Density_2019_06_14.tex

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\date{June 14, 2019}


\newtheorem{thm}{Theorem}[section]
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\title
[]
{}
%\author[E.\ Rains]{E.\ Rains}
%\address{}
%\email{}
%\author[?]{?}
%\address{}
%\email{}
%\author[A.\ Silverberg]{A.\ Silverberg}
%\address{Department of Mathematics, University of California, Irvine, CA 92697, USA}
%\email{asilverb@uci.edu}
%\subjclass[2010]{??}
%\keywords{abelian varieties}
%\thanks{Support for the research was provided by the Alfred P.~Sloan Foundation
%and the National Science Foundation.}

\begin{document}


%\begin{abstract} 
%\end{abstract}



\maketitle

\section{Introduction}

Let $\AA_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. 

\begin{thm}
\label{mainthm}
Every rational function on $\AA_g$ that is constant on every isomorphism class of unpolarized abelian varieties is constant on every product of elliptic curves.
\end{thm}


\section{Definitions and notation}

\begin{defn}
Let $\SS_g$ denote the set of $(A,B) \in \AA_g \times \AA_g$ such that $A$ and $B$ are isomorphic as unpolarized abelian varieties.



Let $\SS_{2,1} = S_2 \cap \AA_1^4$, i.e., $\SS_{2,1}$ is the set of $(E_1,E_2,E_3,E_4) \in \AA_1^4$ such that $E_1\times E_2$ and $E_3\times E_4$ are isomorphic as unpolarized abelian varieties.
\end{defn}

\begin{defn}
For us, a {\em supersingular} elliptic curve will mean that its endomorphism algebra is a quaternion algebra.
\end{defn}

\begin{defn}
If $p$ is a prime, let $SS_p$ denote the set of supersingular points in $\AA_1^4$ (over $\overline{\F}_p$).

Let $SS \subset \AA_1^4$ denote the union of the sets $SS_p$, as the prime $p$ varies.
\end{defn}

\begin{defn}
If $p$ is a prime, let $T_p \subset \SS_{2,1} \subset \AA_1^4$ denote the set of $(E_1,E_2,E_3,E_4) \in \SS_{2,1} = S_2 \cap \AA_1^4$ such that the $E_i$ are ordinary (over $\overline{\F}_p$) and isogenous.
Let $T \subset \SS_{2,1} \subset \AA_1^4$ denote the union of the sets $T_p$, as the prime $p$ varies.

\end{defn}

\begin{defn}
Suppose that $m$ and $n$ are relatively prime positive integers.
Let $X_0(mn)$, as usual, be the moduli space of pairs $(E,\phi)$ such that $E \in \AA_1$
and $\phi$ is an isogeny on $E$ whose kernel is a cyclic group of order $mn$. 
View $X_0(mn)$ as a subset of $\AA_1^4$ via the map 
$(E,\phi) \mapsto (E,E/m\ker \phi,E/n\ker \phi,E/\ker \phi)$.

If $p$ is a prime, let $Y_p \subset \AA_1^4$ denote the union of the sets $X_0(mn) \subset \AA_1^4$, running over relatively prime positive integers $m$ and $n$ that are prime to $p$.
\end{defn}

Note that $Y_p \subset \SS_2$ via the map 
$$(E,E/m\ker \phi,E/n\ker \phi,E/\ker \phi) \mapsto (E \times (E/\ker \phi),(E/m\ker \phi) \times (E/n\ker \phi)).$$

If $X$ is a subset of $\AA_g^k$ for some positive integers $g$ and $k$, write $\overline{X}$ for the Zariski closure of $X$ in $\AA_g^k$.

\begin{defn}
By a {\bf square} we mean $(E,E_A,E_B,E_AB) \in \AA_1^4$ and a commutative diagram
$$
\xymatrix@C=15pt{
& E \ar[ld]_{\varphi_A}  \ar[rd]^{\varphi_B}  \\
E_A \ar[rd]_{\psi_B}  & &  \ar[ld]^{\psi_A} E_B \\
 & E_{AB}
}
$$
where  
the maps  
$\varphi_A$, $\varphi_B$, $\psi_A$, and $\psi_B$
are isogenies such that 
\begin{enumerate}
\item
$(\deg(\varphi_A),\deg(\varphi_B)) = 1$,
\item
$\ker \psi_B = \varphi_A(\ker \varphi_B)$,
\item
$\ker \psi_A = \varphi_B(\ker \varphi_A)$.
\end{enumerate}
\end{defn}



\section{Some background}

\begin{thm}[Deligne]
\label{DeligneSSthm}
If $g \ge 2$ and $E_1,\ldots,E_g,E_1',\ldots,E_g'$ are supersingular elliptic curves in characteristic $p$, then $E_1\times\cdots\times E_g$ and $E_1'\times\cdots\times E_g'$ are isomorphic over every extension over which all the endomorphisms are defined.
\end{thm}



\section{Some density results}

\begin{conj}
If $g > 1$, then the set $\SS_g$ is Zariski-dense in $\AA_g \times \AA_g$.
\end{conj}

The strategy for showing density (of $T$ in $\AA_1^4$?) is to first show that the CM points are dense in $X_0(mn)(\overline{\Q})$ for all relatively prime positive integers $m$ and $n$. (Perhaps the finite field case suffices?) Then show that the union of the images of the sets $X_0(mn)$ is dense (and thus CM points in there are dense). Note that the union of the sets $X_0(p)$ is dense in $(\P^1)^2$ since it's a hypersurface of degree that goes to infinity, so the set of tuples $(E,E/p\ker \phi,E/q\ker \phi)$ is dense in $(\P^1)^3$ (the closure of the union has dimension at least 3, since it's dense or a hypersurface).




\begin{prop}
\label{S2A2}
\begin{enumerate}
\item
$\overline{SS} = \AA_1^4$.
\item
$\overline{\SS_2} = \AA_2 \times \AA_2$.
\end{enumerate}
\end{prop}

\begin{proof}
The key point is that anything that vanishes on $SS_p$ has multidegree at least ${\frac{p-1}{12}}$ on each factor.
Then vary $p$ to get that $\overline{SS} = \AA_1^4$.

We have $SS_p \subset \SS_{2,1}$, and the set $SS_p$ has multidegree ${\frac{p-1}{12}}$ (in the stacky sense, i.e., adjoin level 5 and level 7 structure, or any level $\ell$ structure for $\ell \ge 5$; then the number of points is ${\frac{p-1}{12}}|\Sp_2(\F_\ell)|$; this number might not be an integer, but that's OK since it's a Deligne-Mumford stack so it's nice...).
It follows that any function that vanishes on $\SS_{2,1}$ has multidegree at least ${\frac{p-1}{12}}$. This holds for all primes $p$. Over $\Z$, this gives a lower bound on the degree of any function $f$ that vanishes on $\SS_{2,1}$.
Varying $p$ shows that no such function exists, giving that $\SS_2$ is Zariski-dense in $\AA_2 \times \AA_2$.
\end{proof}




\begin{prop}
\label{TYp}
$\overline{T_p} = \overline{Y_p}$.
\end{prop}

\begin{proof}
It suffices to show that $X_0(mn) \cap T_p$ is dense in $X_0(mn)$.
This holds since if $x \in X_0(mn)(\bar{k})$ is such that the corresponding elliptic curve $E$ is not supersingular, then $x\in T$ (since $T$ consists of ordinary elliptic curves). If $x \in X_0(mn)(\bar{k})$ is geometrically CM (to avoid supersingular), then the $mn$ isogeny comes from an ideal $\a_{mn}$. If $\gcd(m,n)=1$, then $\a_{mn}=\a_{m} \times \a_{n}$. This gives the square. So $x \in T$. 
Can construct a square from any point in $X_0(mn)$. A CM point in $X_0(mn)$ comes nominally from $T$. 

The point is that if $x \in X_0(mn)({\F_q})$, this gives a cyclic $mn$-isogeny whose domain is CM, since it's over $\F_q$ (if you exclude the finitely many supersingular points). Isogenies correspond to ideals in $\End(E)$ (modulo whether $\End(E)$ is maximal...). Thus $x \in T(\F_q) \subset T_p$.

These $x$ are dense (since they're all but finitely many of the points in $\overline{\F_q}$).

We now have that $X_0(mn) \cap \overline{T_p}$ is the fiber over $p$ of the $\Z$-scheme $X_0(mn)$.

The above shows that $\overline{T_p} = \overline{Y_p}$.
\end{proof}



\begin{prop}
\label{SSYp}
$SS_p \subset \overline{Y_p}$.
\end{prop}

\begin{proof}
\end{proof}

\begin{prop}
\label{TA1}
$\overline{T} = \AA_1^4$.
\end{prop}

\begin{proof}
Suppose $p$ is a prime and $h$ is an algebraic function that vanishes on $T_p$.
Then $h$ vanishes on $\overline{T_p}$, so $h$ vanishes on $\overline{Y_p}$ by Proposition \ref{TYp}, so $h$ vanishes on $SS_p \subset \overline{Y_p}$ by Proposition \ref{SSYp}, giving that $h$ has multidegree at least ${\frac{p-1}{12}}$ on each factor.
Varying $p$ and applying Proposition \ref{S2A2}(i) gives the desired result.
\end{proof}

(The next result isn't used in the proof of the main results.)

\begin{cor}
\label{SS21A14}
$\overline{\SS_{2,1}} = \AA_1^4$.
\end{cor}

\begin{proof}
This follows from Proposition \ref{S2A2}(i) since $SS \subset \SS_{2,1}$.
\end{proof}


\section{Proof of Theorem \ref{mainthm}}


One can hopefully reduce everything to the case $n=2$.

The next result (which isn't really used) says that if $\invar(E_1,\ldots,E_g)$ is an isomorphism invariant and is ``algebraic'', then $\invar$ factors through $j$.

\begin{prop}
Suppose $\invar(E_1,\ldots,E_g)$ is an isomorphism invariant and is ``algebraic''. Then there exists $F \in k(x_1,...,x_g)^{S_g}$ such that
$$
\invar(E_1,\ldots,E_g) = F(j(E_1),\ldots,j(E_g))
$$
for {\em all} tuples $(E_1,\ldots,E_g)$ of elliptic curves over $k$.
\end{prop}

\begin{proof}
If $\invar(E_1,\ldots,E_g)$ is an isomorphism invariant and is ``algebraic'', then it will work for all tuples $(E_1,\ldots,E_g)$ (not just ordinary isogenous ones). For generic $E_1,\ldots,E_g$, the product $E_1\times \ldots\times E_g$ has a unique principal polarization, so $E_1\times \ldots\times E_g$ and $E_1'\times \ldots\times E_g'$ are isomorphic if and only if the unordered sets $\{ E_1,\ldots,E_g\}$ and $\{ E_1',\ldots,E_g'\}$ are the same.
\end{proof}

So for ordinary isogenous elliptic curves, if $E_1\times E_2$ and $E_1'\times E_2'$ are isomorphic (as unpolarized abelian varieties), then 
$$
F(j(E_1),j(E_2),j(E_3),\ldots,j(E_g)) = F(j(E_1'),j(E_2'),j(E_3),\ldots,j(E_g)).
$$
However over $\overline{\Q}$, the sets $\{ \{ j(E_1),j(E_2)\},\{j(E_3),j(E_4)\} : E_1\times E_2 \cong E_3\times E_4 \}$
are Zariski-dense in $\Sym^2(\P^2)$.

(We will consider $F(x,y)-F(z,w)$ as a function on $(\P^1)^2 \times (\P^1)^2$. It vanishes on a dense set, so it is the zero map, so $h$ is constant.)


Theorem \ref{mainthm} in the case $g=2$ is an immediate consequence of the following result.

\begin{thm}
\label{mainthmT}
If $\invar : \AA_g \to \P^1$ is a rational function that is constant on each isomorphism class of unpolarized abelian varieties of the form $\prod_{i=1}^g E_i$ for which the $E_i$ are isogenous and ordinary, then $\invar$ is constant on $\AA_1^g$.
\end{thm}

\begin{proof}
(This is just for $g=2$.)
Define $h : \AA_1^4 \to \P^1$ by
$$
h(E_1,E_2,E_3,E_4) = \invar(E_1\times E_2)-\invar(E_3\times E_4).
$$
The hypotheses imply that $h$ vanishes on $T$. So $h$ vanishes on $\AA_1^4$, by Proposition \ref{TA1}. Thus, $\invar$ is constant on $\AA_1^2 \subset \AA_2$.
\end{proof}




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