Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019
Antonella Perucca^{[a]}
The problem of detecting linear dependence
Pages: 99116
Received: 6 January 2019
Accepted in revised form: 6 June 2019
Mathematics Subject Classification (2010): Primary: 11G10; Secondary 14L10, 14K15.
Keywords: Number fields, localglobal principle, detecting linear dependence.
Authors address:
[a]: University of Luxembourg, Mathematics Research Unit, 6 av. de la Fonte, 4364 EschsurAlzette, Luxembourg
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Abstract:
Let \( A \) be an algebraic group defined over a number field \( \ K \ \), let \( \ P \ \) be a point in \( \ A(K) \ \),
and let \( \ G \ \) be a finitely generated subgroup of \( \ A(K) \ \). If \( \ P \ \) belongs to \( \ G \ \), then clearly
its reduction \( \ (P \bmod \mathfrak p) \ \) belongs to \( \ (G \bmod \mathfrak p) \ \) for all but finitely
many primes \( \ \mathfrak p \ \) of \( \ K \ \) (notice that we only consider those primes \( \ \mathfrak p \ \) such
that the reductions are welldefined, and are ''good'' reductions). The problem of detecting linear
dependence asks whether the converse holds, so whether we have a localglobal principle.
In this survey article we also investigate the problem of detecting linear dependence for torsion points.
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