Research PaperNo Access

A short note on Manin–Mumford

Harry Schmidt

Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland

https://doi.org/10.1142/S1793042123500094Cited by:0 (Source: Crossref)

Abstract

We give a short proof of Manin–Mumford in the multiplicative group based on the pigeon-hole principle and a simple structure theorem for tori contained in subvarieties of $𝔾_{m}^{n}$ due to Bombieri and Gubler. The arguments appear to be new and are applicable in other situations. Our proof is quite elementary and we compare it to previous proofs that rely on machinery such as intersection theory or $o$ -minimality.

Keywords:

AMSC: 14L10

References

1. E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Mathematical Monographs, Vol. 4 (Cambridge University Press, Cambridge, 2006). Google Scholar
2. E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59(2) (1989) 337–357. Crossref, Web of Science, Google Scholar
3. M. Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94(3) (1988) 575–603. Crossref, Web of Science, Google Scholar
4. K. S. Kedlaya, A. Kolpakov, B. Poonen and M. Rubinstein, Space vectors forming rational angles, preprint (2020), arXiv:2011.14232. Google Scholar
5. S. Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965) 229–234. Crossref, Google Scholar
6. S. Lang, Fundamentals of Diophantine Geometry (Springer-Verlag, New York, 1983). Crossref, Google Scholar
7. M. Laurent, Équations diophantiennes exponentielles, Invent. Math. 78(2) (1984) 299–327. Crossref, Google Scholar
8. H. B. Mann, On linear relations between roots of unity, Mathematika 12 (1965) 107–117. Crossref, Web of Science, Google Scholar
9. J. Pila, O-minimality and the André--Oort conjecture for $ℂ^{n}$ , Ann. of Math. (2) 173(3) (2011) 1779–1840. Crossref, Web of Science, Google Scholar
10. J. Pila and A. Wilkie, The rational points of a definable set, Duke Math. J. 133(3) (2006) 591–616. Crossref, Web of Science, Google Scholar
11. J. Pila and U. Zannier, Rational points in periodic analytic sets and the Manin–Mumford conjecture, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(2) (2008) 149–162. Crossref, Google Scholar
12. P. Sarnak, Torsion points on varieties and homology abelian covers (available on the homepage of the IAS) (1988). Google Scholar
13. P. Sarnak and S. Adams, Betti numbers of congruence groups, Israel J. Math. 88(1–3) (1994) 31–72. With an appendix by Ze’ev Rudnick. Crossref, Web of Science, Google Scholar
14. H. Schmidt, Polynomial dynamics and local analysis of small and grand orbits, preprint (2020), arXiv:2009.07609. Google Scholar
15. J. Tsimerman, The André–Oort conjecture for $𝒜_{g}$ , Ann. of Math. (2) 187(2) (2018) 379–390. Crossref, Web of Science, Google Scholar