# Intersecting a plane with algebraic subgroups of multiplicative groups

• Volume: 7, Issue: 1, page 51-80
• ISSN: 0391-173X

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## Abstract

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Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed elsewhere.

## How to cite

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Bombieri, Enrico, Masser, David, and Zannier, Umberto. "Intersecting a plane with algebraic subgroups of multiplicative groups." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.1 (2008): 51-80. <http://eudml.org/doc/272261>.

@article{Bombieri2008,
abstract = {Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed elsewhere.},
author = {Bombieri, Enrico, Masser, David, Zannier, Umberto},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Bounded Height Conjecture; Torsion Finiteness Conjecture},
language = {eng},
number = {1},
pages = {51-80},
publisher = {Scuola Normale Superiore, Pisa},
title = {Intersecting a plane with algebraic subgroups of multiplicative groups},
url = {http://eudml.org/doc/272261},
volume = {7},
year = {2008},
}

TY - JOUR
AU - Bombieri, Enrico
AU - Masser, David
AU - Zannier, Umberto
TI - Intersecting a plane with algebraic subgroups of multiplicative groups
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2008
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 1
SP - 51
EP - 80
AB - Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed elsewhere.
LA - eng
KW - Bounded Height Conjecture; Torsion Finiteness Conjecture
UR - http://eudml.org/doc/272261
ER -

## References

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4. [4] E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Internat. Math. Res. Notices20 (1999), 1119–1140. Zbl0938.11031MR1728021
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7. [7] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - structure theorems and applications, Int. Math. Res. Not. IMRN 19 (2007), 33 pages. Zbl1145.11049MR2359537
8. [8] E. Bombieri and J. Vaaler, On Siegel’s Lemma, Invent. Math.73 (1983), 11–32. Zbl0533.10030MR707346
9. [9] E. Bombieri and U. Zannier, Algebraic points on subvarieties of ${𝔾}_{m}^{n}$, Internat. Math. Res. Notices7 (1995), 333–347. Zbl0848.11030MR1350686
10. [10] J. W. S. Cassels, “An Introduction to Diophantine Approximation”, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 45, Cambridge, 1965. Zbl0077.04801MR120219
11. [11] T. Loher and D. Masser, Uniformly counting points of bounded height, Acta Arith.111 (2004), 277–297. Zbl1084.11034MR2039627
12. [12] R. Pink, A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, manuscript dated 17th April 2005.
13. [13] A. Schinzel, “Polynomials with Special Regard to Reducibility”, Encyclopaedia of Mathematics and its Applications, Vol. 77, Cambridge, 2000. Zbl0956.12001MR1770638
14. [14] U. Zannier, Proof of Conjecture $1$, Appendix to [13], 517-539.
15. [15] B. Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc.65 (2002), 27–44. Zbl1030.11073MR1875133

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