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Integral division points on curves. (English) Zbl 1292.11072
Let \(k\) be a number field and \(S\) a set of primes containing all the infinite ones. Let \(A/k\) be a semi-abelian variety, \(\Gamma_0\) a finitely generated subgroup of \(A(\overline k)\) and \(\Gamma \subseteq A(\overline k)\) be the division group attached to \(\Gamma_0\), i.e the set of points \(P\in A(\overline k)\) such that there exists a integer \(n\) such that \(nP\in \Gamma_0\).
If \(X/k\) is any variety and \(\overline X\) its completion, define \(\partial X:=\overline X - X\). Let \(T\) be any subset of \(\overline X\), and let \(\overline T\) be its Zariski closure in \(\overline X\). Any \(P\in X(\overline k)\) is said to be \(S\)-integral relative to \(T\) if it is \((\overline T \cup \partial X, S)\)-integral on \(\overline X\).
The authors pose the following conjecture:
Conjecture Let \(k\) and \(S\) be as above and let \(A/k\) be a semi-abelian variety and \(\Gamma\) a division group in \(A(\overline k)\). Suppose that \(D\) is a non-zero effective divisor on \(A\) which is not the translate of any torsion divisor by any point of \(\Gamma\). Then the set \[ \{P\in \Gamma: P \text{ is \(S\)-integral relative to }D\} \] is not Zariski dense in \(A\).
The authors then prove the conjecture for \(1\)-dimensional semi-abelian varieties, i.e. for elliptic curves and \(1\)-dimensional tori.

MSC:
11G35 Varieties over global fields
11G05 Elliptic curves over global fields
11G50 Heights
14G05 Rational points
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P35 Arithmetic properties of periodic points
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References:
[1] doi:10.1007/978-1-4612-0851-8
[2] doi:10.1007/978-0-387-69904-2
[3] doi:10.1016/0022-314X(88)90019-4 · Zbl 0654.10019
[4] doi:10.1007/BF01405086 · Zbl 0235.14012
[6] doi:10.1090/S0002-9947-2011-05350-X · Zbl 1300.37052
[7] doi:10.1016/j.jnt.2010.10.003 · Zbl 1214.11070
[8] doi:10.1112/blms/bdn053 · Zbl 1243.11073
[9] doi:10.1007/978-1-4612-1210-2
[11] doi:10.1007/BF01394276 · Zbl 0638.14026
[12] doi:10.1007/978-1-4612-5485-0
[14] doi:10.1007/978-3-662-07010-9
[15] doi:10.1112/plms/pdm022 · Zbl 1130.11035
[16] doi:10.2307/2944319 · Zbl 0734.14007
[17] doi:10.1142/S1793042110003356 · Zbl 1258.37077
[18] doi:10.2307/3062133 · Zbl 1026.11038
[19] doi:10.1112/jlms/jdq097 · Zbl 1244.37052
[20] doi:10.1007/BF01393743 · Zbl 0662.10012
[22] doi:10.2140/ant.2008.2.217 · Zbl 1182.11030
[23] doi:10.1353/ajm.1999.0014 · Zbl 1018.11027
[25] doi:10.2140/ant.2008.2.183 · Zbl 1158.14029
[26] doi:10.1016/j.jnt.2011.01.005 · Zbl 1246.37102
[27] doi:10.1215/S0012-7094-93-07129-3 · Zbl 0811.11052
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