\input amstex \def\bfR{\text{\bf R}} \def\bfZ{\text{\bf Z}} \documentstyle{amsppt} \topmatter \title Siegel's theorem for complex function fields\endtitle \author Jos\'e Felipe Voloch\endauthor \subjclass 11G05, 11G30, 14G25\endsubjclass \keywords algebraic curves, function fields, integral points\endkeywords \abstract We give a short proof of the finiteness of the set of integral points on an affine algebraic curve of genus at least one, defined over a function field of characteristic zero.\endabstract \address Dept. of Mathematics, University of Texas, Austin, TX 78712-1082, USA\endaddress \email voloch\@math.utexas.edu\endemail \endtopmatter \document Siegel [Si] 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 curves of genus at least two, one even has the Mordell conjecture (proved by Faltings [F] in the number field case and by Manin [M] in the function field case) that there are only finitely many rational points. \par For genus one, Manin [M] gave a proof of a strenghtening of Lang's result as a by-product of his work on the Mordell conjecture. Mason [Ms] then gave an effective proof by more elementary considerations. In this note, we give a short proof of Manin's (and hence Lang's) result for genus one. The proof can be adapted to higher genus as well (see the remark below).\par Let $K$ be a function field with constant field of characteristic zero and $E/K$ an elliptic curve with non-constant $j$-invariant. The reader may consult [S] for definitions and results about elliptic curves. In particular we shall use the following results. The group $E(K)$ is a finitely generated abelian group by the Mordell-Weil theorem and there is a height function $h: E(K) \to {\bfR}$ with the property that there are only finitely many points of bounded height ([L], Proposition 2). The height can be written as a sum of local heights $\sum \lambda_v (P) $, where $v$ ranges through the places of $K$. The local heights satisfy $\lambda_v (P) = max \{0,v(t(P))\} + \beta_v (P)$, where $\beta_v$ is bounded for all $v$ and it is identically zero for all but finitely many $v$, and $t$ is a uniformizer at $0 \in E$. For example, $t = x/y$, where $x,y$ are coordinates of a Weierstrass equation for $E$.\par Now, Lang's result for genus one can be reduced to the case of a Weierstrass equation ([S], Corollary IX.3.2.2) and in this case we argue as follows. Let $S$ be a finite set of places of $K$. Then $\sum_{v \notin S} \lambda_v (P)$ is bounded independently of $P$, if $P$ is $S$-integral and, since there are only finitely many points of bounded height, if there are infinitely many $S$-integral points, then $\lambda_v$ is unbounded for some $v \in S$. It suffices thus to prove the following result: \proclaim{Theorem (Manin)} Let $K$ be a function field with constant field of characteristic zero and $E/K$ an elliptic curve with non-constant j-invariant and $v$ a place of $K$. Then the local height function $\lambda_v$ is bounded on $E(K)$. \endproclaim \demo{Proof} The points on $E(K_v)$ that reduce to $0 \bmod v$ form a subgroup $E_1(K_v)$ isomorphic to the group of points of a formal group. Choosing an uniformizer $t$, as above, on $E$ at 0, then $E_1(K_v) = \{ P \in E(K_v) | t(P) \in {\Cal M}_v \}$, where ${\Cal M}_v$ is the maximal ideal of the local ring at $v$. Moreover, $\lambda_v(P)$ differs from $v(t(P))$ by a bounded amount. Hence, it suffices to show that $v(t(P))$ is bounded above on $E(K)$. Suppose not and choose $P_n, n=1,2, \ldots$ in $E(K)$ such that $v(t(P_{n+1})) > v(t(P_n)) > 0$. We claim that $P_1, P_2, \ldots$ are linearly independent over $\bfZ$. Recall that $t$ induces a group isomorphism between $E_r/E_{r+1}$, where $E_r =E_r(K_v) = \{ P \in E(K_v) | t(P) \in {\Cal M}_v^r \}$, and ${\Cal M}_v^r / {\Cal M}_v^{r+1}$. If $n_iP_i = \sum_{j>i} n_jP_j , n_i \ne 0$ and $r = v(t(P_i))$ then $n_iP_i$ is 0 in $E_r/E_{r+1}$, but $t(n_iP_i) \equiv n_it(P_i) \not\equiv 0 \pmod{ {\Cal M}_v^{r+1}}$, which proves the claim. On the other hand, the claim contradicts the Mordell-Weil theorem and this completes the proof.\par \enddemo \demo{Remark} On a curve of genus greater than one, if a sequence of points $P_1, P_2, \ldots$ approach rapidly a point $P_\infty$, then a similar argument shows that the $P_i - P_\infty$ are linearly independent over {\bfZ} in the Jacobian of the curve, and Lang's result follows from this. The author and A. Buium [BV] have recently proved a conjecture of Lang to the effect that an affine open subset of an abelian variety of any dimension over a function field of characteristic zero has finitely many integral points. \enddemo \widestnumber\key{Ms} \Refs \ref \key{BV} \by A. Buium, J. F. Voloch \paper Integral points of abelian varieties over function fields of characteristic zero \jour Math. Annalen, to appear \endref \ref \key{F} \by G. Faltings \paper Endlichkeitss\"atze f\"ur abelsche Variet\"aten \"uber Zahlk\"orpen \jour Invent. Math. \vol 73 \yr 1983 \pages 349--366 \endref \ref \key{L} \by S. Lang \paper Integral points on curves \jour Publ. Math. IHES \vol 6 \yr 1960 \pages 27--43 \endref \ref \key{M} \by Yu. I. Manin \paper Rational points on an algebraic curve over function fields \jour Transl. Am. Math. Soc. II Ser. \vol 50 \pages 189--234 \yr 1966 \paperinfo (Russian original: Izv. Acad. Nauk U.S.S.R. 1963.) and Letter to the editor. Math. USSR Izv. {\bf34} (1990), 4650-466. \endref \ref \key{Ms} \by R. C. Mason \book Diophantine equations over function fields \bookinfo London Math. Soc. Lecture Notes \vol 96 \publ CUP \publaddr Cambridge \yr 1984 \endref \ref \key{S} \by J. H. Silverman \book The arithmetic of elliptic curves \bookinfo GTM 106 \publ Springer \publaddr New York \yr 1986 \endref \ref \key{Si} \by C. L. Siegel \paper Einige Anwendungen diophantischer Approximationen \jour Abh. Preuss. Akad. Wiss. Phys. Math. Kl. \vol 1 \yr 1929 \pages 41--69 \endref \endRefs \enddocument \bye