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The isotrivial case in the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  13 January 2025

Dragos Ghioca Open the ORCID record for Dragos Ghioca [Opens in a new window]
Dragos Ghioca*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada
*
e-mail: dghioca@math.ubc.ca
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Abstract

Let G be a semiabelian variety defined over a finite subfield of an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of $G(K)$.

Keywords

Semiabelian varietiesfinite fieldsMordell–Lang conjecture

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 14G17: Positive characteristic ground fields
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Corvaja, P., Ghioca, D., Scanlon, T., and Zannier, U., The dynamical Mordell-Lang conjecture for endomorphisms of semiabelian varieties defined over fields of positive characteristic . J. Inst. Math. Jussieu 20(2021), no. 2, 669–698.CrossRefGoogle Scholar
Faltings, G., The general case of S. Lang’s conjecture. In: Barsotti symposium in algebraic geometry (Abano Terme, 1991), Perspectives in Mathematics, 15, Academic Press, San Diego, CA, 1994, 175–182.CrossRefGoogle Scholar
Ghioca, D., The isotrivial case in the Mordell-Lang theorem . Trans. Amer. Math. Soc. 360(2008), no. 7, 3839–3856.CrossRefGoogle Scholar
Ghioca, D. and Yang, S., The Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic . Bull. Aust. Math. Soc. 109(2024), no. 2, 254–264.CrossRefGoogle Scholar
Ginsburg, S. and Spanier, E. H., Semigroups, Presburger Formulas, and Languages . Pac. J. Math. 16(1966), 285–296.CrossRefGoogle Scholar
Hieronymi, P. and Schulz, C., A strong version of Cobham’s theorem. In: STOC’22-Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, 2022, 1172–1179.CrossRefGoogle Scholar
Hrushovski, E., The Mordell-Lang conjecture for function fields . J. Amer. Math. Soc. 9(1996), no. 3, 667–690.CrossRefGoogle Scholar
Laurent, M., Équations diophantiennes exponentielles . Invent. Math. 78(1984), 299–327.CrossRefGoogle Scholar
Moosa, R. and Scanlon, T., F-structures and integral points on semiabelian varieties over finite fields . Amer. J. Math. 126(2004), 473–522.CrossRefGoogle Scholar
Skolem, T., Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen. C. r. 8 congr. scand. á Stockholm (1934), 163–188.Google Scholar
Vojta, P., Integral points on subvarieties of semiabelian varieties. I . Invent. Math. 126(1996), no. 1, 133–181.CrossRefGoogle Scholar