\batchmode \documentstyle[12pt]{article} \title{Integral points of abelian varieties over function fields of characteristic zero} \author{Alexandru Buium and Jos\'e Felipe Voloch} \date{\ } \begin{document} \maketitle \vspace{3 mm} This paper studies the question of integral points on affine open subsets of Abelian varieties over function fields of characteristic zero. Siegel [Sie] has shown that an affine algebraic curve of genus at least one defined over a number field, has only finitely many integral points. Lang [L] has proven an analogous result for curves defined over a function field of characteristic zero, not defined over the constant field. For higher dimensions, in the classical case of number fields, Lang has conjectured and Faltings [F] proved that for $A$ an abelian variety over the number field $K$, if $V$ is an affine open subset of $A$ and $S$ is a finite set of places of $K$, then the set of $S$-integral points of $V$ is finite. Faltings has also a quantitative (but non-effective) result. Parshin [P] also obtained a result for function fields of characterisic zero, under the hypothesis that the complement of $V$ does not contain translates of abelian subvarieties of $A$. In this paper, we remove this hypothesis and also obtain a quantitative result. Our main theorem is a boundedness result for the local height associated to a subvariety of an abelian variety defined over a discrete valuation field with residue characteristic zero; it is a higher dimensional analogue of a result of Manin [M] about elliptic curves. Our method is quite different from those in [M] and [P]; it is based on a differential algebraic argument from [B] plus an argument involving formal groups. {\it Acknowledgements.} The first author wishes to express his gratitude to the Humboldt Foundation for support during 1992-93 and to the University of Essen, Germany, for hospitality. \vspace{3 mm} Assume $K$ is any field and $v:K_{a} \rightarrow (- \infty, \infty] $ is a real valuation on its algebraic closure. For any projective variety $V$ over $K$ and any closed subscheme $X \subset V$ we dispose of a function $\lambda_{v}(X,.):V(K_{a}) \rightarrow [0,\infty] $ which satisfies the following property: for any affine open set $U \subset X$ and any system of generators $g_{1},...,g_{m} \in {\cal O}(U)$ of the ideal defining $X \cap U$ in $U$, we may write $ \lambda_{v}(X,P) = min\{v(g_{1}(P)),..., v(g_{m}(P))\}+b(P)$ with $b$ bounded on any bounded subset of $U(K_{a})$. The function $\lambda_{v}(X,.)$ is uniquely determined by the above property up to the addition of a bounded function and is called the {\it local height function} associated to $X$. This notion is developed in detail in [Sil]. Let us fix an algebraically closed ground field $k$ of characteristic zero. All valuations are assumed to be trivial on $k$. Here are our main results: \vspace{3 mm} \proclaim Theorem. Assume $K$ is a field extension of $k$. Let $A$ be an abelian variety over $K$ with $K/k$ trace zero. Let $X \subset A$ be a closed subvariety and let $\Gamma \subset A(K_{a})$ be a finite rank subgroup. Let $v$ be a real valuation on $K_{a}$ which is discrete on $K$. Then $\lambda_v (X,P)$ is bounded for $P$ in $\Gamma \setminus X$. \proclaim Corollary. Assume $K$ is a function field over $k$. Let $A$ be an abelian variety over $K$ with $K/k$ trace zero. Let $X \subset A$ be an ample divisor. Then for any finite set $S$ of places of $K$, the set of $S$-integral points of $A \setminus X$ is finite. {\it Proof of Corollary:} The height $h(P)$, for a $S$-integral point of $A \backslash X$, is the sum of $\lambda_v (X,P)$ over the elements of $S$, and it follows that the height is bounded, which proves the corollary. \vspace{3 mm} {\it Remarks.} 1) A few comments about terminology. We say that $A$ has $K/k$ {\it trace zero} if $A \otimes _{K} K_{a}$ contains no abelian subvariety defined over $k$. We say that $\Gamma$ is of {\it finite rank} if $\Gamma \otimes_{\bf Z} {\bf Q}$ has finite dimension over $\bf Q$. So we do not assume that $\Gamma$ in the Theorem is finitely generated; in particular if we let $K$ in the Theorem be a function field over $k$, we may take $\Gamma$ to be the group of division points of $A(K)$ i.e. the group of all points $P \in A(K_{a})$ for which there exists a non-zero integer $n_{P}$ with $n_{P}P \in A(K)$. 2) What the Theorem says is that there is no sequence in $\Gamma \backslash X$ converging $v-$adically to $X$ (a sequence $P_{n}$ in a projective variety is said to {\it converge $v-$adically} to $X$ if $\lambda_{v}(X,P_{n})$ goes to infinity as $n$ goes to infinity.) It is worth recalling here that even if $K$ in the Theorem is $v-$adically complete, $A(K)$ is not compact and a sequence in $A(K)$ converging $v-$adically to $X$ may have no subsequence converging to a point of $X$. \vspace{5 mm} To prove the Theorem we need some preparation. Let $F$ be any field with a real valuation $v$. \vspace{3 mm} \noindent 1. We will need to talk about convergence of sequences of $F-$points of arbitrary (e.g. infinite dimensional) affine schemes. This is somewhat non-standard so we prefer to ``recall'' the definitions. Let $U=Spec\ R$ be an arbitrary affine $F-$scheme. A sequence $P_{n} \in U(F)$ will be called bounded (or bounded in $U$ if there is any danger of confusion) if there exists a family $(f_{i})$ of $F-$algebra generators of $R={\cal O}(U)$ such that for each $i$ the sequence of real numbers $v(f_{i}(P_{n}))$ is bounded from below. If this holds then it is trivial to check that for any $f \in {\cal O}(U)$ the sequence $v(f(P_{n}))$ is bounded from below. Let $X \subset U$ be a closed subscheme defined by an ideal $I$. A sequence $P_{n} \in U(F)$ is said to converge $v-$adically to $X$ (in $U$, if there is any danger of confusion) if it is bounded and there exists a family $(g_{j})$ of generators of $I$ with the property that for each $j$, $v(g_{j}(P_{n}))$ goes to infinity as $n$ goes to infinity. If this holds then it is trivial to check that for any $g \in I$, $v(g(P_{n}))$ goes to infinity as $n$ goes to infinity. It is also trivial to check that if $P_{n}$ converges $v-$adically to $X$ then it also converges $v-$adically to $X_{red}$. \vspace{3 mm} \noindent 2. Let $U$ be an affine $F-$scheme and $X,Y \subset U$ closed subschemes. If a sequence $P_{n} \in U(F)$ coverges $v-$adically to both $X$ and $Y$ then it converges $v-$adically to $X \cap Y$. Let $\pi:U \rightarrow U'$ be a morphism of affine $F-$schemes and let $X \subset U$ and $X' \subset U'$ be closed subschemes such that $\pi(X) \subset X'$ set-theoretically. Let $P_{n} \in U(F)$ be a sequence converging $v-$adically to $X$. Then $\pi(P_{n})$ converges $v-$adically to $X'$. \vspace{3 mm} \noindent 3. Now we recall (and give some complements to) a construction done in [B]. Let $\delta$ be any derivation on $F$. Then for any $F-$scheme $V$ there exists a pair $(jet_{\infty}(V), \tilde{\delta})$ where $jet_{\infty}(V)$ is a $V-$scheme and $\tilde{\delta}$ is a derivation on ${\cal O}_{jet_{\infty}(V)}$ prolonging $\delta$, having the following universality property: for any pair $(W,\partial)$ consisting of a $V-$scheme $W$ and of a derivation $\partial$ of ${\cal O}_{W}$ which prolongs $\delta$ there exists a unique horizontal morphism of $V-$schemes $W \rightarrow jet_{\infty}(V)$. Here {\it horizontal} means commuting with the corresponding derivations. Note that $jet_{\infty}(V)$ is a ``huge'' scheme; e.g. it is always infinite dimensional for $V$ an algebraic variety over $F$, of positive dimension. If $U \subset V$ is an open subscheme then $jet_{\infty}(U)$ naturally identifies with the inverse image of U in $jet_{\infty}(V)$. If $G$ is any algebraic $F-$group then $jet_{\infty}(G)$ has a natural structure of $F-$group scheme. \vspace{3 mm} \noindent 4. It is particularly useful to see how $jet_{\infty}(V)$ looks like in case \[V=Spec\ F[y_{1},...,y_{N}]/(g_{1},...,g_{m})\] is an affine algebraic scheme. Indeed it follows from the universality property that we have \[jet_{\infty}(V)=Spec\ F\{y_{1},...,y_{N}\}/ [g_{1},...,g_{m}]\] where $ F\{y_{1},...,y_{N}\}$ is the ``ring of $\delta-$polynomials'' and $[g_{1},...,g_{m}]$ is the ``$\delta-$ideal generated by $g_{1},...,g_{m}$''. Recall that by definition the {\it ring of $\delta-$polynomials} is the usual ring of polynomials with coefficients in $F$ in the infinite family of variables $(y_{ij}),\ 1 \leq i \leq N,\ j \in {\bf N}$ equipped with the unique derivation which prolongs $\delta$ and sends each $y_{ij}$ into $y_{i,j+1}$ (call this unique derivation $\tilde{\delta}$). We always identify $y_{i0}$ with $y_{i}$; in particular $g_{1},...,g_{m}$ are viewed as elements in the ring of $\delta-$polynomials. Now the ``$\delta-${\it ideal generated by} $g_{1},...,g_{m}$'' is by definition the ideal generated by the infinite family $(\tilde{\delta}^{j}g_{p}),\ 1 \leq p \leq m,\ j \in {\bf N}$. The description above shows that if $X \subset V$ is a closed subscheme of an $F-$variety then $jet_{\infty}(X)$ identifies with a closed horizontal subscheme of $jet_{\infty}(V)$; here {\it horizontal} means ``whose ideal sheaf is preserved by $\tilde{\delta}$''. \vspace{3 mm} \noindent 5. Let $V$ be any $F-$scheme. Then by the universality property, any $F-$point $P$ of $V$ lifts to an $F-$point $jet_{\infty}(P)$ of $ jet_{\infty}(V)$. Explicitly, if $V$ is as in (4) and $P$ is defined by $ y_{i} \mapsto \alpha_{i} \in F$ then $jet_{\infty}(P)$ is defined by $y_{ij} \mapsto \delta^{j} \alpha_{i} \in F$. Assume $U$ is an affine $F-$variety and let $f \in {\cal O}(U)$ be a regular function. Denote the image of $f$ in $ {\cal O}(jet_{\infty}(U))$ by the same letter $f$. We may consider the element $\tilde{\delta}^{j}f \in {\cal O}(jet_{\infty}(U))$ and evaluate it at the $F-$point $jet_{\infty}(P)$. On the other hand we may evaluate $f$ at $P$ and then take the $j-$th derivative in $F$. The description in (4) and the explicit form of $jet_{\infty}(P)$ given above show that what we obtain in both cases is the same: $(\tilde{\delta}^{j}f)(jet_{\infty}(P))=\delta^{j}(f(P))$. \vspace{3 mm} \noindent 6. Assume now $\delta$ is a bounded derivation on $F$, by which we mean that $v(\delta x) \geq v(x)$ for all $x \in F$. Let $U$ be an affine $F-$variety and $P_{n} \in U(F)$ a sequence of points. We claim that if $P_{n}$ is bounded in $U$ then $jet_{\infty}(P_{n})$ is bounded in $jet_{\infty}(U)$. Indeed by the description in (4), ${\cal O}(jet_{\infty}(U))$ is generated by elements of the form $\phi_{ij}:=\tilde{\delta}^{j}f_{i}$ with $f_{i} \in {\cal O}(U)$ and $j \in {\bf N}$. By the formula at the end of (5) we get for each $i$ and $j$: \[v(\phi_{ij}(jet_{\infty}(P_{n}))) =v(\delta^{j}(f_{i}(P_{n}))) \geq v(f_{i}(P_{n})) \] and our claim is proved. We also claim that if $X \subset U$ is a closed subscheme and if $P_{n}$ converges $v-$adically to $X$ then $jet_{\infty}(P_{n})$ converges $v-$adically to $jet_{\infty}(X)$. Indeed, by (4) again, the ideal defining $jet_{\infty}(X)$ in $jet_{\infty}(U)$ is generated by elements of the form $\psi_{ij}:=\tilde{\delta}^{j}g_{i}$ with $g_{i}$ in the ideal defining $X$ and we conclude exactly as above. \vspace{3 mm} The next result is an ``approximation analogue'' of Lang's conjecture on intersections of subvarieties of abelian varieties with finite rank subgroups. A similar result was also obtained by E. Hrushovski, (personal communication) using different methods. \vspace{3 mm} \proclaim Proposition 7. Let $F$ be an algebraically closed extension of $k$, $v$ a real valuation on $F$, and $\delta$ a bounded derivation on $F$ whose field of constants is $k$ (i.e. $Ker \ \delta = k$). Let $A$ be an abelian variety over $F$ with $F/k$ trace zero, let $X \subset A$ be a closed subvariety and $\Gamma \subset A(F)$ a finite rank subgroup. Then there exists in $X$ a finite union $Y$ of translates of abelian subvarieties with the property that any sequence $P_n$ in $\Gamma$ converging $v$-adically to $X$ also converges $v-$adically to $Y$. \vspace{3 mm} {\it Proof.} Consider the scheme $jet_{\infty}(A)$, cf. (3) and let $\pi:jet_{\infty}(A) \rightarrow A$ be the canonical projection. Theorem 2 in [B] says that there is a horizontal, irreducible, closed $F-$subgroup scheme $H \subset jet_{\infty}(A)$ which is of finite type over $F$ such that for any $P \in \Gamma$ we have $jet_{\infty}(P) \in H(F)$. So we dispose of two closed subschemes $H$ and $jet_{\infty}(X)$ in $jet_{\infty}(A)$. Let $Z$ be their scheme-theoretic intersection and let $Y$ be the Zariski closure of $\pi(Z)$ in $A$. Then Theorem 1 in [B] says that any variety of general type dominated by a component of $Y$ must have its Albanese variety descending to $k$. This plus our trace hypothesis easily implies (see the last page of [B] for the argument) that $Y$ is a union of translates of abelian subvarieties. Now $A$ can be covered by finitely many affine open sets $U_{i}$ such that the sequence $P_{n}$ is a union of subsequences $P_{in}$, each contained in the corresponding $U_{i}$ and bounded in $U_{i}$. So to prove the Proposition we may assume there is an open affine subset $U \subset A$ such that $P_{n} \in \Gamma \cap U(F)$ converges $v-$adically to $X \cap U$ in $U$ (in the sense of (1)) and we have to check that $P_{n}$ also converges $v-$adically to $Y \cap U$ in $U$. Now by (6), $jet_{\infty}(P_{n})$ converges $v-$adically to $jet_{\infty}(X \cap U)=jet_{\infty}(X) \cap jet_{\infty}(U)$ in $jet_{\infty}(U)$. On the other hand $jet_{\infty}(P_{n})$ converges $v-$adically to $H \cap jet_{\infty}(U)$ (because it is bounded and contained in it). By (2), $jet_{\infty}(P_{n})$ converges $v-$adically to $Z \cap jet_{\infty}(U)$. By (2) again, $\pi(jet_{\infty}(P_{n}))= P_{n}$ converges $v-$adically to $Y \cap U$ and we are done. \vspace{3 mm} Actually we proved more. Assume we are in the hypotheses of the Proposition. Let $\Gamma^{*} \subset A(F)$ be the $\delta-$closure of $\Gamma$ (cf. [B], p.560 for the definition of ``$\delta-$closure''.) Then we actually proved that any sequence $P_{n}$ of points in $\Gamma^{*}$ which converges $v-$adically to $X$ also converges $v-$adically to $Y$. This might have some interest in its own because the rank of $\Gamma^{*}$ is generally infinite. \vspace{3 mm} \proclaim Proposition 8. Let $L$ be a complete real valued field with residue characteristic zero. Let $G$ be a commutative analytic group over $L$. Let $\Gamma \subset G$ be a finite rank subgroup. Then $\Gamma$ is discrete in the $v-$adic topology of $G$. \vspace{3 mm} {\it Proof.} We refer to [Se], Chapter 4, for background. For any real $\alpha \geq 0$ let $I_{\alpha}$ be the additive group of all elements of $L$ whose valuation is $\geq \alpha$. Since $G$ contains an open subgroup which is standard (in the sense of [Se], i.e. it is the group of points of a formal group over the valuation ring of $L$) we may assume $G$ itself is standard. Then $G$ has a filtration $(G_{i})_{i \in {\bf N}}$ with open subgroups such that $G_{i}/G_{i+1}$ is isomorphic to the group $(I_{i}/I_{i+1})^{g}$, for all $i \in {\bf N}$, where $g$ is the dimension of $G$, and $\bigcap_{i \in {\bf N}} G_{i} = \{ 0 \}$. Assume there exists a sequence $P_{i}$ in $\Gamma \backslash \{ 0 \}$ converging to $0$. Then we may assume there exists a sequence of integers $0i} n_{j}P_{j}$, $n_{i} \neq 0$. Then $n_{i}P_{i} \in G_{k_{i+1}} \subset G_{k_{i}+1}$. Since $(I_{k_{i}}/I_{k_{i}+1})^{g}$ is torsion free, it follows that $P_{i} \in G_{k_{i}+1}$, a contradiction, and we are done. \vspace{3 mm} \noindent 9. {\it Proof of the Theorem.} Embed $K$ into its $v-$adic completion $k_{1}((t))$, where $k_{1}$ is the residue field, $t$ is some variable and $v$ on $K$ is the restriction of the valuation $v_{t}=$``order of series in $t$''. There exists a $K-$embedding of valued fields $(K_{a},v) \subset (k_{1a}((t))_{a},v_{t})$; note that $k_{1a}((t))_{a}= \bigcup_{q \in {\bf N}} k_{1a}((t^{1/q}))$. Now countably many series in $k_{1a}((t))_{a}$ are enough to define our data $A,X,\Gamma$ so these data are defined over $k_{2}((t))_{a}$, where $k_{2}$ is some countably generated extension of $k$ contained in $k_{1a}$. We may embed $k_{2}$ over $k$ into $k((s))_{a}=\bigcup_{p \in {\bf N}} k((s^{1/p}))$ where $s$ is a new variable. We get an embedding $k_{2}((t))_{a} \subset F:=k((s))_{a}((t))_{a} $. Then $F$ is an algebraically closed real valued field (with valuation $v_{t}$). Consider on $F$ the bounded derivation $\delta :=s^{2} \partial_{s} + t \partial_{t}$ where $\partial_{s}:=\partial/\partial s, \ \partial_{t}:= \partial / \partial t$. We claim that $Ker\ \delta=k$. Indeed if $\delta f=0$ for some series $f=\sum f_{n}t^{n/q} \in F, \ \ f_{n} \in k((s^{1/p_{n}}))$, then for any $n$ such that $f_{n} \neq 0$ we get $s^{2}(f_{n}^{-1} \partial_{s} f_{n}) =-n/q$. Let $v_{s}:k((s))_{a}^{*} \rightarrow {\bf Q}$ be the valuation defined by ``order of series in $s$''. Since $ v_{s} ( f_{n}^{-1} \partial_{s} f_{n}) \geq -1$, we must have $n=0$. So we must have $f=f_{0}$ and $\partial f_{0} / \partial s =0$, hence $f \in k$ and our claim is proved. Take now any sequence $P_{n}$ in $\Gamma \backslash X$. We claim that $P_{n}$ cannot converge $v-$adically to $X$, and this will close the proof of the Theorem. Assume it does. Then the same will hold over $k_{1a}((t))_{a}$, hence over $k_{2}((t))_{a}$, hence over $F$. By Chow's rigidity theorem the abelian variety $A_{F}$ over $F$ corresponding to $A$ will still have $F/k$ trace zero. By Proposition 7, $P_{n}$ converges $v_{t}-$adically to a finite union of translates of abelian subvarieties of $A_{F}$. Passing to a subsequence we may assume $P_{n}$ converges $v_{t}-$adically to a translate of an abelian subvariety $B \subset A_{F}$. Let $\pi:A_{F} \rightarrow C:=A_{F}/B$ be the canonical projection. Then $\pi(P_{n})$ converges $v_{t}-$adically to a point of $C$. Let $L$ be the completion of $F$. Then the group $C(L)$ is an analytic group over $L$ and $\pi(\Gamma)$ is not discrete in it, this contradicting Proposition 8. Our Theorem is proved. \vspace{40 mm} \centerline {\bf References} \bigskip \noindent [B] A. Buium, {\it Intersections in jet spaces and a conjecture of S.Lang}, Ann. Math. {\bf 136} (1992), 557-567.\\ \noindent [F] G. Faltings, {\it Diophantine approximation on abelian varieties}, Ann. Math. {\bf 133} (1991) 549-576.\\ \noindent [L] S. Lang, {\it Integral points on curves}, Publ. Math. IHES {\bf 6} (1960), 27-43.\\ \noindent [M] Yu.I.Manin, {\it Rational points on an algebraic curve over function fields}, Transl. A.M.S. II, Ser. 50, (1966), 189-234 (Russian original: Izv. Akad. Nauk. U.S.S.R. 1963) and Letter to the editor Izv. Akad. Nauk. U.S.S.R.34 (1990), 465-466.\\ \noindent [Se] J.P.Serre, {\it Lie Algebras and Lie Groups}, W.A.Benjamin, N.Y.1965.\\ \noindent [Sil] J. H. Silverman, {\it Arithmetic distance functions and height functions in Diophantine geometry}, Math. Ann. {\bf 279} (1987) 193-216.\\ \noindent [Sie] C. L. Siegel, {\it Einige Anwendungen diophantischer Approximationen}, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. {\bf 1} (1929) 41-69.\\ \noindent [P] A. N. Parshin, {\it Finiteness theorems and hyperbolic manifolds}, in { \it The Grothendieck festschrift}, P. Cartier et al., eds., Birka\"user, Basel, 1990, vol. 3,pp 163-178.\\ \ \\ \ \\ \noindent Institute of Math. of the Romanian Academy and\\ Univ. of Essen, Germany,\\ (current address: Max Plank Inst. f\"{u}r Math.\\ Gottfried-Claren-Str. 26, 5300 Bonn 3, Germany)\\ \ \\ \ \\ \noindent Dept. of Mathematics, Univ. of Texas,\\ Austin, TX 78712, USA\\ e-mail: voloch@math.utexas.edu \end{document}