\documentstyle{article} \author{Ehud Hrushovski \footnote{Hebrew University at Jerusalem. Supported by the Miller Institute, University of California, Berkeley} \\ Thomas Scanlon \footnote{Mathematical Sciences Research Institute. Supported by an NSF MSPRF}} \title{Lascar and Morley ranks differ in differentially closed fields} \renewcommand{\baselinestretch}{1.4} \setlength{\topmargin}{-0.7cm} \setlength{\textwidth}{14.8cm} \setlength{\textheight}{23cm} \setlength{\oddsidemargin}{0.2cm} \setlength{\parindent}{0mm} \setlength{\rightmargin}{0cm} \setlength{\leftmargin}{0cm} %----- \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{cor}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{question}[theorem]{Question} \newtheorem{definition}[theorem]{Definition} \newtheorem{ntn}[theorem]{Notation} \newtheorem{example}[theorem]{Example} \newtheorem{rem}[theorem]{Remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{problem}[theorem]{Problem} \newtheorem{construction}[theorem]{Construction} \newtheorem{exercise}[theorem]{Exercise} \newcommand{\proof}{\noindent \it Proof \hbox{ }} \newtheorem{fact}[theorem]{Fact} \newtheorem{claiM}[theorem]{Claim} \newcommand{\claim}{{\noindent \bf Claim }} \newcommand{\REM}{\begin{remark} \ \end{remark} } %------- %\font\msbmten cmtt10 \font\msbmten msbm10 \def \Bbb#1{\hbox{\msbmten {#1} }} \def \iso \simeq \def\acl{{\rm acl}} \def\mod{{\rm mod}} \def\implies{\Rightarrow} \def\si{\sigma} \def\k{\bf \it k} \def\proof{ { {\noindent \it Proof} \hbox{ } } } \def\ms{\medskip} \def\tensor{{\otimes}} \def\meet{\cap} \def\union{\cup} \def\iso{\simeq} \def\acl{{\rm acl }} \def\Pp {\Bbb P} \def\Aa {\Bbb A} \def\Cc{\Bbb C} \def\Uu{\Bbb U} \def\Qq{\Bbb Q} \def \tQ{\tilde{\Bbb Q}} \def\ti{\tilde} \def\AND{\ \wedge \ } \def\OR{\ \vee \ } \def\tp{ \rm{tp}} \def\<{\begin} \def\>{\end} \begin{document} \maketitle We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. \S~\ref{DefMR} consists of some general lemmas relating the above issues. \S~\ref{nondef} points out a family of sets of finite Morley rank, whose Morley rank exhibits discontinuous upward jumps. To make the base of the family itself have finite Morley rank, we use a theorem of Buium. We thank John Baldwin, Anand Pillay, and Wai Yan Pong for reading an earlier version of this note and suggesting improvements. \<{section} {Definability of Morley rank} \label{DefMR} We will say that Morley rank is \emph{definable} (respectively \emph{upward}, resp. \emph{downward semi-definable}) if for every set of parameters $A$ and $A$-definable family of definable sets $E_b (b \in B)$ and $b \in B$, there is an $A$-definable set $B' \subseteq B$ such that $b \in B'$ and $MR(E_{b'})= MR(E_b) $ (resp. $\geq$, $\leq$) for $b' \in B'$. \<{lemma} \label{defMR} Let $T$ be a theory of finite Morley rank. If Morley and Lascar ranks coincide on definable sets, then Morley rank is downward semi-definable. \>{lemma} \proof Suppose Morley rank is not downward semi-definable, and let $E_b$ ($b \in B$) be an $A$-definable family demonstrating this. That is, there is some $b^* \in B$ so that for every $A$-definable set $B' \subseteq B$ with $b \in B'$ there is some $b' \in B'$ with $MR(E_{b'}) > MR(E_{b^*})$. Replace $B$ with an $A$-definable set of minimal Morley rank and degree containing $b$; so that now $MR(\tp (b^*/A)) = MR(B) := m$. Let $d := MR(E_{b^*})$. For $b \in B$, let $E'_b := {E_b}^m$. Then $MR(E'_b) = mMR(E_b)$. So it is always a multiple of $m$. We have $MR(E'_{b^*})=md$, while for many $b'$, $MR(E'_{b'}) \geq md+m$. For $B' \subset B$, let $$X_{B'} := \{(e,b): b \in B', e \in E'_b \}$$ and $X := X_B$. Replace $B$ with some $A$-definable $B' \subseteq B$ with $b^* \in B'$ and the property that for every $A$-definable $B'' \subseteq B'$ with $b^* \in B''$, $(MR(X_{B''}), dM(X_{B''}) ) = (MR(X_{B'}), dM(X_{B'}) )$. If $(e,b) \in X$ and $b$ is not generic in $B$, then $b \in B'$ for some $A$-definable subset of $B$ with $MR(B') < MR(B)$ and $(e,b) \in X_{B'}$. Since $X_{B \setminus B'}$ and $X_B$ have the same Morley rank and degree by the above reduction, $MR(X_B') < MR(X)$; so, $MR(\tp(e,b/A)) < MR(X)$. On the other hand, if $(e,b) \in X$, and $b \in B$ is generic, then $U(\tp(e,b/A)) \leq md+m$. But $MR(E'_b) \geq md+m$ for infinitely many $b \in B$; so $MR(X) > md+m$. Thus $U(\tp(e,b/A)) < MR(X)$ for any $(e,b) \in X$. So $U(X) < MR(X)$, a contradiction. \>{section} \<{section}{A non-definable family } \label{nondef} We now work with differential fields of characteristic $0$, and fix a universal domain $\Uu$ (a saturated differentially closed field.) Our plan is to produce a finite rank definable family of abelian varieties whose Manin kernels exhibit non-definable jumps in Morley rank. One difficulty is that there does not exist a definable family of abelian varieties containing a copy of every abelian variety of a given dimension. However, there are definable families of abelian varieties containing isomorphic copies of every \emph{principally polarized} abelian variety of a given dimension. For every abelian variety $A$ there is another abelian variety $\check{A}$, called the dual abelian variety, which parametrizes the line bundles on $A$ algebraically equivalent to zero. A polarization is an isogeny $\lambda: A \rightarrow \check{A}$. A polarization is principal if it is an isomorphism. A \emph{principally polarized abelian variety} is an abelian variety $A$ given together with a principal polarization $\lambda: A \rightarrow \check{A}$. Not all abelian varieties admit a principal polarization, but elliptic curves always do. \<{theorem}[\cite{GIT} VII \S2]\label{universal} Let $L$ be an algebraically closed field, and $g$ a positive integer. There exists a definable family $\{ (A_b, \lambda_b) : \ b \in F \}$ of $g$-dimensional principally polarized abelian varieties, such that every principally polarized $g$-dimensional abelian variety over $L$ is isomorphic to some $(A_\alpha, \lambda_\alpha)$. \>{theorem} Let $E_t$ be an elliptic curve with $j$-invariant $t$. Given $t, t'$, let $E(t,t') := E_t \times E_{t'}$. Given also an integer $n$, there exist group-theoretic isomorphisms $\iota $ between the finite $n$-torsion subgroups of $E_t$ and of $E_{t'}$. Let $E(t,t', \iota, n)$ be the quotient of $E_t \times E_{t'}$ by the graph of $\iota$. Then $A := E(t,t',\iota, n)$ is an Abelian variety of dimension $2$. When $E_t,E_{t'}$ are not isogenous, $A$ has precisely two connected definable subgroups of Morley rank 1, namely the images of $E_t$ and of $E_{t'}$. Their intersection has order $n^2$. For a general choice of $\iota$, $E(t,t', \iota, n)$ need not admit a principal polarization. However, if we choose $\iota$ to be anti-symplectic (ie $\langle \iota(x) , \iota (y) \rangle_{E_{t'}} = \langle y, x \rangle_{E_{t}}$ where $\langle \cdot, \cdot \rangle$ is the Weil pairing (See~\cite{Mil} \S 16 for the general theory of the Weil pairing) then $E(t,t', \iota, n)$ is self-dual. \<{lemma} \label{ppav} If $\iota : E_t[n] \rightarrow E_{t'}[n]$ is an anti-symplectic isomorphism of the $n$-torsion points, then $A := E(t,t',\iota, n)$ has a natural principal polarization. \>{lemma} \proof Since $\iota$ is an anti-symplectic map, the graph of $\iota$ is isotropic for the pairing on $E_t \times E_{t'}$ : \begin{eqnarray*} \langle (x,\iota(x)), (y, \iota(y)) \rangle_{E_t \times E_{t'}} & = & \langle x, y \rangle_{E_t} \cdot \langle \iota(x), \iota(y) \rangle_{E_{t'}} \\ & = & \langle x, y \rangle_{E_t} \langle y, x \rangle_{E_{t}} \\ & = & 1 \end{eqnarray*} Since $\# (E_t \times E_{t'}) [n] = n^4$ and the pairing is perfect, a maximal isotropic space has size $n^2$ which is the size of the graph of $\iota$. Hence, the graph of $\iota$ is a maximal isotropic subspace. The lemma now follows from the more general lemma: \<{lemma} \label{ppiso} Let $A$ be a principally polarized abelian variety identified with its dual via the polarization. Let $\Gamma \subseteq A[n]$ be a maximal isotropic subgroup of the $n$-torsion subgroup of $A$. Let $B = A/\Gamma$. Then $B$ also admits a principal polarization. \>{lemma} \proof Let $\pi: A \to B$ be the quotient map. Let $\phi: B \to A$ be defined by $b \mapsto [n] a$ where $a$ is any choice of a pre-image of $b$ under $\pi$. $\phi$ induces a dual exact sequence $$ 0 \longrightarrow \textrm{Hom} (\ker{\phi}, \mu_n) \longrightarrow A \longrightarrow \check{B} \longrightarrow 0$$ where $\mu_n$ is the group of $n$-th roots of unity (See~\cite{Mum} III \S 15). The kernel of $\phi$ is $A[n]/\Gamma$. Since $\Gamma$ is isotropic, the pairing on $A[n]$ descends to a pairing $(A[n]/\Gamma) \times \Gamma \rightarrow \mu_n$. Since the pairing is perfect and $\# A[n] = (\# \Gamma)^2$, via this pairing $\Gamma = \textrm{Hom} (A[n]/\Gamma, \mu_n)$. Thus, the above exact sequence is $$0 \longrightarrow \Gamma \longrightarrow A \longrightarrow \check{B} \longrightarrow 0 $$ That is, $\check{B}$ is the quotient $A/\Gamma = B$. \<{lemma}\label{simp} Let $F'$ be a Zariski (resp. Kolchin) closed subset of $F$, the definable parameter space for two dimensional principally polarized abelian varieties of Theorem~\ref{universal}. Assume $F'$ has a Zariski (resp. Kolchin) dense subset $\{t_1,t_2,\ldots\}$, such that $A_{t_n}$ is isomorphic to some $E(t,t',\iota, n )$ with $\iota$ anti-symplectic as above. Then for generic $t \in F'$, $A_t$ is a simple abelian variety. \>{lemma} \proof Otherwise, a generic $A_t$ contains two elliptic curves. Their intersection is necessarily finite, say of order $m$. But then infinitely many $A_{t_n}$ must contain two elliptic curves with intersection of order $m$. For $n>m$, this contradicts the remarks above. At this point it is quite easy to see that exists in DCF$_0$ a definable family of definable sets, whose generic element is strongly minimal, but with densely many sets of Morley rank 2. Thus: \<{cor} In $DCF_0$, Morley rank is not downwards semi-definable. \>{cor} However, since $DCF_0$ does not have finite Morley rank, lemma \ref{defMR} does not directly apply. At this point we quote a theorem from \cite{Bui}. \<{theorem}[Buium~\cite{Bui}]\label{buium} Let $(A, \lambda)$ be any principally polarized abelian variety of maximal $\delta$-rank. There exists a definable family $\{ (A_t, \lambda_t): \ t \in F_1 \}$, containing a definably isomorphic copy of every principally polarized abelian variety isogenous to $A$, and such that $F_1$ has finite Morley rank. \>{theorem} We leave the notion of \emph{$\delta$-rank} undefined here since we need only the facts that: \begin{itemize} \item A generic elliptic curve has maximal $\delta$-rank. \item The property of having maximal $\delta$-rank is isogeny invariant. \item The product of two abelian varieties each of maximal $\delta$-rank is also of maximal $\delta$-rank. \end{itemize} It seems likely that the $\delta$-rank condition is unnecessary in Buium's theorem, but we leave this issue aside. \<{cor} There exists a finite Morley rank definable subset $Y$, such that Morley rank is not downwards semi-definable. \>{cor} \proof Pick $t,t'$ algebraically independent over $\k$, the field of differential constants of $\Uu$. Let $J_t, J_{t'}$ be elliptic curves with $j$-invariants $t, t'$. Let $A := J_t \times J_{t'}$, and let $F_1$ be a family as guaranteed to exist by Theorem~\ref{buium}. Given $n$, pick $c=c(n) \in F_1$ with $A_c$ isomorphic to $E(t,t', \iota, n)$. Let $F_2$ be the Kolchin closure of the set $\{c(1),c(2),...\}$. Let $b$ be a generic element of $F_2$. By Lemma~\ref{simp}, $A_{b}$ is a simple Abelian variety. If $A_b$ were isogenous to an Abelian variety defined over $\k$, this would be guaranteed by a certain formula true of $b$, and the same formula would hold of infinitely many $c(n)$; hence $A$ would also have this property, contradicting the choice of $t, t'$. Thus $A_b$ is a simple, non-isotrivial Abelian variety. For $t \in F_2$, let $M_t$ be the Manin kernel of $A_t$. $M_t$ is uniformly definable over $t$ (cf. \cite{hru-ita}). Then (cf. \cite{hru-sok}) $M_t$ has Morley rank 1 for generic $t \in F'$ (when $A_t$ is a nonisotrivial simple Abelian variety.) But it has Morley rank 2 for each $t=c(n)$ (when $A_t$ is isogenous to a product of elliptic curves.) Thus Morley rank is not downward semi-definable in $Y = \{(a,t) : t \in F^\#, a \in M_t \}$. \<{cor} Morley and Lascar rank do not agree on definable sets in DCF$_0$. \>{cor} \proof Since $Y$ has finite Morley rank, with the structure induced from the ambient differentially closed field, Lemma~\ref{defMR} applies. \<{question} \ \>{question} Marker and Pillay have noted that on $0$-definable sets of differential order $2$, Lascar and Morley ranks are the same. Examples similar to the one produced above have order at least $5$. Is there a theorem responsible for this gap? \>{section} \begin{thebibliography}{[99]} \bibitem{Bui} {\sc A. Buium}, Geometry of differential polynomial functions III: Moduli spaces, \emph{American Journal of Mathematics}, 117 (1995), no. 1, 1--73. \bibitem{hru-ita} {\sc E. Hrushovski} and {\sc Itai M.}, On model complete differential fields, preprint, 27 November 1997. \bibitem{hru-sok} {\sc E. Hrushovski} and {\sc \v{Z}. Sokolovi\'{c}}, Minimal subsets of differentially closed fields, \emph{Transactions of the American Mathematical Society}, to appear. \bibitem{Mil} {\sc J. S. Milne}, Abelian varieties, in Arithmetic Geometry, {\sc G. Cornell} and {\sc J. Silverman}, eds., Springer-Verlag, New York, 1986. \bibitem{Mum} {\sc D. Mumford}, Abelian Varieties, 1970, Oxford University Press, Oxford. \bibitem{GIT} {\sc D. Mumford}, {\sc J. Fogarty}, and {\sc F. Kirwan}, Geometric Invariant Theory, 3rd ed., 1994, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, New York. \end{thebibliography} \>{document}