Cyclotomic torsion points in elliptic schemes. (English) Zbl 1491.11058
Let \(E\) be an elliptic curve defined over a number field \(k\). By a work of Serre about Galois representations attached to elliptic curves, there are only a finitely many torsion points of \(E\) which are defined over the cyclotomic closure \(k^{\mathrm{c}}\). The author of the paper under review proves the following for an elliptic scheme: Let \(\mathcal E/C\) be an elliptic scheme over a curve \(C\). Let \(s : C\to \mathcal E\) be a section which is not identically torsion. Let \(f : C \to \mathbb A^1\) be a non constant rational function, and all are defined over \(k\). If \(n\) is a natural number, then there are only finitely many points \(P\in C\) such that \(f(P)\) is the sum of \(n\) roots of unity and such that \(s(P)\in \mathcal E_P\) is torsion.
The author believes that the restriction on the length \(n\) is a technical condition, and that the result is true in general without the restriction. The author follows the methods used by Masser, Pila, and Zannier to prove the Manin-Mumford conjecture and the relative Manin-Mumford conjecture, but he makes further contributions in this work for handling the presence of semi-algebraic sets of dim \(\ge 1\) in the transcendental variety defined using logarithms.
The author believes that the restriction on the length \(n\) is a technical condition, and that the result is true in general without the restriction. The author follows the methods used by Masser, Pila, and Zannier to prove the Manin-Mumford conjecture and the relative Manin-Mumford conjecture, but he makes further contributions in this work for handling the presence of semi-algebraic sets of dim \(\ge 1\) in the transcendental variety defined using logarithms.
Reviewer: Sungkon Chang (Savannah)
References:
| [1] | Ax, J., On Schanuel’s Conjectures, Ann. Math., 93, 2, 252-268 (1971) · Zbl 0232.10026 · doi:10.2307/1970774 |
| [2] | Bertrand, D.; Baldassarri, F.; Bosch, S.; Dwork, B., Extensions de D-modules et groupes de Galois différentiels, p-Adic Analysis, Lecture Notes in Mathematics, 67-85 (1989), Berlin: Springer, Berlin |
| [3] | Bertrand, D.; Masser, D.; Pillay, A.; Zannier, U., Relative Manin-Mumford for semi-Abelian surfaces, Proc. Edinb. Math. Soc. (2), 59, 4, 837-875 (2016) · Zbl 1408.14139 · doi:10.1017/S0013091515000486 |
| [4] | Bombieri, E.; Gubler, W., Heights in Diophantine Geometry. New Mathematical Monographs 4 (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1115.11034 |
| [5] | Brownawell, WD; Kubota, KK, Algebraic independence of Weierstrass functions, Acta Arith., 33, 111-149 (1977) · Zbl 0356.10027 · doi:10.4064/aa-33-2-111-149 |
| [6] | van den Dries, L.; Macintyre, A.; Marker, D., The elementary theory of restricted analytic fields with exponentiation, Ann. Math., 140, 1, 183-205 (1994) · Zbl 0837.12006 · doi:10.2307/2118545 |
| [7] | Lang, S., Algebra. Graduate Texts in Mathematics 211 (2002), Berlin: Springer, Berlin · Zbl 0984.00001 |
| [8] | Masser, D.; Zannier, U., Torsion anomalous points and families of elliptic curves, Am. J. Math., 132, 1677-1691 (2010) · Zbl 1225.11078 |
| [9] | Masser, D.; Zannier, U., Torsion points on families of squares of elliptic curves, Math. Ann., 352, 453-484 (2012) · Zbl 1306.11047 · doi:10.1007/s00208-011-0645-4 |
| [10] | Masser, D.; Zannier, U., Torsion points on families of products of elliptic curves, Adv. Math., 259, 116-133 (2014) · Zbl 1318.11075 · doi:10.1016/j.aim.2014.03.016 |
| [11] | Orr, M.: Introduction to abelian varieties and the Ax-Lindemann-Weierstrass theorem. In: O-minimality and Diophantine Geometry, London Math. Soc. Lecture Note Ser. vol. 421, pp. 100-128. Cambridge University Press, Cambridge (2015) · Zbl 1351.11037 |
| [12] | Pila, J.; Wilkie, AJ, The rational points of a definable set, Duke Math. J., 133, 3, 591-616 (2006) · Zbl 1217.11066 · doi:10.1215/S0012-7094-06-13336-7 |
| [13] | Pila, J.; Zannier, U., Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 19, 2, 149-162 (2008) · Zbl 1164.11029 · doi:10.4171/RLM/514 |
| [14] | Serre, J-P, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., 15, 4, 259-331 (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086 |
| [15] | Tarski, A., A Decision Method for Elementary Algebra and Geometry (1948), Santa Monica: RAND Corporation, Santa Monica · Zbl 0035.00602 |
| [16] | Zannier, U., Some Problems of Unlikely Intersections in Arithmetic and Geometry. With Appendixes by David Masser. Annals of Mathematics Studies 181 (2012), Princeton: Princeton University Press, Princeton · Zbl 1246.14003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
