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Geometry of \(p\)-jets. (English) Zbl 0882.14007

In the author’s previous work on diophantine problems over function fields, methods of differential algebra were used. In this present work, the author concentrates on number fields where, of course, no non-trivial derivations exist. The author replaces these with certain “natural” non-additive maps called “\(p\)-derivations”. The main result shown is the following quantitative version of the Manin-Mumford conjecture:
Theorem: Let \(X\) be a curve defined over a number field \(K\) and let \(X\to J\) be the associated Abel map into the Jacobian of \(X\). Let \({\mathfrak p}\) be a prime of \(K\) with \(p= \text{char} {\mathfrak p} >2g\) with \(g\) the genus of \(X\). Assume that \(K/ \mathbb{Q}\) is unramified at \({\mathfrak p}\) and that \(X/K\) has good reduction at \({\mathfrak p}\). Let \(K^{ac}\) be the algebraic closure of \(K\). Then \[ \# (X(K^{ac}) \cap J(K^{ac})_{\text{tors}}) \leq p^{4g} 3^g[p(2g-2) +6g]\cdot g!. \]

MSC:

14G25 Global ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
14H40 Jacobians, Prym varieties
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
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[1] A. Buium, Geometry of differential polynomial functions. I. Algebraic groups , Amer. J. Math. 115 (1993), no. 6, 1385-1444. JSTOR: · Zbl 0797.14016 · doi:10.2307/2374970
[2] A. Buium, Geometry of differential polynomial functions. II. Algebraic curves , Amer. J. Math. 116 (1994), no. 4, 785-818. JSTOR: · Zbl 0829.14018 · doi:10.2307/2375002
[3] A. Buium, Intersections in jet spaces and a conjecture of S. Lang , Ann. of Math. (2) 136 (1992), no. 3, 557-567. JSTOR: · Zbl 0817.14021 · doi:10.2307/2946600
[4] A. Buium, Effective bound for the geometric Lang conjecture , Duke Math. J. 71 (1993), no. 2, 475-499. · Zbl 0812.14029 · doi:10.1215/S0012-7094-93-07120-7
[5] A. Buium and F. Voloch, The Mordell conjecture of characteristic \(P\): an explicit bound , to appear in Compositio Math. · Zbl 0885.14010
[6] R. Coleman, Torsion points on curves and \(p\)-adic abelian integrals , Ann. of Math. (2) 121 (1985), no. 1, 111-168. JSTOR: · Zbl 0578.14038 · doi:10.2307/1971194
[7] R. Coleman, Ramified torsion points on curves , Duke Math. J. 54 (1987), no. 2, 615-640. · Zbl 0626.14022 · doi:10.1215/S0012-7094-87-05425-1
[8] W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[9] M. Greenberg, Schemata over local rings , Ann. of Math. (2) 73 (1961), 624-648. JSTOR: · Zbl 0115.39004 · doi:10.2307/1970321
[10] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV , Inst. Hautes Études Sci. Publ. Math. (1967), no. 32, 361. · Zbl 0153.22301
[11] E. Kolchin, Differential algebra and algebraic groups , Academic Press, New York, 1973. · Zbl 0264.12102
[12] S. Lang, On quasi algebraic closure , Ann. of Math. (2) 55 (1952), 373-390. JSTOR: · Zbl 0046.26202 · doi:10.2307/1969785
[13] S. Lang, Some applications of the local uniformization theorem , Amer. J. Math. 76 (1954), 362-374. JSTOR: · Zbl 0058.27201 · doi:10.2307/2372578
[14] M. Martin-Deschamps, Propriétés de descente des variétés à fibré cotangent ample , Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 39-64. · Zbl 0535.14013 · doi:10.5802/aif.977
[15] M. Raynaud, Courbes sur une variété abélienne et points de torsion , Invent. Math. 71 (1983), no. 1, 207-233. · Zbl 0564.14020 · doi:10.1007/BF01393342
[16] M. Raynaud, Around the Mordell conjecture for function fields and a conjecture of Serge Lang , Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 1-19. · Zbl 0525.14014
[17] J. F. Ritt, Differential Algebra , American Mathematical Society Colloquium Publications, Vol. XXXIII, American Mathematical Society, New York, N. Y., 1950. · Zbl 0037.18402
[18] J.-P. Serre, Local fields , Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979. · Zbl 0423.12016
[19] J. H. Silverman, The arithmetic of elliptic curves , Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
[20] H. Tango, On the behavior of extensions of vector bundles under the Frobenius map , Nagoya Math. J. 48 (1972), 73-89. · Zbl 0239.14007
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