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• W2022021021 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. 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. 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). Let K be a function field with constant field of characteristic zero and E/K an elliptic curve with nonconstant 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) -) R 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 E A (P), where v ranges through the places of K. The local heights satisfy Av(P) = max{O, v(t(P))}+,fv(P), where f3v is bounded for all v and is identically zero for all but finitely many v, and t is a uniformizer at 0 E E. For example, t = x/y, where x, y are coordinates of a Weierstrass equation for E. 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 Ev i S Av (P) is bounded independently of P. if P is S-integral and, since there are only finitely many points of bounded Received by the editors October 5, 1992 and, in revised form, March 20, 1993. 1991 Mathematics Subject Classification. Primary 1 1G05, 1 1G30, 14G25." @default.
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• W2022021021 date "1994-04-01" @default.
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• W2022021021 title "Siegel’s theorem for complex function fields" @default.
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• W2022021021 doi "https://doi.org/10.1090/s0002-9939-1994-1209430-9" @default.
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