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On the conjectures of Mordell and Lang in positive characteristics. (English) Zbl 0735.14019
Let \(k\) be a number field embedded in a fixed algebraic closure \(\overline \mathbb{Q}\) of \(\mathbb{Q}\). Let \(A\) be an abelian variety defined over \(k\) and let \(V\subset A\) be a sub-variety. Let \(\Gamma\) be a finitely generated subgroup of \(A(\overline\mathbb{Q})\) and set \(\overline\Gamma=\{x\in A(\overline\mathbb{Q})\mid\exists m\in\mathbb{N} \hbox{ with } mx\in\Gamma\}\). Then one has the following (conjecture of Lang-Manin-Mumford) theorem (Faltings, Hindry): The variety \(V\) contains a finite number of translates \(\gamma_ i+B_ i\) of abelian subvarieties of \(A\) such that \(V(\overline\mathbb{Q})\cap\overline\Gamma=\bigcup_{1\leq i\leq t}(\gamma_ i+B_ i\cap\overline\Gamma)\).
The case \(\Gamma=\{0\}\) of the above theorem is due to Raynaud and Hindry. — The author considers a variant of this result in characteristic \(p\) and where \(V\) is a curve. Thus, let \(\mathcal C\) be a curve of genus \(g\) over an algebraically closed field \(\mathcal K\) of finite characteristic \(p\). We assume that \(\mathcal C\) is ordinary in that its Jacobian has \(p^ g\) points of \(p\)-torsion. Moreover, \(\mathcal C\) can be defined over a field \(K\subseteq{\mathcal K}\) which is finitely generated over the prime field; thus \(K\) is the function field of a variety \(W\). The curve \(\mathcal C\) is then classified by a morphism from \(W\) to the moduli space of curves of genus \(g\); we also assume that \(\mathcal C\) is non- isotrivial in that we require this map to be non-constant. Let \(J\) be the Jacobian of \(\mathcal C\) and \(\Gamma\subset J({\mathcal K})\) a finitely generated subgroup. Let \(\overline\Gamma=\{x\in J({\mathcal K})\mid\exists m\in\mathbb{N} \hbox{ with } (m,p)=1 \hbox{ and }mx\in\Gamma\}\). The author then proves the following theorem: Embed \(\mathcal C\) into its Jacobian via the Albanese map. Then \({\mathcal C}({\mathcal K})\cap\overline\Gamma\) is finite.
Reviewer: D.Goss (Columbus)

MSC:
14G05 Rational points
14G27 Other nonalgebraically closed ground fields in algebraic geometry
14H40 Jacobians, Prym varieties
11R58 Arithmetic theory of algebraic function fields
14H99 Curves in algebraic geometry
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References:
[1] Coleman, R.F.: Torsion points on curves. Adv. Stud. Pure Math.12, 235-247 (1987) · Zbl 0653.14015
[2] Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math.73, 349-366 (1983) · Zbl 0588.14026
[3] Grauert, H.: Mordell’s Vermutung über Punkte auf algebraischen Kurven und Funktionenkörper. Publ. Math. Inst. Hautes Étud. Sci.25, 131-149 (1965) · Zbl 0137.40503
[4] Hochschild, G.: Simple algebras with purely inseparable splitting fields of exponent 1. Trans. Am. Math. Soc.79, 477-489 (1955) · Zbl 0065.01902
[5] Lang, S.: Division points on curves. Ann. Mat. Pura Appl., IV. Ser. LXX 229-234 (1965) · Zbl 0151.27401
[6] Manin, Yu.I.: Rational points on an algebraic curve over function fields. Transl. Am. Math. Soc. II. Ser.50, 189-234 (1966); letter to the editor, Math. USSR Izv.34, 465-466 (1990)
[7] Mumford, D., Fogarty, J.: Geometric Invariant Theory, 2. ed. Berlin Heidelberg New York: Springer 1982 · Zbl 0504.14008
[8] Raynaud, M.: Courbes sur une variété abélienne et points de torsion. Invent. Math.71, 207-233 (1983) · Zbl 0564.14020
[9] Rudakov, A.N., Shafarevich, I.R: Inseparable morphisms of algebraic surfaces. Math. USSR, Izv.10, 1205-1237 (1976); Shafarevich, I.R.: Coll. Math. Papers, pp. 577-609. Berlin Heidelberg New York: Springer 1989 · Zbl 0379.14006
[10] Samuel, P.: Lectures on old and new results on Algebraic Curves. Bombay: Tata Inst. Fund. Res. 1966 · Zbl 0165.24102
[11] Seshadri, C.S: L’operation de Cartier. Applications. In: Chevalley, C. (ed.) Varietés de Picard. Sémin. 3. année 1958/59 (Éc. Norm. Sup., Exposé 6, pp. 101-115). Paris: 1960
[12] Szpiro, L.: Propriétés numériques du faisceau dualisant relatif. Astérisque86, 44-78 (1981) · Zbl 0517.14006
[13] Szpiro, L.: Small points and torsion points. In: Sundararaman, D., et al. (eds.) Lefschetz centennial Conference. (Contemp. Math. vol. 58, pp. 251-260). Providence: Am. Math. Soc. 1986
[14] Voloch, J.F.: Explicitp-descent for elliptic curves in characteristicp, Compos. Math.74, 247-258 (1990) · Zbl 0715.14027
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