## Functiones et Approximatio Commentarii Mathematici

### Hasse principle for linear dependence in Mordell-Weil groups

Stefan Barańczuk

#### Abstract

We establish a local-global principle for linear dependence of points in Mordell--Weil groups of abelian varieties over number fields. We give a complete characterization, in terms of a relation between the rank and the dimension, of abelian varieties with endomorphism ring equal to $\mathbb{Z}$ for which the principle holds. In the case of elliptic curves we prove the result in full generality, i.e., without the assumption on the endomorphism ring.

#### Article information

Source
Funct. Approx. Comment. Math., Advance publication (2019), 5 pages.

Dates
First available in Project Euclid: 26 October 2019

https://projecteuclid.org/euclid.facm/1572055503

Digital Object Identifier
doi:10.7169/facm/1792

Subjects
Secondary: 11H52

#### Citation

Barańczuk, Stefan. Hasse principle for linear dependence in Mordell-Weil groups. Funct. Approx. Comment. Math., advance publication, 26 October 2019. doi:10.7169/facm/1792. https://projecteuclid.org/euclid.facm/1572055503

#### References

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• [3] Y. Flicker, P. Krasoń, Multiplicative relations of points on algebraic groups, Bull. Pol. Acad. Sci. Math. 65 (2017), no. 2, 125–138.
• [4] J.-P. Serre, A course in Arithmetic, Graduate Texts in Mathematics, Springer 1996.
• [5] J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Volume 106, 2009.