Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019

Antonella Perucca[a]

The problem of detecting linear dependence

Pages: 99-116
Accepted in revised form: 6 June 2019
Mathematics Subject Classification (2010): Primary: 11G10; Secondary 14L10, 14K15.
Keywords: Number fields, local-global principle, detecting linear dependence.
[a]: University of Luxembourg, Mathematics Research Unit, 6 av. de la Fonte, 4364 Esch-sur-Alzette, Luxembourg

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 well-defined, and are ''good'' reductions). The problem of detecting linear dependence asks whether the converse holds, so whether we have a local-global principle. In this survey article we also investigate the problem of detecting linear dependence for torsion points.

References
[1]
G. Banaszak, On a Hasse principle for Mordell-Weil groups, C. R. Math. Acad. Sci. 347 (2009), 709-714. MR2543968
[2]
G. Banaszak and D. Blinkiewicz, Commensurability in Mordell-Weil groups of abelian varieties and tori, Funct. Approx. Comment. Math. 58 (2018), 145-156. MR3816070
[3]
G. Banaszak, W. Gajda and P. Krasoń, Detecting linear dependence by reduction maps, J. Number Theory 115 (2005), 322-342. MR2180505
[4]
G. Banaszak and P. Krasoń, On arithmetic in Mordell-Weil groups, Acta Arith. 150 (2011), 315-337. MR2847263
[5]
S. Barańczuk, On reduction maps and support problem in $${K}$$-theory and abelian varieties, J. Number Theory 119 (2006), 1-17. MR2228946
[6]
S. Barańczuk and K. Górnisiewicz, On reduction maps for the étale and Quillen K-theory of curves and applications, J. K-Theory 2 (2008), 103-122. MR2434168
[7]
[8]
W. Gajda and K. Górnisiewicz, Linear dependence in Mordell-Weil groups, J. Reine Angew. Math. 630 (2009), 219-233. MR2526790
[9]
C. Hall and A. Perucca, Characterizing abelian varieties by the reduction of the Mordell-Weil group, Pacific J. Math. 265 (2013), 427-440. MR3096508
[10]
P. Jossen, Detecting linear dependence on an abelian variety via reduction maps, Comment. Math. Helv. 88 (2013), 323-352. MR3048189
[11]
P. Jossen, On the arithmetic of $$1$$-motives, Ph.D. thesis, Central European University Budapest, July 2009. Article
[12]
P. Jossen and A. Perucca, A counterexample to the local-global principle of linear dependence for abelian varieties, C. R. Math. Acad. Sci. Paris 348 (2010), 9-10. MR2586734
[13]
C. Khare, Compatible systems of mod $$p$$ Galois representations and Hecke characters, Math. Res. Lett. 10 (2003), 71-83. MR1960125
[14]
E. Kowalski, Some local-global applications of Kummer theory, Manuscripta Math. 111 (2003), 105-139. MR1981599
[15]
M. Larsen and R. Schoof, Whitehead's lemma and Galois cohomology of abelian varieties, preprint, 2003. Article
[16]
The LMFDB Collaboration, The L-functions and Modular Forms Database, 2013, http://www.lmfdb.org, [Online; accessed 16 September 2013].
[17]
A. Perucca, On the problem of detecting linear dependence for products of abelian varieties and tori, Acta Arith. 142 (2010), 119-128. MR2601054
[18]
A. Perucca, On the reduction of points on abelian varieties and tori, Int. Math. Res. Not. IMRN 2011 (2011), 293-308. MR2764865
[19]
A. Perucca, Two variants of the support problem for products of abelian varieties and tori, J. Number Theory 129 (2009), 1883-1892. MR2522711
[20]
K. A. Ribet, Kummer theory on extensions of abelian varieties by tori, Duke Math. J. 46 (1979), 745-761. MR0552524
[21]
P. Rzonsowski, Linear relations and arithmetic on abelian schemes, Funct. Approx. Comment. Math. 52 (2015), 83-107. MR3326126
[22]
M. Sadek, On dependence of rational points on elliptic curves, C. R. Math. Acad. Sci. Soc. R. Can. 38 (2016), 75-84. MR3559359
[23]
A. Schinzel, On power residues and exponential congruences, Acta Arith. 27 (1975), 397-420. MR0379432
[24]
M. Sha and I. E. Shparlinski, Effective results on linear dependence for elliptic curves, Pacific J. Math. 295 (2018), 123-144. MR3778329
[25]
Th. Skolem, Anwendung exponentieller Kongruenzen zum Beweis der Unlösbarkeit gewisser diophantischer Gleichungen, Avh. Norske Vid. Akad. Oslo 1937 (1937), no. 12, 1-16. zbMATH
[26]
T. Weston, Kummer theory of abelian varieties and reductions of Mordell-Weil groups, Acta Arith. 110 (2003), 77-88. MR2007545

Home Riv.Mat.Univ.Parma