Local to global principle for semiabelian varieties isogenous to the product of an abelian variety and a torus. (English) Zbl 1489.11091
Let \(F\) be a number field, \(F'\) an extension, \(A\) a semi-abelian variety over \(F\) such that it’s base change to \(F'\) splits as a product \(\mathbb G_m\times G_1^{e_1}\times\dots\times G_k^{e_k}\) with \(G_1,\dots,G_k\) pairwise non isogenous, simple abelian varieties such that \(e_r \leq \dim_{\mathrm{End}^0(G_r)}(H_1(G_r(\mathbb C), \mathbb Q)\). For a finite place \(v\) of \(F'\), let \(A_v\) denote reduction of \(A \pmod v\).
Let \(P \in A(F), \Lambda \subset A(F)\) a finite generated subgroup. If \(P \in \Lambda \pmod v\) for almost all primes \(v\), the paper shows that \(P \in \Lambda + A(F)_{\mathrm{tor}}\). In fact, the paper shows that it is sufficient to check inclusion at a finite set of carefully selected primes.
The paper builds on the results and methods of G. Banaszak and P. Krasoń [Acta Arith. 150, No. 4, 315–337 (2011; Zbl 1281.11061)] and can be seen as a combination of the results of that paper with the classical paper of A. Schinzel [Acta Arith. 27, 397–420 (1975; Zbl 0342.12002)].
Let \(P \in A(F), \Lambda \subset A(F)\) a finite generated subgroup. If \(P \in \Lambda \pmod v\) for almost all primes \(v\), the paper shows that \(P \in \Lambda + A(F)_{\mathrm{tor}}\). In fact, the paper shows that it is sufficient to check inclusion at a finite set of carefully selected primes.
The paper builds on the results and methods of G. Banaszak and P. Krasoń [Acta Arith. 150, No. 4, 315–337 (2011; Zbl 1281.11061)] and can be seen as a combination of the results of that paper with the classical paper of A. Schinzel [Acta Arith. 27, 397–420 (1975; Zbl 0342.12002)].
Reviewer: Asvin G (Madison)
Keywords:
semiabelian variety; reduction map; linear dependence; commensurability; extensions of commutative algebraic groupsReferences:
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