text:
\begin{document}
\begin{abstract}
In this paper, we give a recipe for producing infinitely many non-divisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of ``field of definition" for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension $2$ cycles we call ``primitive cycles," the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori's isogeny method \cite{N} (modified by Totaro \cite{T}) to produce infinitely many non-divisible cycles. As an application, we prove the Chow group modulo a prime of a product of $3$ or more very general elliptic curves is infinite, generalizing work of Schoen.
\end{abstract}
\maketitle

\section{Introduction}
\noindent A recent result of Totaro \cite{T} shows that there exist smooth complex projective varieties whose Chow group modulo $\ell$ is infinite for all primes $\ell$, generalizing earlier work of Rosenschon and Srinivas in \cite{RS}. Equivalently, there are examples of such varieties whose Chow group is not an $\ell$-divisible group for any prime $\ell$. The methods employed in \cite{RS} and \cite{T} show the non-divisibility of the Ceresa cycle on the very general Abelian threefold. Then, an adaptation of Nori's isogeny technique is used to produce the required infinitely many cycles. (Note that in the case of \cite{RS}, the authors use Atkin-Lehner correspondences as in \cite{S1}).\\
\indent Our goal here will be to extend the existing techniques in the literature for producing infinitely many cycles modulo $\ell$ to prove similar results for products of two or more very general Abelian varieties. In particular, we are able to use the method developed here to prove the following result, which extends work of Schoen in \cite{S1}:
\begin{Thm}\label{main} Let $E_{1}$, $E_{2}, \ldots E_{n}$ be very general complex elliptic curves. Then,
$$CH^{d} (E_{1}\times E_{2} \times \dots \times E_{n}) \otimes \zit/\ell$$
is infinite for all primes $\ell$, $2 \leq d \leq n-1$ and $n\geq 3$.
\end{Thm}
\noindent As noted in the sequel, it will be sufficient to show this for $d=2$ and $n=3$. We briefly describe the method involved, which works best in codimension $2$ (mostly because of the Merkurjev-Suslin theorem). In general, we consider a family of smooth complex projective varieties $f: \X \to S$ whose fibers are products of two or more very general Abelian varieties. We introduce the notion of a {\em primitive correspondence} $P$, which is a correspondence on $\X$ which kills the first level of the coniveau filtration on $H^{3}$ of the very general fiber of $f$. A codimension $2$ cycle on $\X$ is said to be {\em primitive with respect to $P$} if the normal function of $P_{*}(\gamma)$ is non-torsion. Our first order of business will be to use the techniques of \cite{RS} and \cite{T} to show that any primitive cycle $\gamma$ is non-divisible in the Chow group of the very general fiber of $f$. In the case of Theorem \ref{main}, $\gamma$ is an ersatz Ceresa cycle which was considered in \cite{BST} and \cite{S1}. We spread out this cycle to a larger family $f: \X \to S$ and using some rather elementary geometric observations about the total space $\X$, we are able to deduce the required statement about the normal function of $\gamma$. (We observe that in the case of \cite{RS} and \cite{T}, the total space of the universal Abelian threefold with level structure is much more complicated and some deep results of Hain \cite{H} must be invoked.)\\
\indent In order to produce infinitely many cycles, we use Nori's isogeny method. To implement his method, we introduce the notion of a ``field of definition" for a cycle in the Chow group modulo a power of $\ell$. This is defined as the minimum field for which the absolute Galois group acts trivially on $\gamma$. (In general, this notion does not seem to be well-understood and reflects the general failure of Galois descent for cycles modulo $\ell$). We will need to show that the field of definition of $\gamma$ is a ramified extension of a Siegel modular variety (the infinitude is then detected by the action of isogenies on the ramification locus). What is interesting is that the ramification locus is quite generally non-empty, as shown in Proposition \ref{long}, even when it is unclear what the field of definition is. In the case of the Ceresa cycle on the very general Abelian threefold \cite{N}, \cite{RS} and \cite{T}, the field of definition and the ramification locus can be described rather explicitly (see Remark \ref{definition}).\\
\indent As a consequence of Theorem \ref{main}, one obtains the result below (as in \cite{RS} Theorem 1.4 and \cite{T} Corollary 0.2) using Schoen's external product maps (\cite{S2} Theorem 0.2), which also involves very general elliptic curves:
\begin{Cor} Let $E_{1}$, $E_{2}, \ldots E_{n}$ be very general complex elliptic curves. Then, the $\ell$-torsion
\[CH^{d} (E_{1}\times E_{2} \times \dots \times E_{n})[\ell]\]
is infinite for all primes $\ell$, $3 \leq d \leq n-1$ and $n\geq 4$.
\end{Cor}

\subsection*{Acknowledgements}

The author would like to thank Burt Totaro for his interest in early drafts of this paper. He would also like to thank Chad Schoen for some useful comments and Robert Laterveer for calling his attention to a certain method of Claire Voisin \cite{V} described in the sequel.

\section{The Bloch-Esnault criterion}
\noindent We begin with the following deep result which is central to the arguments that follow and which is a fixture in arguments of non-divisibility of cycles (c.f., \cite{S1},\cite{RS}, \cite{T}).
\begin{Thm}[Bloch-Esnault, \cite{BE} Theorem 1.2] Let $X$ be a smooth projective variety over a complete discrete valuation field $K$ with perfect residue field $k$ of mixed characteristic $(0,p)$. Suppose further that $X$ has good ordinary reduction and that
\begin{enumerate}[label=(\alph*)]
\item\label{no-tors-2} The crystalline cohomology of the special fiber $Y$ has no torsion.
\item\label{forms} $H^{0} (Y, \Omega^{m}_{Y}) \neq 0$.
\end{enumerate}
Then, $N^{1}H^{m}_{\text{\'et}} (X_{\overline{K}}, \zit/p) \neq H^{m}_{\text{\'et}} (X_{\overline{K}}, \zit/p)$, where $N^{*}$ denotes the coniveau filtration in \'etale cohomology.
If the \'etale cohomology $H^{m+1}_{\text{\'et}} (X_{\overline{K}}, \zit_{p})$ has no torsion, then for all $r\geq 1$,
\[ N^{1}H^{m}_{\text{\'et}} (X_{\overline{K}}, \zit/p^{r}) \neq H^{m}_{\text{\'et}} (X_{\overline{K}}, \zit/p^{r})\]
\end{Thm}
\noindent Note that the original result only gives the first statement, but the second statement may easily be deduced from a coefficients theorem. We can apply this result in the case that $V$ is an Abelian variety of dimension $g$ defined over the $p$-adic field $\qit_{p}$ with good ordinary reduction since there is no torsion in the crystalline cohomology of an Abelian variety. In particular, there is the following consequence:
\begin{Cor} Let $A_{1}, A_{2}, \ldots A_{s}$ be very general complex Abelian varieties of dimension $g_{1}, \ldots g_{s}$; set $A= A_{1} \times \ldots \times A_{s}$ and $g:= g_{1}+\ldots +g_{s}$. Then,
\[ N^{1}H^{m}_{\text{\'et}} (A, \zit/p^{r}) \neq H^{m}_{\text{\'et}} (A, \zit/p^{r})\]
for all $1\leq m \leq g$, primes $p$ and $r\geq 1$.
\begin{proof} By the previous result, it will suffice to show that for all primes $p$, there are very general complex Abelian variety of dimension $g\geq 1$ admiting a model over $\qit_{p}$ with good ordinary reduction. To this end, fix a prime $p$ and let $N\geq 3$ be coprime to $p$. Then, there is a fine moduli scheme $\mathcal{A}_{g}(N)$ representing principally polarized Abelian varieties of dimension $g$ with full level $N$ structure. In fact, $\mathcal{A}_{g}(N)$ is smooth as a scheme over $\zit[\zeta_{N}, \frac{1}{N}] \subset \qit_{p}$. Certainly, there are very general complex Abelian varieties of dimension $g$ defined over $\qit_{p}$ (since $\overline{\qit_{p}}$ is non-canonically isomorphic to $\cit$). Moreover, the set $\A_{g}(N)(\qit_{p})'$ of $A \in \A_{g}(N)(\qit_{p})$ with good reduction is a Zariski open subset of $\A_{g}(N)(\qit_{p})$. Then, we consider the reduction map
\[ \mathcal{A}_{g}(N)(\qit_{p})' \to \mathcal{A}_{g}(N)(\mathbb{F}_{p})\]
which certainly contains ordinary Abelian varieties. In fact, since this map is Zariski-continuous, the set of $A \in \mathcal{A}_{g}(N)(\qit_{p})'$ which do not have ordinary reduction is a proper Zariski-closed set in $\mathcal{A}_{g}(N)(\qit_{p})$. It follows that for all primes $p$, there exist very general complex Abelian varieties defined over $\qit_{p}$ which have good ordinary reduction.
\end{proof}
\end{Cor}
\begin{Rem} Note that the above corollary implicitly uses the fact that for a very general complex Abelian variety $A$ of dimension $g$, there exists a non-canonical isomorphism of fields $\cit \cong \overline{\cit(\A_{g}(N))}$ for which $A$ is isomorphic as a scheme over $\cit \cong \overline{\cit(\A_{g}(N))}$ to the geometric generic fiber of the universal Abelian variety over $\A_{g}(N)$. In particular, for $A, A'$ very general complex Abelian varieties of dimension $g$, there exists a non-canonical automorphism $\sigma$ of $\cit$ and a corresponding isomorphism of schemes over $\cit$:
\[ A' \cong A \times^{\sigma} \cit \]
This will be used implicitly throughout this paper, whenever necessary.
\end{Rem}
\section{Primitive correspondences}
\begin{Not}\label{notation} In the sequel, we will consider smooth varieties $X$ over $\cit$ and for any Abelian group $G$ we let $H^{*} (X, G)$ be the singular cohomology group with coefficients in $G$ of the underlying complex manifold. Moreover, let $N^{*}H^{*}(X, G)$ denote the corresponding coniveau filtration. For $S$ a smooth quasi-projective variety over $\cit$ and $f: \mathcal{X} \to S$ be a smooth projective morphism, let $f^{-1} (s) = \mathcal{X}_{s}$ for $s \in S(\cit)$. Moreover, let $\cit(S)$ be the function field of $S$, $\X_{\cit(S)}$ the generic fiber and for any field extension $L/\cit(S)$, let $\X_{L}:= \X_{\cit(S)} \times_{\cit(S)} L$.\\
\indent We let $\M_{S}$ denote the category of relative Chow motives over $S$ with integral coefficients. Moreover, for $i\geq 0$ let $R^{i}f_{*}G$ denote the local system whose stalk over $s \in S(\cit)$ is $H^{i} (\mathcal{X}_{s}, G)$.
\end{Not}
\noindent With the above notation, consider a relative correspondence
\[ P \in CH^{g} (\mathcal{X} \times_{S} \mathcal{X}) = \End_{\M_{S}} (\h(\mathcal{X})) \]
\begin{Def} We say that $P$ is a {\em primitive correspondence in codimension $i$} if the following hold:
\begin{enumerate}[label=(\alph*)]
\item For some integer $m \geq 1$, $P^{\circ 2} = m\cdot P$.
\item\label{coniveau} For the very general $s \in S(\cit)$, the complementary correspondence $Q = m\cdot\Delta_{\mathcal{X}} - P$ satisfies
\[ Q_{*}H^{2i-1} (\mathcal{X}_{s}, \zit) \subset N^{1}H^{2i-1} (\mathcal{X}_{s}, \zit) \]
\item\label{local} For each prime $\ell$, there exists a field $F \subset \overline{\cit(S)}$ (depending on $\ell$) for which $\overline{F} = \overline{\cit(S)}$ and for which
\[P_{*}H^{2i-1}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r})\]
is stable and irreducible as a $Gal(\overline{\cit(S)}/F)$-module.\footnote{Here, ``irreducible" means having no $\zit/\ell^{r}[Gal(\overline{\cit(S)}/F)]$-submodules.}
\end{enumerate}
\end{Def}
\begin{Rem} The prototypical case of this notion is in codimension $2$ for families of principally polarized Abelian varieties with sufficiently large monodromy and $P$ an idempotent giving the primitive cohomology in degree $3$. A fully fleshed-out example involving families of products of elliptic curves (but not quite corresponding to primitive cohomology) will be given in the sequel.
\end{Rem}

\noindent Our interest in primitive correspondences is the following lemma, which is nothing but a general statement of analogous results in \cite{S1}, \cite{RS} and \cite{T}.
\begin{Lem}\label{kill-tors} Let $f: \X \to S$ be a family as in Notation \ref{notation} and fix a prime $\ell$. Suppose that for the very general $s \in S(\cit)$
\begin{enumerate}[label=(\alph*)]
\item $N^{1}H^{3} (\mathcal{X}_{s}, \zit/\ell^{r}) \neq H^{3} (\mathcal{X}_{s}, \zit/\ell^{r})$
\item $H^{4} (\mathcal{X}_{s}, \zit)$ is $\ell$-torsion-free.
\end{enumerate}
Finally, let $P \in CH^{d} (\X \times_{S} \X)$ be a primitive correspondence in codimension $2$. Then,
\[ P_{*}CH^{2} (\mathcal{X}_{s})[\ell^{\infty}] = 0\]
for the very general $s \in S(\cit)$.
\begin{proof} Consider the Bloch cycle class map \cite{B}
\[ CH^{2} (\mathcal{X}_{s})[\ell^{\infty}] \to H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}(2))\]
By the Merkurjev-Suslin theorem \cite{MS}, this map is injective and its image is
\[N^{1}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}(2))\]
so it will suffice to show that $P_{*}(N^{1}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell})) = 0$. To this end, note that by assumption, $H^{4} (\mathcal{X}_{s}, \zit)$ is $\ell$-torsion-free, so we have
\begin{equation} H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}) = H^{3} (\mathcal{X}_{s}, \zit) \otimes \qit_{\ell}/\zit_{\ell}\label{divisible}\end{equation}
In particular, since $\qit_{\ell}/\zit_{\ell}$ is a divisible group, multiplication by $m$ is invertible and we have
\[ H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}) = P_{*}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}) \oplus Q_{*}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell})\]
Applying $Q_{*}$ to (\ref{divisible}) gives
\[ \begin{split} Q_{*}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}) & \subset N^{1}H^{3} (\mathcal{X}_{s}, \zit) \otimes \qit_{\ell}/\zit_{\ell}\\ & \subset N^{1}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}) \end{split}\]
Since $N^{1}H^{3} (\mathcal{X}_{s}, \zit/\ell^{r}) \neq H^{3} (\mathcal{X}_{s}, \zit/\ell^{r})$ and since
\[P_{*}H^{3} (\mathcal{X}_{s}, \zit/\ell^{r}) \cong P_{*}H^{3}_{\text{\'et}} (\mathcal{X}_{\overline{\cit(S)}}, \zit/\ell^{r})\]
is irreducible as a $Gal(\overline{\cit(S)}/F)$-module for $r>>0$, it follows that
\[P_{*}H^{3} (\mathcal{X}_{s}, \zit/\ell^{r}) \cap  N^{1}H^{3} (\mathcal{X}_{s}, \zit/\ell^{r}) = 0\]
from which we deduce that
\[ Q_{*}H^{3} (\mathcal{X}_{s}, \zit/\ell^{r}) = N^{1}H^{3} (\mathcal{X}_{s}, \zit/\ell^{r})\]
for $r>>0$. Taking a direct limit gives
\[Q_{*}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}) = N^{1}H^{3} (\mathcal{X}_{s}, \qit_{\ell}/\zit_{\ell}) \]
from which the desired result follows since $P\circ Q=0$.
\end{proof}
\end{Lem}
\noindent As an application, fix a prime $\ell$ and let $P$ be a primitive correspondence in codimension $2$. Now, let
\[ CH^{2} (\mathcal{X})_{0}  \]
denote the group of ``vertically" nulhomologous cycles; i.e., the group of $\gamma \in CH^{2} (\mathcal{X})$ which restrict to a cycle in the kernel of the cycle class map
\[ CH^{2} (\X_{s}) \to H^{4} (\X_{s}, \zit(2)) \]
for $s \in S$. Then, there is the Abel-Jacobi map:
\begin{equation} CH^{2} (\X)_{0} \to H^{1} (S, R^{3}f_{*}\zit(2))\label{abel} \end{equation}
arising from the Leray spectral sequence
\[ E_{2}^{p,q}= H^{p} (S, R^{q}f_{*}\zit(n)) \Rightarrow H^{p+q} (\X,\zit(n)) \]
\begin{Def} With the above notation, we say that $\gamma \in CH^{2} (\mathcal{X})_{0}$ is {\em primitive with respect to $P$} if $P_{*}(\gamma)$ is non-torsion under (\ref{abel}).\end{Def}
\begin{Prop}\label{non-trivial} With the assumptions of Lemma \ref{kill-tors}, suppose that $\gamma \in CH^{2} (\mathcal{X})_{0}$ is primitive with respect to $P$. Then, for $\ell$ a prime and $r>>0$,
\[ \restr{\gamma}{\X_{s}} \neq 0 \in CH^{2} (\X_{s}) \otimes \zit/\ell^{r} \]
\begin{proof} It will suffice to prove that
\[ \gamma \neq 0 \in CH^{2} (\X_{\overline{\cit(S)}}) \otimes \zit/\ell^{r} \]
for $r>>0$. To this end, we suppose by way of contradiction that $\gamma = \ell^{r}\cdot\gamma'$
for some $\gamma' \in CH^{2} (\X_{\overline{\cit(S)}})$. Let $E/\cit(S)$ be a finite extension such that
\[ \gamma' \in CH^{2} (\X_{E}) \]
and let $G_{E}:= Gal(\overline{\cit(S)}/E)$. Since $H^{4}$ is $\ell$-torsion-free, it follows that $\gamma'$ is homologically trivial and we can consider its image under the $\ell$-adic Abel-Jacobi map:
\begin{equation} CH^{2}_{hom} (\X_{E}) \to  H^{1} (G_{E}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))\label{Abel}\end{equation}
arising from the Hochschild-Serre spectral sequence
\[ E_{2}^{p,q}= H^{p} (G_{E}, H^{3}_{\text{\'et}} (Y_{\overline{E}}, \zit/\ell^{r}(2))) \Rightarrow H^{p+q}_{\text{\'et}} (Y_{E},\zit/\ell^{r}(n)) \]
for $Y$ a smooth projective variety over $E$. Now, there is the following claim:
\begin{Lem}\label{non-trivial-lem} For any Galois extension $E/\cit(S)$ as above, set $G' := Gal(E/\cit(S))$. Then, the image of $P_{*}(\gamma')$ under (\ref{Abel}) lies in
\begin{equation} H^{1} (G_{\cit(S)}, P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2))) = H^{1} (G_{E}, P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))^{G'}\label{equality}  \end{equation}
where (\ref{equality}) holds for $r>>0$. In particular, for $r>>0$
\[P_{*}(\gamma) = 0 \in H^{1} (G_{\cit(S)}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2))) \]
\begin{proof} Since $\gamma$ is defined over $\cit(S)$, we have for all $g\in G'$
\[0 = \gamma - g(\gamma) = \ell^{r}(\gamma' - g(\gamma')) \]
so that $\gamma' - g(\gamma')$ is torsion and hence annihilated by $P_{*}$ by Lemma \ref{kill-tors}. Thus, the Abel-Jacobi image of $P_{*}(\gamma')$ in $H^{1} (G_{F}, P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))$ is Galois-invariant. That (\ref{equality}) holds follows from an inflation-restriction argument. Indeed, since $H^{4} (\X_{s}, \zit)$ is $\ell$-torsion-free, it follows from condition \ref{local} in the definition of a primitive correspondence that
\begin{equation} P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit_{\ell}(2))\label{primitive} \end{equation}
is irreducible as a $G_{F}$-module, for some field $F$ such that $\overline{F} = \overline{\cit(S)}$. Since $G_{F}$ contains a subgroup of $G_{\cit(S)}$ of finite index, it follows that (\ref{primitive}) is also irreducible as a $G_{\cit(S)}$-module, as well as as a $G_{E}$-module. Since taking Galois invariants commutes with taking inverse limits, we have
\[ \mathop{\lim_{\longleftarrow}}_{r\geq 0} P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2))^{G_{E}} =P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit_{\ell}(2))^{G_{E}}= 0 \]
It follows that for $r>>0$, we have
\begin{equation} P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2))^{G_{E}} =0\label{primitive-2}\end{equation}
Then, the Hochschild-Serre spectral sequence gives a short exact sequence:
\[ \begin{split}H^{1} (G', P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2))^{G_{E}}) & \hookrightarrow H^{1} (G_{\cit(S)}, P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))\\ & \to H^{1} (G_{E}, P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))^{G'} \\ & \to H^{2} (G', P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2))^{G_{E}}) \end{split}\]
Since the left and bottomost groups are $0$ by (\ref{primitive-2}), we deduce that:
\[ H^{1} (G_{\cit(S)}, P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2))) = H^{1} (G_{E}, P_{*}H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))^{G'} \]
\end{proof}
\end{Lem}
\noindent We will obtain a contradiction, provided that we can show that
\[ P_{*}(\gamma) \neq 0 \in H^{1} (G_{\cit(S)}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit_{\ell}(2)))/\ell^{r} \subset H^{1} (G_{\cit(S)}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))\]
To this end, we observe that since the image under the analogous Abel-Jacobi map (\ref{abel}) is non-torsion and $H^{1} (S, R^{3}f_{*}\zit(2))$ is a finitely-generated group, it follows that the image of $P_{*}(\gamma)$ is non-vanishing in
\[ H^{1} (S, R^{3}f_{*}\zit(2))/\ell^{r} \subset H^{1} (S, R^{3}f_{*}\zit/\ell^{r}(2))) \]
So, it suffices to show the natural map:
\[H^{1} (S, R^{3}f_{*}\zit/\ell^{r}(2))) \to \mathop{\lim_{\longrightarrow}}_{U} H^{1} (U, R^{3}f_{U*}\zit/\ell^{r}(2))) \cong H^{1} (G_{\cit(S)}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(S)}}, \zit/\ell^{r}(2)))  \]
is injective, where $U \subset S$ ranges over all open subsets of $S$ and $f_{U}:f^{-1}(U) \to U$ is the corresponding family. However, using the Gysin exact sequence, it follows immediately that the map
\[H^{1} (S, R^{3}f_{*}\zit/\ell^{r}(2))) \to  H^{1} (U, R^{3}f_{U*}\zit/\ell^{r}(2))) \]
is injective for any open $U \subset S$.
\end{proof}
\end{Prop}
\section{Nori's recipe for infinitude}
\subsection*{Fields of definition}
\noindent Nori's isogeny technique (which was adapted by Totaro to the mod $\ell$ case) can be used quite generally to produce infinitely many distinct cycles in the situation described here. As we will show, this method works so long as ``field of definition" of the cycles being used (defined below) is a ramified extension of the function field of a modular variety. By using isogenies, one can generate infinitely many distinct cycles since the isogenies end up permuting the branch locus.
\begin{Def} Let $L$ be a field (of characteristic $0$) and $n, j$ positive integers, $X$ a smooth projective variety over $L$ and
\[ \gamma \in CH^{j} (X_{\overline{L}}) \otimes \zit/n \]
The {\em field of definition of $\gamma$ over $L$} is the minimum field extension $K/L$ for which
\[ \gamma \in  (CH^{j} (X_{\overline{L}}) \otimes \zit/n)^{Gal(\overline{L}/K)}\]
When there is no confusion, we will omit ``over $L$."
\end{Def}
\begin{Rem}\label{note} An immediate observation is that if $\gamma$ lies in the image of the map
\begin{equation} CH^{j} (X_{F}) \otimes \zit/n \to CH^{j} (X_{\overline{L}}) \otimes \zit/n \label{extens} \end{equation}
for some finite extension $F/L$, then $F$ contains the field of definition of $\gamma$ over $L$. Note that the converse need not hold, however, because of Galois descent issues.
\end{Rem}
\begin{Set}\label{setup} Now, suppose that $A_{1}, \ldots A_{m}$ are complex Abelian varieties of dimensions $g_{1}, \ldots g_{m}$ for $m\geq 2$ and set $g:= g_{1}+ \ldots +g_{m}$. Suppose that $f_{k}: \mathcal{X}_{k} \to S$ are families as in Notation \ref{notation} for which the geometric generic fibers of the corresponding family
\[ f: \X:= \X_{1} \times_{S} \ldots \times_{S} \X_{m} \to S \]
is isomorphic as a scheme over $\cit \cong \overline{\cit(S)}$ to the product
\[ A:= A_{1} \times \ldots \times A_{m} \]
Suppose for simplicity that $S$ has the same dimension as the moduli space
\[\A:= \A_{g_{1}} \times \ldots \times \A_{g_{m}} \]
By the assumption on the geometric generic fiber of $f$, it follows that the corresponding moduli map $S \to \A$ is dominant. The assumption on the dimension ensures that this map is in fact generically finite.
\end{Set}
\noindent Let $\gamma \in CH^{2} (\X)$ be a primitive cycle with respect to some primitive correspondence $P \in CH^{g} (\X \times_{S} \X)$, as in the previous section. In fact, for simplicity, we consider correspondences satisfying the following condition:
\begin{Ass}\label{ass} With the notation of Set-up \ref{setup}, $P$ is of the form
\[ P_{1} \times_{S} \ldots \times_{S} P_{m} \in CH^{g} (\X_{1}^{\times_{S} 2} \times_{S} \ldots \times_{S} \X_{m}^{\times_{S} 2}) \cong CH^{g} (\X \times_{S} \X) \]
where for each $k$:
\[P_{k*}CH^{0} (\X) = 0, \ P_{k*}NS(\X_{k, \overline{\cit(S)}}) = 0\]
where $NS(\X_{k, \overline{\cit(S)}})$ denotes the Ner\'on-Severi group of the geometric generic fiber.
\end{Ass}
\noindent Fix a prime $\ell$ and $r>>0$ and view
\[ \gamma \in CH^{2} (\X_{\overline{\cit(S)}}) \otimes \zit/\ell^{r} \]
We observe that $\overline{\cit(S)} = \overline{\cit(\A)}$ (since $S \to \A$ is generically finite), and so it is possible for the field of definition of $\gamma$ to be contained in what we call {\em a congruence subfield}, which we define to be a function field of an \'etale cover of the fine moduli scheme
\[\A(N) := \A_{g_{1}}(N) \times \ldots \times \A_{g_{m}}(N) \]
for some $N\geq 3$. The following result shows that this is not the case.
\begin{Prop}\label{long} With the conditions of Set-up \ref{setup} and Assumption \ref{ass}, the field of definition of $\gamma$ over $\cit(\A(N))$ is not contained in any congruence subfield for $N$ prime to $\ell$.
\begin{proof} We first identify the congruence subfields. So, let $Y \to \A(N)$ be an \'etale cover for some $N\geq 3$.

\begin{Lem}\label{neat} There is some \'etale cover of $Y$ of the form $\Y_{1} \times \ldots \times \Y_{m}$, where each $\Y_{i}$ is an \'etale cover of $\A_{g_{i}}(N)$.
\begin{proof}[Proof of Lemma] Observe that
\[ \pi_{1}(Y) \leq \pi_{1}(\A(N)) = \prod_{i=1}^{m} \Gamma_{g_{i}} (N)\]
where $\Gamma_{g_{i}} (N) \leq Sp_{2g_{i}} (\zit)$ is the full level-$N$ congruence subgroup of the symplectic group. It suffices to show that there exist lattices
\[ \Gamma_{i} \leq Sp_{2g_{i}} (\zit) \]
such that $\Gamma := \prod_{i=1}^{m} \Gamma_{i} \leq \pi_{1}(Y)$ is a subgroup of finite index.
To see this, note that $\pi_{1}(Y)$ is a lattice in the semi-simple Lie group
\[\prod_{i=1}^{m} Sp_{2g_{i}} (\rit)  \]
Since none of the factors of this group are compact, it follows by \cite{WM} Proposition 4.3.3 (a consequence of the Borel Density theorem) that the desired lattices exist.
\end{proof}
\end{Lem}
\noindent Now, suppose by way of contradiction that the field of definition of $\gamma$ over $\cit(\A(N))$ is contained in some congruence subfield $K$. By Lemma \ref{neat} we can take $K$ to be the function field of some variety of the form
\[ \Y:= \Y_{1} \times \ldots \times \Y_{m} \]
where $\Y_{i}$ is an \'etale cover of $\A_{g_{i}} (N)$. Of course, a priori $\gamma$ lies in the image of
\[ CH^{2} (\X_{E}) \to CH^{2} (\X_{\overline{\cit(\A(N))}})\]
for some finite Galois extension $E/\cit(\A(N))$. So, by Remark \ref{note}, it follows that $K \subset E$.
\begin{Lem} Suppose that
\begin{equation} \gamma \in (CH^{2} (\X_{\overline{\cit(\A(N))}}) \otimes \zit/\ell^{r})^{Gal(\overline{\cit(\A(N))}/K)}\label{inv} \end{equation}
Then, for any Galois extension $E/K$ as above, the Abel-Jacobi image of $P_{*}(\gamma)$ represents a cycle in
\begin{equation}\begin{split}  & H^{1}(Gal(\overline{\cit(\A)}/E), H^{3}_{\text{\'et}} (\X_{\overline{\cit(\A)}}, \zit/\ell^{r}(2)))^{Gal(E/K)}\\
=& H^{1}(Gal(\overline{\cit(\A)}/K), H^{3}_{\text{\'et}} (\X_{\overline{\cit(\A)}}, \zit/\ell^{r}(2)))\label{inv-2} \end{split} \end{equation}
for $r>>0$.
\begin{proof}[Proof of Lemma] If (\ref{inv}) holds, then applying the argument from Lemma \ref{non-trivial-lem} to the cycle
\[ \gamma - g(\gamma) \in CH^{2} (\X_{E}) \]
for $g \in Gal(E/K)$, it follows that
\[P_{*}(\gamma)- g(P_{*}(\gamma)) = P_{*}(\gamma- g(\gamma)) = 0 \in H^{1}(G_{E}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(\A(N))}}, \zit/\ell^{r}(2))) \]
for $r>>0$. Since $Gal(E/K)$ is finite, the first line of (\ref{inv-2}) then holds for $r>>0$. The argument in Lemma \ref{non-trivial-lem} also gives the second line in (\ref{inv-2}).
\end{proof}
\end{Lem}
\noindent On the other hand, since $\gamma$ is primitive, the proof of Proposition \ref{non-trivial} also shows that the Abel-Jacobi image of $\gamma$ does not vanish in
\[ H^{1}(G_{K}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(\A)}}, \zit/\ell^{r}(2))) \]
In fact, we have the following claim:
\begin{claim} The Abel-Jacobi image of $\gamma$ lies in the image of the natural map
\[ H^{1} (\Y, R^{3}f_{\Y}, \zit/\ell^{r}(2)) \to H^{1}(G_{K}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(\A(N))}}, \zit/\ell^{r}(2))) \]
where $f_{\Y}: \X \times_{S} \Y \to \Y$ is the induced family.
\begin{proof}[Proof of claim] By abuse of terminology, we will use $\gamma$ in place of ``the Abel-Jacobi image of $\gamma$." Then, it suffices by the Gysin exact sequence to show that for every divisor $D$ on $\Y$, $\gamma$ lies in the kernel of the residue map:
\begin{equation} H^{1}(G_{K}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(\A(N))}}, \zit/\ell^{r}(2))) \to H^{0} (G_{\cit(D)}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(D)}}, \zit/\ell^{r}(1)))\label{residue}
\end{equation}
But since $\gamma \in CH^{2} (\X)$, it already lies in the kernel of all the residue maps
\[H^{1}(G_{\cit(S)}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(\A)}}, \zit/\ell^{r}(2))) \to H^{0} (G_{\cit(D')}, H^{3}_{\text{\'et}} (\X_{\overline{\cit(D')}}, \zit/\ell^{r}(1)))
\]
where $D'$ ranges over the divisors on $S$. Since $K=\cit(\Y) \subset \cit(S)$ by Remark \ref{note}, it follows that $\gamma$ lies in the kernel of all the residue maps (\ref{residue}), as desired.
\end{proof}
\end{claim}
\noindent To complete the proof, we may view $P_{*}\gamma$ as a non-zero cycle in
\[ H^{1} (\Y, P_{*}R^{3}f_{\Y*}\zit/\ell^{r}(2))\]
The following claim now gives the desired contradiction:
\begin{claim} $H^{1} (\Y, P_{*}R^{3}f_{\Y*}\zit/\ell^{r}(2)) = 0$.
\begin{proof}[Proof of Claim] For each $k=1, \ldots m$, let $\V_{k}(N) \to \A_{g_{k}} (N)$ be the universal Abelian variety and set
\[ \V(N):= \prod_{k=1}^{m} \V_{k} \]
Then, note that since each $\A_{g_{k}} (N)$ is a fine moduli scheme, the natural map
\[ \Y \to \A(N) \]
corresponding to the family of Abelian varieties $f_{\Y}: \X \times_{S} \Y \to \Y$ coincides with the product of \'etale covers:
\[ \Y_{k} \to \A_{g_{k}} (N) \]
It follows by universality that
\[ \X \times_{S} \Y \cong \V(N) \times_{\A(N)} \Y = \prod_{k=1}^{m} \V_{k, \Y}(N) \]
as schemes over $\Y$, where $\V_{k, \Y}(N) = \V_{k}(N) \times_{\A_{g_{k}} (N)} \Y_{k}$. Now, let $\pi_{k}:\Y \to \Y_{k}$ and $f_{k, \Y}: \V_{k, \Y}(N) \to\Y_{k}$ be the projections. Thus, we have:
\[R^{3}f_{\Y*}\zit/\ell^{r}(2) \cong \bigoplus \pi_{1}^{*}R^{j_{1}}f_{1,\Y*}\zit/\ell^{r} \otimes \ldots \otimes \pi_{m}^{*}R^{j_{m}}f_{m,\Y*}\zit/\ell^{r}  \]
where the sum ranges over $j_{1}+ \ldots + j_{m} =3$. By the K\"unneth theorem, this means that
\[\begin{split} H^{1} (\Y, R^{3}f_{\Y*}\zit/\ell^{r}) \cong & \bigoplus_{k=1}^{m} \pi_{k}^{*}H^{1} (\Y_{k}, R^{3}f_{k,\Y*}\zit/\ell^{r}(2))\\
& \bigoplus_{j,k=1}^{m} \pi_{j}^{*}H^{1} (\Y_{j}, R^{1}f_{j,\Y*}\zit/\ell^{r}(1)) \otimes \pi_{k}^{*}H^{0} (\Y_{k}, R^{2}f_{k,\Y*}\zit/\ell^{r}(1)) \end{split}\]
Indeed, all the other terms vanish since
\[H^{0} (\Y_{k}, R^{1}\zit/\ell^{r}) = 0\]
(recall that $\Y_{k} \to \A_{g_{k}}(N)$ is a finite \'etale cover and that $N$ and $\ell$ are coprime by assumption). Moreover, $P_{*}$ kills each of the summands on the first line of the above K\"unneth decomposition since
\[P_{j*}(H^{0} (\Y_{j}, f_{j,\Y*}\zit/\ell^{r})) = 0\]
by Assumption \ref{ass} (note that $m\geq 2$, so $H^{0} (\Y_{k}, f_{k,\Y*}\zit/\ell^{r}))$ should appear in the first line. Finally,
\[ H^{0} (\Y_{k}, R^{2}f_{k,\Y*}\zit/\ell^{r}(1)) \subset NS(\mathcal{X}_{k, \overline{\cit(S)}}) \otimes \zit/\ell^{r} \]
by a ``big" monodromy argument. By Assumption \ref{ass}, $P_{k*}$ kills $NS(\mathcal{X}_{k, \overline{\cit(S)}})$; hence, $P_{*}$ kills all the summands on the second line as well. This gives the desired result.
\end{proof}
\end{claim}
\end{proof}
\end{Prop}
\begin{Rem}\label{definition} When $\gamma$ is the Ceresa cycle on the very general Abelian threefold, the field of definition of $\gamma$ is the function field of $\M_{3}(N)$, the fine moduli scheme of curves of genus $3$ with full level $N$-structure ($N\geq 3$), which is a ramified quadratic extension of the function field of $\cit(\A_{3}(N))$, the fine moduli scheme of Abelian threefolds with full level $N$-structure. (By the Torelli theorem, the extension $\cit(\M_{3}(N))/\cit(\A_{3}(N))$ is ramified along the hyperelliptic locus.)
\end{Rem}
\begin{Cor}\label{needed-cor} With the above notation, fix $N\geq 3$ relatively prime to $\ell$. Then, the field of definition of $\gamma$ over $\cit(\A(N))$, is a ramified extension, $K_{\gamma} (N)/\cit(\A(N))$. Moreover, for any positive integer $N'$ divisible by $N$, we have
\[ K_{\gamma}(N') = K_{\gamma}(N)\cdot\cit(\A(N')) \]
the compositum of $K_{\gamma}(N)$ and $\cit(\A(N'))$.
\begin{proof} The first statement is just a restatement of Proposition \ref{long}. For the second statement, note that by definition $K_{\gamma} (N)$ is the field corresponding to the stabilizer in the Galois group $G_{\cit(\A(N))} = Gal(\overline{\cit(\A(N))}/\cit(\A(N)))$ of
\[ \gamma \in CH^{2} (\X_{\overline{\cit(\A(N))}})\otimes \zit/\ell^{r} \]
On the other hand, the absolute Galois group of the compositum $K_{\gamma}(N)\cdot\cit(\A(N'))$ is
\[ G_{\cit(\A(N'))} \cap G_{K_{\gamma}(N)} \]
which is precisely the stabilizer of $\gamma$ in $\cit(\A(N'))$. The corresponding field is $K_{\gamma}(N')$.
\end{proof}
\end{Cor}
\subsection*{Nori's method}
\noindent To implement Nori's recipe, we will need some more set-up. For each positive integer $n\geq 3$ prime to $\ell$ (see Proposition \ref{long}), set $K(n) = \cit(\A (n))$
and consider the field
\[L := \mathop{\lim_{\longrightarrow}}_{n} K(n)\]
where the limit ranges over all $n$ prime to $\ell$. Also, for each $k=1,\ldots m$, consider the  groups
\[ G_{k} = GSp_{2g_{k}} (\qit), \ M_{k} = M_{2g_{k}} (\zit) \]
and set
\[ G= G_{1} \times \ldots \times G_{m}, \ M = M_{1} \times \ldots \times M_{m}\]
Furthermore, we set
\[ W:= \cit^{2g_{1}} \times \ldots \times \cit^{2g_{m}}, \ \mathfrak{h}:=\mathfrak{h}_{g_{1}} \times \ldots \times \mathfrak{h}_{g_{m}} \]
We observe that $G$ acts naturally on $\mathfrak{h} \times W$, the universal cover of the universal Abelian variety $\mathcal{V}(r) \xrightarrow{f_{r}} \mathcal{A}(r)$, with the center
\[Z(G) = Z(G_{1}) \times \ldots \times Z(G_{m})\]
acting trivially. Indeed, each
\[\phi \in G \cap M \]
certainly acts on $\mathfrak{h} \times W$ (see, for instance, \cite{N}), and one can extend this action to all of $G$ by taking the action of each $Z(G_{k})$ (consisting of scalar matrices) to be trivial.\\
\indent Moreover, this action descends to an action on universal Abelian varieties; i.e., for each $\phi \in G$ and positive integer $r$ prime to $\ell$, there is some $s \mid r$ and a commutative diagram:
\[ \begin{CD} \V(r) @>{\phi}>> \V(s)\\
@V{f_{r}}VV @V{f_{s}}VV\\
\A(r)@>{\phi}>> \A(s)  \end{CD}\]
Taking the inverse limit over $r$ we obtain a commutative diagram:
\begin{equation} \begin{CD} \V_{L} @>{\phi}>> \V_{L}\\
@VVV @VVV\\
\text{Spec } L@>{\phi}>> \text{Spec } L\label{generic}  \end{CD}\end{equation}
Here, we set $\V_{K_{r}}$ to be the generic fiber of $f_{r}$ and
\[ \V_{L} := \mathop{\lim_{\longleftarrow}}_{r} \V_{K_{r}}\]
where the limit ranges over $r$ prime to $\ell$. This is an Abelian variety of dimension $g$ over the field $L$, and we observe that by construction, each $\phi \in G$ acts by an isogeny (in the diagram (\ref{generic})). As in Nori's approach, we will need a way of keeping track of all the lifts of a given $\phi \in G$ to an action on the geometric generic fiber:
\[ \mathcal{V}_{\overline{L}} := \mathcal{V}_{L} \times_{L} \overline{L} \]
So, we consider the group
\[\G := \{ (\psi, \phi)\in  Aut(\overline{L}) \times G| \restr{\psi}{L} = \restr{\phi}{L} \} \] where the notation $\restr{\phi}{L}$ means the restriction of $\phi \in G$ to its action on $L$. Note that there is a natural short exact sequence
\[ 1 \to Gal(\overline{L}/L) \to \mathcal{G} \xrightarrow{\pi_{2}} G \to 1 \]
As in op cit. p. 194, one has
\begin{Lem}\label{easy} There is a homomorphism
\[ \rho:\G \to \End(\mathcal{V}_{\overline{L}}) \to \End (CH^{2}(\mathcal{V}_{\overline{L}})\otimes \zit/\ell^{r}) \]
for which the composition
\[ Gal(\overline{L}/L) \hookrightarrow \G \xrightarrow{\rho} \End (CH^{2}(\mathcal{V}_{ \overline{L}})\otimes \zit/\ell^{r}) \] is the usual Galois action.
\end{Lem}
\noindent Let $\End(\mathcal{V}_{\overline{L}})_{\ell}$ be the subgroup of isogenies of degree prime-to-$\ell$, let $\G_{\ell} \leq \G$ be the corresponding subgroup in $\G$ and
\[ G_{\ell} := \pi_{2}(\G_{\ell}) \]
It follows from the argument of \cite{T} Lemma 3.2 that
\[\rho(\G_{\ell}) \subset \Aut(CH^{2}(\mathcal{V}_{\overline{L}})\otimes \zit/\ell^{r})\]
\indent Now, with the conditions and notation of the previous subsection, consider a primitive cycle $\gamma \in CH^{2} (\X)$ with respect to some primitive correspondence $P$ on $\X$ satisfying Assumption \ref{ass}. Then, let
\[ G' \leq G_{\ell} \cap (Sp_{2g_{1}} (\qit) \times \ldots \times Sp_{2g_{m}} (\qit)) \]
be an infinite subgroup and select
\[\phi_{1}, \phi_{2}, \ldots \in G'\]
giving distinct cosets in $G/\Gamma$ for
\[ \Gamma := Sp_{2g_{1}} (\zit) \times \ldots \times Sp_{2g_{m}} (\zit) \]
Lift these to $\tilde{\phi}_{1}, \tilde{\phi}_{2}, \ldots \in \G$ via $\mathcal{G}' \xrightarrow{\pi_{2}} G'$. Note by assumption that each $\rho(\tilde{\phi}_{j})$ is an automorphism on
\[CH^{2}(\mathcal{V}_{\overline{L}})\otimes \zit/\ell^{r}\]
Thus, we have the following:
\begin{Prop}\label{Nori} With the above notation, there exist infinite subgroups $G'$ as above and $\phi_{1}, \phi_{2}, \ldots \in G'$ for which the corresponding set
\[ \{ \rho(\tilde{\phi}_{1})(\gamma), \rho(\tilde{\phi}_{2})(\gamma), \ldots \} \subset CH^{2}(\mathcal{V}_{\overline{L}})\otimes \zit/\ell^{r} \]
is infinite (where $r>>0$).
\begin{proof} It follows by Proposition \ref{long} that the field of definition of
\[ \gamma \in CH^{2} (\X_{\overline{\cit(S)}}) \otimes \zit/\ell^{r} = CH^{2} (\X_{\overline{L}}) \otimes \zit/\ell^{r} \]
over $L$ is the field
\[\displaystyle \mathop{\lim_{\longrightarrow}}_{n} K_{\gamma}(n)\]
where the limit ranges over $n$ prime to $\ell$. For each such $n$, let
\[B_{\gamma}(n) \subset \mathcal{A}(n)\]
denote the corresponding branch locus of the extension $K_{\gamma}(n)/K(n)$. We note that for $n \mid n'$, $B_{\gamma}(n')$ is the pull-back of $B_{\gamma}(n)$ along $\A(n') \to \A(n)$. This follows from Corollary \ref{needed-cor} and the fact that $\A(n') \to \A(n)$ is \'etale. Thus, the pull-back of $B_{\gamma}(n)$ along $\mathfrak{h} \to \A(n)$ is a closed subset of $\mathfrak{h}$ for the analytic topology and does not depend on $n$. We will call this set the absolute branch locus, $B_{\gamma} \subset \mathfrak{h}$. \\
\indent Each of the cycles $\gamma_{j}:=\rho(\tilde{\phi}_{j})(\gamma)$ has a field of definition (over $L$) $K_{\gamma_{j}}$, and set $H_{j}:= Gal(\overline{L}/K_{\gamma_{j}})$. This latter is the stabilizer of
\[ \gamma_{j} \in CH^{2} (\X_{\overline{L}}) \otimes \zit/\ell^{r} \] in $Gal(\overline{L}/L)$. In order to show that $\gamma_{j}$ are all distinct, it will suffice to show that the $H_{j}$ are all distinct. Now, by Lemma \ref{easy} (and a basic property of stabilizer subgroups), we have
\[ H_{j} = \tilde{\phi}_{j}Gal(\overline{L}/K_{\gamma})\tilde{\phi}_{j}^{-1} \leq \mathcal{G}\]
and as a consequence, $K_{\gamma_{j}} = \tilde{\phi}_{j}(K_{\gamma})$. In particular, it follows that $K_{\gamma_{j}}$ is a ramified extension of $L$; as in the first paragraph of the proof, one then has a corresponding absolute branch locus $B_{\gamma_{j}}$ and this will satisfy:
\[B_{\gamma_{j}} = \phi_{j}^{-1}(B_{\gamma}) \subset \mathfrak{h}\]
Hence, it will suffice (as in \cite{N} and \cite{T}) to show that all the $B_{\gamma_{j}}$ are distinct. Since $\phi_{j}$ were chosen to have distinct cosets in $G/\Gamma$, it will suffice to prove the lemma below.
\end{proof}
\end{Prop}
\begin{Lem} Consider the group
\[\Lambda= \{ \phi \in Sp_{2g_{1}} (\rit) \times \ldots \times Sp_{2g_{m}} (\rit)  \ | \  \phi(B_{\gamma}) = B_{\gamma}\} \]
Then, there exist infinite subgroups $G'$ as above such that $G' \cap \Lambda \leq \Gamma$.
\begin{proof} Note that since $B_{\gamma}$ is obtained by pulling back along $\mathfrak{h} \to \A_{g}(N)$, it is stable under the action of a subgroup of finite index of $\Gamma$. Thus, the corresponding Lie algebra $\text{Lie}(\Lambda)$ is stable under the adjoint action $\Gamma$, which is Zariski dense in
\[ K(\rit):= Sp_{2g_{1}} (\rit) \times \ldots \times Sp_{2g_{m}} (\rit) \]
from which it follows that $\text{Lie}(\Lambda)$ is invariant under the adjoint action of $K(\rit)$. Since $K(\rit)$ is a semi-simple Lie group, it follows that $\text{Lie}(\Lambda)$ is a product of the Lie algebras
\[ \mathfrak{s}\mathfrak{p}_{2g_{1}} (\rit), \ldots, \mathfrak{s}\mathfrak{p}_{2g_{m}} (\rit) \]
Since $B_{\gamma} \neq \mathfrak{h}$, we cannot have $\text{Lie}(\Lambda) =\mathfrak{k}$, from which it follows that there is some $j$ for which
\[ \text{Lie}(\Lambda) \cap \mathfrak{s}\mathfrak{p}_{2g_{j}} (\rit) = 0 \]
Since $\Lambda$ is closed in the analytic topology in $K(\rit)$, it follows that
\[ \Lambda \cap Sp_{2g_{j}} (\rit)\]
is discrete. By the Borel-Density Theorem \cite{B}, it follows that
\[\Lambda \cap Sp_{2g_{j}} (\rit) \leq Sp_{2g_{j}} (\zit) \]
Thus, we can take $G' \leq Sp_{2g_{j}} (\qit) \cap G_{\ell}$, for instance.
\end{proof}
\end{Lem}

\section{Application}
\noindent We would like to apply the method described in the previous sections to obtain a proof of Theorem \ref{main}. For this we will first need families $f_{k}: \X_{k} \to S$ as in Set-up \ref{setup} for which $m=3$ and for which $g_{1}=g_{2}=g_{3}=1$. In particular, this means that $S$ will have dimension $3$. Furthermore, consider the corresponding $j$-invariant maps:
\[ J_{k}: S \to \A_{1} = \pit^{1} \setminus \{0, 1, \infty \} \]
We would like for the product:
\[ J:= J_{1} \times J_{2} \times J_{3}: S \to \A:= \A_{1} \times \A_{1} \times \A_{1} \]
to be dominant. While there are certainly many possible choices, we will consider the following extension of a family of elliptic curves considered in \cite{BST} and \cite{S1}.\\
\indent Let $T \subset \pit(H^{0} (\pit_{\cit}^{2}, \mathcal{O}(2)))$ be the projective linear subspace corresponding to the family of conics in $\pit^{2}$:
\[ x^{2}+y^{2}+z^{2}+axy+byz+cxz = 0 \]
Let $S$ be an open subset of $T$ consisting of smooth conics. Then, we consider the universal family of conics over $S$:
\begin{equation} \begin{CD}  \mathcal{Q} @>>> & \pit^{2} \\ @VhVV & \\ S &  \end{CD}\label{fiber}\end{equation}
There is a finite morphism $\phi: \pit^{2} \to \pit^{2}$ defined by
\begin{equation} [x, y, z] \mapsto [x^{2}, y^{2}, z^{2}] \label{phi}\end{equation}
whose Galois group is $G:= \zit/2 \times \zit/2$ and is generated by the involutions:
\[ [x, y, z] \xmapsto{\sigma_{1}} [-x, y, z], \ [x, y, z] \xmapsto{\sigma_{2}} [x, -y, z]  \]
and denote by $\sigma_{3} := \sigma_{1}\circ\sigma_{2} = \sigma_{2}\circ\sigma_{1}$. We then consider the Cartesian product:
\begin{equation}\label{cart-diag} \begin{CD}  \mathcal{C} @>q>> &\mathcal{Q}  \\ @VVV & @VVV\\ \pit^{2} @>\phi>> & \pit^{2} \end{CD}\end{equation}
There is a natural morphism $g: \mathcal{C} \xrightarrow{} S$, and after possibly shrinking $S$, we may assume that all the fibers of $g$ are smooth. Explicitly, the total space $\C$ is the family of degree $4$ curves in $\pit^{2}$
\begin{equation} x^{4}+y^{4}+z^{4}+ax^{2}y^{2}+by^{2}z^{2}+cx^{2}z^{2} = 0 \label{3-family}\end{equation}
The involutions above then induce an action on $\mathcal{C} \xrightarrow{g} S$, which gives rise to the quotients:
\[ p_{k}: \C \to \X_{k}:= \mathcal{C}/\sigma_{k}, \ f_{k}: \X_{k} \to S, \ k=1,2,3 \]
where $f_{k} \circ p_{k} = g$. It is easy to see that for the general fiber of $g$, each $\sigma_{k}$ fixes precisely $4$ points; using a Riemann-Hurwitz argument, it follows that the fibers of $f_{k}$ all have genus 1. Upon shrinking $S$ further, we can assume that all the fibers of $f_{k}$ are genus $1$. Moreover, there are induced quotient maps:
\[ q_{k} : \X_{k} \to \mathcal{Q} = \C/G \]
as well as involutions, $(-1)_{\X_{k}}: \X_{k} \to \X_{k},$ which induce the usual action by $-1$ on the fibers of $f_{k}$ (since the quotient is a family of conics). Then, we consider the fiber product
\[ f:= f_{1} \times_{S} f_{2}  \times_{S} f_{3}: \X:= \X_{1} \times_{S} \X_{2} \times_{S} \X_{3} \to S \]
There is then an action by \[\tilde{G} = \zit/2 \times \zit/2 \times \zit/2\] on $f: \X \to S$ generated by involutions
\[ \tilde{\sigma}_{k} : \X \to \X, \ k=1,2,3 \]
which act by $(-1)_{\X_{k}}$ on the $k^{th}$ factor and the identity on the remaining factors. We also use the notation:
\[ \tilde{\sigma}_{jk} := \tilde{\sigma}_{j}\circ\tilde{\sigma}_{k}, \ \tilde{\sigma}_{123}:= \tilde{\sigma}_{1}\circ\tilde{\sigma}_{2}\circ\tilde{\sigma}_{3} \]
Moreover, there is a morphism:
\[ \rho: \C \xrightarrow{\Delta^{(3)}} \C^{3} := \C \times_{S} \C \times_{S} \C \xrightarrow{p} \X \]
where $p:= p_{1} \times_{S} p_{2} \times_{S} p_{3}$. Then, we have the following lemma and its consequence, the desired moduli property:
\begin{Lem}\label{basics} There is a natural map of $S$-schemes
\[ \X \to Pic^{0} (\C/S) \]
which is an isogeny for the general $s \in S$, where as usual $Pic^{0}(\C/S)$ denotes the identity component of the relative Picard scheme.
\begin{proof}  The existence of this map follows from the tautological property of the relative Picard variety since for $k=1,2,3$, there are
\[ \mathcal{L}_{k} \neq 0 \in CH^{1} (\C \times_{S} \X_{k})/\pi_{1}^{*}CH^{1} (\C) \oplus \pi_{2}^{*}CH^{1} (\X_{k})\]
(for instance, one takes $\mathcal{L}_{k} = \Gamma_{q_{k}}$) and pulling these back gives
\[ \mathcal{L} \neq 0 \in CH^{1} (\C \times_{S} \X)/\pi_{1}^{*}CH^{1} (\C) \oplus \pi_{2}^{*}CH^{1} (\X)\]
That this is an isogeny, it suffices by upper semi-continuity of dimension to check this for some fiber. But this was already checked for the Fermat quartic in \cite{Bl} Lemma 3.1 (in fact, the isogeny has degree a power of $2$).
\end{proof}
\end{Lem}
\begin{Cor} The period map $J: S \to \A$ corresponding to $f: \X \to S$ is dominant.
\begin{proof} By the previous lemma, it suffices to show that that the moduli map $S \to \A_{3}$, sending a curve in $S$ to its Picard variety, has dimension $3$. By upper semi-continuity of dimension, it suffices to show that there is a zero-dimensional fiber of this map; i.e., that for some curve in (\ref{3-family}), the Jacobian is isomorphic to only finitely many other Jacobians in the family. For this, we consider the Fermat quartic and note that there are only finitely many other Jacobians with CM type $\qit(i) \oplus \qit(i) \oplus \qit(i)$.
\end{proof}
\end{Cor}
\noindent Now, we need to produce a primitive correspondence
\[ P \in CH^{3} (\X \times_{S} \X) \]
satisfying Assumption \ref{ass}, as well as some $\gamma \in CH^{2} (\X)$ which is primitive with respect to $P$. To define $P$, retain the notation from earlier in the section and set
\[ \Pi_{k} := \Delta_{\X_{k}} - \Gamma_{\tilde{\sigma}_{k}} \in CH^{1} (\X_{k} \times_{S} \X_{k}) \]
and then set
\[ P:= \Pi_{1} \times_{S} \Pi_{2} \times_{S}\Pi_{3} \in CH^{3} (\X_{1}^{\times_{S} 2} \times_{S} \X_{2}^{\times_{S} 2} \times_{S} \X_{2}^{\times_{S} 3}) \cong CH^{3} (\X \times_{S} \X) \]
The main cycle is then defined as an analogue of the Ceresa cycle:
\begin{equation} \boxed{\gamma := \rho_{*}([\C]) - \tilde{\sigma}_{123}^{*}\rho_{*}([\C]) \in CH^{2} (\X)} \end{equation}
Note that for $\{i,j,k \} = \{1,2,3 \}$ and $j<k$, we have
\[ \tilde{\sigma}_{jk}^{*}\rho([\C]) = \rho([\C])\in CH^{2} (\X)\]
since there is a commutative diagram:
\[ \begin{tikzcd} \C \arrow{r}{\sigma_{i}} \arrow{d}{\rho} & \C \arrow{d}{\rho}\\
\X \arrow{r}{\tilde{\sigma}_{jk}} & \X
\end{tikzcd}\]
by construction. It follows that $\gamma$ is invariant under $\tilde{\sigma}_{jk}$ and anti-invariant under $\tilde{\sigma}_{123}$ and hence is anti-invariant with respect to each $\tilde{\sigma}_{k}$. A computation then shows that
\begin{equation} P_{*}(\gamma) = 2\cdot \label{short-cut}\gamma \in CH^{2} (\X)\end{equation}

\subsection*{Proof that $P$ is primitive} We observe that since $\tilde{\sigma}_{k}$ is an involution for $k=1,2,3$, we have
\[ \begin{split}\Pi_{k}^{\circ 2} = (\Delta_{\X_{k}} - \Gamma_{\tilde{\sigma}_{k}})^{\circ 2} &=  \Delta_{\X_{k}} - 2\Gamma_{\tilde{\sigma}_{k}} + \Gamma_{\tilde{\sigma}_{k}^{2}}\\
&= 2(\Delta_{\X_{k}} - \Gamma_{\tilde{\sigma}_{k}})
\end{split} \]
It follows that $P^{\circ 2} = 8\cdot P$ by a standard computation, which verifies the first condition of the definition. To verify condition \ref{coniveau} note that for $s \in S$, we have
Thus, we see that the complementary correspondence
\[ \Phi_{k} = 2\cdot\Delta_{\X_{k}} - \Pi_{k} = \Delta_{\X_{k}} +\Gamma_{\tilde{\sigma}_{k}}= \prescript{t}{}{\Gamma_{q_{k}}}\circ\Gamma_{q_{k}}  \]
from which it follows that
\[\Phi_{k*}H^{*} (\X_{k,s}, \zit) = q_{k}^{*}q_{k*} H^{*} (\mathcal{Q}_{s}, \zit) \subset N^{1}H^{*} (\X_{k,s}, \zit)\]
Now, we compute
\[ \begin{split} Q & = 8\cdot \Delta_{\X} - \Pi_{1} \times_{S} \Pi_{2} \times_{S} \Pi_{3}\\
& = \Phi_{1} \times_{S} \Phi_{2} \times_{S} \Phi_{3} +(\Phi_{1} \times_{S} \Phi_{2} \times_{S} \Pi_{3} + \text{perm.}) + (\Phi_{1} \times_{S} \Pi_{2} \times_{S} \Pi_{3} + \text{perm.})\end{split}\]
from which we deduce that
\[ Q_{*}H^{*} (\X_{s}, \zit) \subset N^{1}H^{*} (\X_{s}, \zit) \]
by properties of the coniveau filtration (i.e., compatibility with cup product).
We would also like to verify Assumption \ref{ass}. Indeed, we have $\Pi_{k*}H^{j} (\X_{k,s}, \zit) = 0$ for $j\neq 1$, since $\tilde{\sigma}_{k}$ acts by $-1$ on the fibers of $f_{k}$. It follows by the K\"unneth theorem that $P_{*}$ kills $H^{2*} (\X_{s}, \zit)$. Finally, we would like to verify condition \ref{local}. Indeed, note that by construction
 \[ \Pi_{k*}H^{1}(\X_{k,s}, \zit/\ell^{r}) = 2\cdot H^{1} (\X_{k,s}, \zit/\ell^{r}) \]
Using the K\"unneth theorem again, it then follows that
\[ P_{*}H^{3}(\X_{s}, \zit/\ell^{r}) = 8\cdot(H^{1} (\X_{1,s}, \zit/\ell^{r}) \otimes H^{1} (\X_{2,s}, \zit/\ell^{r}) \otimes H^{1} (\X_{3,s}, \zit/\ell^{r}))  \]
For condition \ref{local} in the definition of a primitive correspondence, we can take $F=\cit(\A(N))$ where $N\geq 3$ and prime to $\ell$ and we need to show that
\begin{equation} 8\cdot(H^{1}_{\text{\'et}} (\V_{1}(N)_{\overline{F}}, \zit/\ell^{r})^{\otimes 3})\label{8}  \end{equation}
is irreducible as a $Gal(\overline{F}/F)$-module for $r>>0$. For this, let $\ell \neq 2$, $r\geq 1$ and
\begin{equation} V_{\ell}= H^{1}_{\text{\'et}} (\V_{1}(N)_{\overline{F}}, \zit/\ell^{r}), \ W_{\ell}= V_{\ell}^{\otimes 3}\label{W}  \end{equation}
We then note that the image of the Galois representation
\[ Gal(\overline{F}/F) \to GL(W_{\ell}) \]
contains the image of the reduction mod $\ell^{r}$ map:
\begin{equation} \Gamma(N)^{\times 3} \leq SL_{2}(\zit)^{\times 3} \to SL_{2}(\zit/\ell^{r})^{\times 3}\label{reduction}  \end{equation}
where $\Gamma(N)\leq SL_{2}(\zit)$ is the full congruence subgroup and where we view
\begin{equation} SL_{2}(\zit/\ell^{r})^{\times 3} \subset GL(V_{\ell})^{\times 3} \subset GL(W_{\ell})\label{inclusion}\end{equation}
Since $N$ is prime to $\ell$, (\ref{reduction}) is surjective. Moreover, $V_{\ell}$ is irreducible as an $SL_{2}(\zit/\ell^{r})$-module, from which it follows that the $W_{\ell}$ is irreducible as an $SL_{2}(\zit/\ell^{r})^{\times 3}$-module. Hence, $W_{\ell}$ is also irreducible as a $Gal(\overline{F}/F)$-module, as desired. Finally, for $\ell=2$, the verification is essentially the same, except that we must take $r\geq 4$ and
\[ V_{\ell} = H^{1}_{\text{\'et}} (\V_{1}(N)_{\overline{F}}, \zit/\ell^{r-3}) \]
in (\ref{W}) so that $W_{\ell}$ coincides with (\ref{8}).
\subsection*{Proof that $\gamma$ is primitive} Certainly,
\[\restr{\gamma}{\X_{s}} \in CH^{2} (\X_{s})\]
is homologically trivial for $s\in S$. We would like to show that the Abel-Jacobi image of $P_{*}(\gamma)$ in $H^{1} (S, P_{*}R^{3}f_{*}\zit(2))$ is non-torsion or, equivalently that it is non-zero in $H^{1} (S, P_{*}R^{3}f_{*}\qit(2))$. Now, we observe that
\[ P_{*}R^{j}f_{*}\qit(2) = 0\]
for $j\neq 3$ by construction of $P$. From the functoriality of the Leray spectral sequence with respect to correspondences, it follows that
\[ H^{1} (S, P_{*}R^{3}f_{*}\qit(2)) =P_{*}H^{4} (\X, \qit(2)) \]
Hence, it suffices to show that the image of $P_{*}(\gamma) = 2\cdot\gamma$ (see (\ref{short-cut})) under the cycle class map
\begin{equation}CH^{2} (\X) \otimes \qit \to H^{4} (\X, \qit(2))\label{cycle} \end{equation}
is non-zero. To this end, we have the following:
\begin{Lem} Let $\hat{\X}$ be a smooth compactification of $\X$. Then, algebraic equivalence and homological equivalence agree on $\hat{\X}$ ($\otimes \qit$).
\begin{proof} By \cite{BS} Theorem 1 (ii), it suffices to prove that $\X$ is rationally connected. Since $\Y := \C \times_{S} \C \times_{S} \C$ dominates $\X$, it suffices to prove this for $\Y$. To this end, consider the map
\[ \mathcal{Z}:= \mathcal{Q} \times_{S} \mathcal{Q} \times_{S} \mathcal{Q} \xrightarrow{\pi} Z:=\pit^{2} \times \pit^{2} \times \pit^{2} \]
obtained by projecting each $\mathcal{Q} \subset S \times \pit^{2}$ to $\pit^{2}$. For a given $t_{1}, t_{2}, t_{3} \in \pit^{2}$, the fiber
\[ \pi^{-1}((t_{1}, t_{2}, t_{3})) \]
consists of the conics in the family (\ref{3-family}) which pass through $t_{1},t_{2}, t_{3}$. For the general $(t_{1}, t_{2}, t_{3})$, there is a unique such conic, from which it follows that $\pi$ is birational (not necessarily surjective). From (\ref{cart-diag}) it follows that there is a Cartesian diagram:
\[\begin{tikzcd}
\Y \arrow{r} \arrow{d} & \mathcal{Z} \arrow{d}\\
Z \arrow{r}{\phi \times \phi \times \phi} & Z
\end{tikzcd}
\]
where $\phi$ was defined in (\ref{phi}). It follows that $\mathcal{Z} \to Z$ is also birational and, hence, that $\mathcal{Z}$ is rationally connected.
\end{proof}
\end{Lem}
\begin{Rem} With a bit more work, one can likely show that (\ref{cycle}) is in fact injective; given the previous lemma, this is tantamount to showing that $CH^{2} (\X) \otimes \qit$ is finitely generated (since the group of cycles algebraically equivalent to $0$ is divisible). It is possible that one could do this using an arguments such as those of \cite{V}.
\end{Rem}

\noindent Now, suppose by way of contradiction that $\gamma = 0 \in H^{4} (\X, \qit)$. Then, let $\hat{\X}$ be a smooth compactification and let $D = \hat{\X} \setminus \X$ and spread out $\gamma$ to a cycle in $CH^{2} (\hat{\X}) \otimes \qit$, also denoted by $\gamma$ by abuse of notation. By \cite{V} Lemma 1.1, the Voisin standard conjecture holds for codimension $2$ cycles. This means that there exists some $\gamma' \in CH^{2} (\hat{\X}) \otimes \qit$ supported on $D$ for which
\[ \gamma' = \gamma = 0 \in  H^{4} (\hat{\X}, \qit)\]
Then, by the previous lemma, $\gamma - \gamma' \in CH^{2} (\hat{\X}) \otimes \qit$ is algebraically equivalent to $0$. Moreover, we note that for $s \in S$, the fiber $\X_{s}$ does not intersect $D$, so
\[ \restr{\gamma'}{\X_{s}} = 0 \in CH^{2} (\X_{s}) \otimes \qit \]
From this, we deduce that
\[ \restr{\gamma}{\X_{s}} \in CH^{2} (\X_{s}) \otimes \qit \]
is algebraically equivalent to $0$. In particular, this is true when $\X_{s}$ is the Fermat quartic, for instance, but this contradicts \cite{Bl} Theorem 4.1.

\subsection*{Proof of Theorem \ref{main}} We first prove Theorem \ref{main} in the case that $d=2$ and $n=3$. In this case, we let $\X \xrightarrow{f} S$ be as at the beginning of this section, which satisfies the conditions of Set-up \ref{setup}. Moreover, by the previous two sub-sections, $P$ is a primitive correspondence satisfying Assumption \ref{ass}, and $\gamma$ is primitive with respect to $P$, Proposition \ref{Nori} shows that for very general $s \in S$,
\begin{equation} CH^{2} (\X_{s}) \otimes \zit/\ell^{r}\label{final-Chow} \end{equation}
is infinite for $r>>0$. Finally, as in \cite{T}, this implies that $CH^{2} (\X_{s}) \otimes \zit/\ell$ is also infinite.\\
\indent For $n>3$ and $2\leq d\leq n-1$, note that for very general complex elliptic curves $E_{1}, \ldots E_{n}$, one can select degree $2$ maps $f_{k}: E_{k} \to \pit^{1}$. Then, one has
\[ CH^{d} (E_{1} \times E_{2} \times E_{3} \times \underbrace{\pit^{1} \times \ldots \times \pit^{1}}_{n-3 \text{ times}}) \otimes \zit/\ell \]
is infinite for all $d$ in the above range. Moreover, the natural map
\[ CH^{d} (E_{1} \times E_{2} \times E_{3} \times \underbrace{\pit^{1} \times \ldots \times \pit^{1}}_{n-3 \text{ times}}) \otimes \zit/\ell \to CH^{d} (E_{1} \times \ldots \times E_{n}) \otimes \zit/\ell \]
induced by pull-back along
\[ \text{id}_{E_{1}} \times \text{id}_{E_{2}}  \times \text{id}_{E_{3}} \times f_{4} \times \ldots \times f_{n}: E_{1} \times \ldots \times E_{n} \to E_{1} \times E_{2} \times E_{3} \times \underbrace{\pit^{1} \times \ldots \times \pit^{1}}_{n-3 \text{ times}} \]
is injective (using the projective bundle formula). This gives the desired infinitude.


TITLE: How many $6$ letter words can be formed of $A, B$ and $C$ such that each letter appears at least once in the word?
QUESTION [3 upvotes]: How many $6$ letter words can be formed of $A,B$ and $C$ such that each letter appears at least once in the word?

My solution goes like this:
The total number of words possible is $3^6$. Now, we do complementary counting. The number of ways to choose 1 letter out of those $3$, such that the word formed has only one letter is $\binom {3}{1}$. Now, if the word is formed only of $2$ letters then, the number of such words is $\binom {3}{2}2^6$. Hence, the required number of words is $3^6-(\binom {3}{1}+\binom {3}{2}2^6)$.
Is the above solution correct? Is it valid? If not, where is it going wrong? I am not quite getting it.There may be many posts concerning the same topic but I can't seem to find it either....

REPLY [3 votes]: Addendum added to respond to the comment question of Franklin.

I think that Inclusion - Exclusion should be used here.  See this article for an
introduction to Inclusion-Exclusion.
Then, see this answer for an explanation of and justification for the Inclusion-Exclusion formula.
Using the syntax of the second link, let $S$ denote the set of all of the possible words that can be formed.
Then, Let $S_A$ denote the set of all of the possible words that can be formed that do not contain the letter A.
Define $S_B$ and $S_C$ similarly.
Then, the desired computation is
$$|S| - |S_A \cup S_B \cup S_C|. \tag1 $$
$|S| = 3^6,$ because there are $3$ choices for each of the $6$ letter-positions.
By Inclusion-Exclusion Theory, you have that
$$|S_A \cup S_B \cup S_C|$$
$$= |S_A| + |S_B| + |S_C|$$
$$- \left( ~|S_A \cap S_B| + |S_A \cap S_C| + |S_B \cap S_C| ~\right)$$
$$= (3 \times 2^6) - (3 \times 1^6).$$
Therefore, the overall enumeration is
$$3^6 - \left[3 \times 2^6\right] + \left[3 \times 1^6\right].$$

Addendum
Responding to the comment question of Franklin.

where is my solution going wrong?

You are mismanaging the situation where only one letter is used.  The easiest way to see this is to
regard the computation of $~\displaystyle \binom{3}{1}~$ as actually representing the union of three sets.

The set consisting of the single word AAAAAA

The set consisting of the single word BBBBBB

The set consisting of the single word CCCCCC

For simplicity, I will examine the specific set containing the single word AAAAAA.  The consideration of the other two words will be similar.
The idea is that this specific unsatisfactory word, AAAAAA should be deducted once overall.
It is actually deducted twice:

Once in the computation of $|S_B|$.
Once in the computation of $|S_C|$.

Then, it is added back once, in the computation of $|S_B \cap S_C|$.  So, the net effect is that it is deducted twice and then added back once.
This type of analysis is the backbone of Inclusion Exclusion.  In order to have a deeper, more confident understanding of Inclusion-Exclusion theory, you need to study the two Inclusion Exclusion links that I provided at the start of my answer, more closely.