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Unlikely intersections for curves in additive groups over positive characteristic


Authors: W. D. Brownawell and D. W. Masser
Journal: Proc. Amer. Math. Soc. 145 (2017), 4617-4627
MSC (2010): Primary 11G20, 14G17, 14H99
DOI: https://doi.org/10.1090/proc/13617
Published electronically: May 26, 2017
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Abstract: The conjectures associated with the names of Zilber-Pink greatly generalize results associated with the names of Manin-Mumford and Mordell-Lang, but unlike the latter they are at present restricted to zero characteristic. Recently the second author made a start on removing this restriction by studying multiplicative groups over positive characteristic, and here we go further for additive groups with extra Frobenius structure. We state a conjecture for curves in general dimension and we prove it in three dimensions. We also give an example where the finite set in question can be explicitly determined.


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  • [1] E. Bombieri, P. Habegger, D. Masser, and U. Zannier, A note on Maurin's theorem, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 21 (2010), no. 3, 251-260. MR 2677603
  • [2] E. Bombieri, D. Masser, and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Internat. Math. Res. Notices 20 (1999), 1119-1140. MR 1728021
  • [3] E. Bombieri, D. Masser, and U. Zannier, Anomalous subvarieties--structure theorems and applications, Int. Math. Res. Not. IMRN 19 (2007), Art. ID rnm057, 33 pages. MR 2359537
  • [4] Enrico Bombieri, David Masser, and Umberto Zannier, Intersecting a plane with algebraic subgroups of multiplicative groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 1, 51-80. MR 2413672
  • [5] E. Bombieri, D. Masser, and U. Zannier, On unlikely intersections of complex varieties with tori, Acta Arith. 133 (2008), no. 4, 309-323. MR 2457263
  • [6] Zoé Chatzidakis, Dragos Ghioca, David Masser, and Guillaume Maurin, Unlikely, likely and impossible intersections without algebraic groups, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 24 (2013), no. 4, 485-501. MR 3129750
  • [7] Paula B. Cohen and Umberto Zannier, Multiplicative dependence and bounded height, an example, Algebraic number theory and Diophantine analysis (Graz, 1998) de Gruyter, Berlin, 2000, pp. 93-101. MR 1770456
  • [8] Dragos Ghioca and Rahim Moosa, Division points on subvarieties of isotrivial semi-abelian varieties, Int. Math. Res. Not. (2006), Art. ID 65437, 23 pp.. MR 2264715
  • [9] Dragos Ghioca, The isotrivial case in the Mordell-Lang theorem, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3839-3856. MR 2386248
  • [10] P. Habegger, On the bounded height conjecture, Int. Math. Res. Not. IMRN 5 (2009), 860-886. MR 2482128
  • [11] Philipp Habegger and Jonathan Pila, O-minimality and certain atypical intersections, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 4, 813-858 (English, with English and French summaries). MR 3552014
  • [12] Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667-690. MR 1333294
  • [13] Dominik J. Leitner, Linear equations over multiplicative groups in positive characteristic, Acta Arith. 153 (2012), no. 4, 325-347. MR 2925376
  • [14] D. Leitner, Linear equations over multiplicative groups in positive characteristic II, submitted.
  • [15] Rahim Moosa and Thomas Scanlon, $ F$-structures and integral points on semiabelian varieties over finite fields, Amer. J. Math. 126 (2004), no. 3, 473-522. MR 2058382
  • [16] D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves, Math. Ann. 352 (2012), no. 2, 453-484. MR 2874963
  • [17] David Masser, Umberto Zannier, and Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings, with an appendix by E. V. Flynn, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 9, 2379-2416. MR 3420511
  • [18] D. Masser, Unlikely intersections for curves in multiplicative groups over positive characteristic, Q. J. Math. 65 (2014), no. 2, 505-515. MR 3230373
  • [19] Guillaume Maurin, Courbes algébriques et équations multiplicatives, Math. Ann. 341 (2008), no. 4, 789-824 (French, with English summary). MR 2407327
  • [20] R. Pink, A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, manuscript dated 17th April 2005 (13 pages).
  • [21] Gaël Rémond, Intersection de sous-groupes et de sous-variétés. III, Comment. Math. Helv. 84 (2009), no. 4, 835-863 (French, with English summary). MR 2534482
  • [22] Evelina Viada, The intersection of a curve with algebraic subgroups in a product of elliptic curves, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 1, 47-75. MR 1990974
  • [23] Umberto Zannier and Some problems of unlikely intersections in arithmetic and geometry, with appendixes by David Masser, Annals of Mathematics Studies, vol. 181, Princeton University Press, Princeton, NJ, 2012. MR 2918151
  • [24] Boris Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65 (2002), no. 1, 27-44. MR 1875133

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Additional Information

W. D. Brownawell
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: wdb@math.psu.edu

D. W. Masser
Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland
Email: David.Masser@unibas.ch

DOI: https://doi.org/10.1090/proc/13617
Received by editor(s): October 9, 2016
Received by editor(s) in revised form: November 30, 2016
Published electronically: May 26, 2017
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

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