. We will show that the Bloch-Kato conjectures for the motive $M = T \tensor \Q$ is equivalent to BSD.
The associated L-function is $L(E,s)$ at the point $s = 1$. The $l$-adic representation is the Tate module $T_p = T_p(E)$, and the Hodge realization is the first homology $H_1(E,\Z)$, which has type $(-1,0)$+$(0,-1)$. This implies that $M_{dR}/F^0 = Lie(E) = (H^0(E,\Omega^1))^*$.
Note that $H^1_f(M) = E(\Q)\tensor \Q$. This shows that
\[ \Xi(M) = \frac{[(E(\Q)/tors)^{*}_{\Q}]}{[(E(\Q)/tors)_{\Q}]}\frac{[Lie(E)]}{[(H_1(E(\C),\Z))^{+})]}. \]
Note that $\Xi(M)$ actually has a $\Z$-integral structure we do not have for the general motive, by using a canonical integral structure on de Rham cohomology. It is generated by $\beta = (\wedge v_i^*) \tensor (\wedge v_i)^{-1} \tensor \omega^* \tensor \gamma^{-1} \in \Xi(M)$, where $\{v_i\}$ is a basis for $E(\Q)/tors$, $\{v_i^*\}$ the dual basis, $\omega^*$ is dual to a Neron form, and $\Z\cdot\gamma = H_1(E(\C),\Z)^{+}$.
With respect to this integral structure, we will (roughly) measure both the real volumes and $v$-adic volumes, and, assuming the BK conjecture, show that their product is $\pm 1$, by comparing $p$-adic valuations.
\subsection{Real Stuff}
We have two maps:
\[\alpha \from H_1(E,\Z)^{+} \to Lie(E) \]
with $\alpha(\gamma) = (\int_{\gamma} \omega) \omega^*$,
and
\[ h: E(\Q)/tors \times E(\Q)/tors \to \R \]
the canonical height pairing.
Together, these give a canonical element $Reg(E)\Omega_{\R} \cdot \beta \in \Xi(M)_{\R}$.
\subsection{Integral Structures}
For the sake of computation, we must find some ad-hoc integral structures on the $H^*_f(V_p)$ groups. Abusing notation, we will denote them as $H^*_f(T_p)$.
We define, for all places $v$, all primes $p$,
\[ H^1_f(\Q_v, T_p) = E(\Q_v)_{\Z_p}, \]
and \[H^1_f(\Q, T_p) = E(\Q)_{\Z_p}. \]
We can also define a ``co-integral structure'' $H^{1}_f(\Q, V_p/T_p)$ to be the direct limit of the Selmer groups \[ Sel_{p^n}(E) = \{ x \in H^1(\Q, E[p^n]) \mid x \in Im(E/p^nE(\Q_v) \to H^1(\Q_v, E[p^n])) \text{ for all places } v\}.\]
Then, using the global duality $H^2_f(\Q,V_p) \times H^1_f(\Q, V_p) \to \Q_p$, we verify that $H^2_f(\Q, T_p) := (H^{1}_f(\Q, V_p/T_p))^{\wedge}$ is an integral structure on $H^2_f(\Q, V_p)$.
We similarly define $H^3_f(\Q, T_p) := H^0(\Q, V_p/T_p)^{\wedge}$.
\begin{rmk}It would have been preferable to have define these integral structures at the level of complexes, but there are issues with doing this when $p$ is a prime of bad reduction.\end{rmk}
%This agrees with natural definitions of integral structures at the level of complexes, unless $v = p$ and there is bad reduction, in which case I am not sure.
%\begin{claim}
%$H^1_f(\Q_v, T_p) = E(\Q_v)\tensor \Z_p$
%\end{claim}
%WHAT DO WE MEAN BY $H^2_f$ if we don't use cones??? THIS IS NOT CLEAR.
%Show that we can define an integral structure on $H^*_f$ by specifying dual Selmer conditions, not at the level of complexes, as long as we are only adding torsion, not rank...
\begin{thm}[\cite{Kings}]
$[H^*_c(\Z_S, T_p)] = [H^*_f(\Q, T_p)][\oplus_{v \in S} H^*_f(\Q_v, T_p)]^{-1}$.
\end{thm}
\begin{proof}
The point is to use local Tate duality for abelian varieties to show that the ad-hoc Selmer conditions above are ``integrally self-dual''. As we have stated it, we are also using the compatibility of Cartier duality with local Tate duality (\cite{BCnrd}), but that is just for convenience.
\end{proof}
%In fact, we also have the following perfect pairing:
%\[ H^{i}_c(T_p) \times H^{3-i}(M_p/T_p(1)) \to \Q_p/\Z_p \text{ (and vice-versa) }\]
%\[ H^{i}_f(\Q, T_p) \times H^{3-i}_f(\Q, V_p/T_p) \to \Q_p/\Z_p. \]
We also need the exact sequence
\[ 0 \to E(\Q)/tors \to H^{1}_f(\Q, V_p/T_p) \to \Sh[p^{\infty}] \to 0,\] noting that the direct limit along $E/p^nE(F) \nmto{[p]} E/p^nE(F)$ is $E(F)/tors \tensor \Q_p/\Z_p$ for any field $F$.
\subsection{Computation}
\begin{align*}
[H^*_c(\Z_S, T_p)] &= \frac{[H^*_f(\Q, T_p)]}{[\oplus_{v \in S} H^*_f(\Q_v, T_p)]} \\
&= \frac{[H^2_f(\Q, T_p)][H^1_f(\Q, T_p)]^{-1}[H^3_f(\Q,T_p)]^{-1}}{[T_p^{+}][\oplus_{v \in \S}H^1_f(\Q_v, T_p)]^{-1}} \\
\end{align*}
Global f-cohomology:
\begin{itemize}
\item $H^0_f(\Q, T_p) = 0$
\item $[H^1_f(\Q, T_p)] = [E(\Q)_{\Z_p}] = [E(\Q)_{tors}][(E(\Q)/tors)_{\Z_p}]$
\item $[H^2_f(\Q, T_p)] = [\Sh(E)][((E(\Q)/tors)_{\Z_p})^*]$
\item $[H^3_f(\Q, T_p)] = [E(\Q)_{tors}]$
\end{itemize}
Local f-cohomology:
\begin{itemize}
\item $v = \infty$: $[H^1_f(\R,T_p)] = [\Phi_{\infty}]$
\item $v \neq p$: $[H^1_f(\Q_v, T_p)] = [E(\Q_v)_{\Z_p}] = [\Phi_v][E^0(\F_v)]$
\item $v = p$: $[H^1_f(\Q_v, T_p)] = [E(\Q_p)_{\Z_p}] = [\Phi_p][E^0(\F_p)][\widehat{E}(p\Z_p)] = \frac{[\phi_p][E^0(\F_p)]}{[p]}[D(T_p)/F^0D(T_p)]$
\end{itemize}
\begin{align*}
\frac{[H^*_c(\Z_S, T_p)]}{\left( \frac{[\Sh(E)][(E(\Q)_{\Z_p}/tors)^*]}{[E(\Q)_{tors})]^2[E(\Q)_{\Z_p}/tors]} \right)} &= [H^1_f(\Q_p,T_p)] \frac{[H^1_f(\R, T_p)] \prod_{v \in S, v \neq p, \infty}[\Phi_v] \cdot [E^{0}(\F_v)]}{[H^0_f(\R, T_p)]} \\
&= \frac{[D(T_p)/F^0D(T_p)]}{[T_p^{+}]} \frac{[\Phi_{\infty}]\prod_{v \in S, v \neq \infty}[\Phi_v] \cdot [E^{0}(\F_v)]}{[\Z/p\Z]} \\
\end{align*}
Note $[L^S(E,1)] = \left(\prod_{v \in S, v \neq \infty} \frac{[E^{0}(\F_v)]}{[\Z/p\Z]}\right)^{-1}$.
\begin{align*}
\frac{[H^*_c(\Z_S, T_p)][L^S(E,1)]}{\left( \frac{[\Sh(E)]\prod_{v \in S}[\Phi_v]}{[E(\Q)_{tors})]^2} \right)} &= \frac{[(E(\Q)/tors)^{*}_{\Z_p}]}{[(E(\Q)/tors)_{\Z_p}]} \frac{[D(T_p)/F^0D(T_p)]}{[T_p^{+}]} \\
&\subset \frac{[(E(\Q)/tors)^{*}_{\Q_p}]}{[(E(\Q)/tors)_{\Q_p}]}\frac{[Lie(E)_{\Q_p}]}{[(H_1(E(\C),\Z))^{+})_{\Q_p}]} \\
&= \Xi(M)_{\Q_p}
\end{align*}
Thus Bloch-Kato reduces to the claim that, for each $p$, the integral structure given by $[H^*_c(\Z_S, T_p)][L^S(E,1)]$ agrees with the integral structure given by $\frac{Reg(E)\Omega_{\R}}{L(E,1)} \cdot \alpha$. This is equivalent to
\[ ord_p\left(\frac{|\Sh(E)|\prod_{v \in S}|\Phi_v|}{|E(\Q)_{tors}|^2}\right) = ord_p\left(\frac{L(E,1)}{Reg(E)\Omega_{\R}}\right) \forall p, \]
which implies the BSD conjecture.
\begin{rmk}
Some formulations of BSD do not use the component group at infinity $\Phi_{\infty}$, combining it into the period integral:
\[ \Omega_{\R}\cdot|\Phi_{\infty}| = \int_{E(\R)} \omega \]\end{rmk}
%Note that changing $\gamma$ or $\omega$ does not change this element, but for $\omega$ this is only true because it may only change by $-1$.
%\section{Comparison with Lichtenbaum Conjecture}
\newpage
\bibliographystyle{plain}
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\newblock With an appendix by C. Greither.
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\newblock The equivariant {T}amagawa number conjecture and the
{B}irch-{S}winnerton-{D}yer conjecture.
\newblock In {\em Arithmetic of {$L$}-functions}, volume~18 of {\em IAS/Park
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\bibitem{NekovarSyntomic}
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\newblock Syntomic cohomology and p-adic regulators.
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\newblock Why is the tate local duality pairing compatible with the cartier
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\newblock MathOverflow.
\newblock URL:http://mathoverflow.net/q/177384 (version: 2014-07-29).
\end{thebibliography}
\end{document}