Torsion anomalous points and families of elliptic curves

By D. Masser, Umberto Zannier. American Journal of Mathematics, 2008

126 citations recorded by the source corpus.

We prove that there are at most finitely many complex $\lambda \neq 0,1$ such that two points on the Legendre elliptic curve $Y^2 = X(X-1)(X-\lambda)$ with coordinates $X = 2,3$ both have finite order. This is a very special case of some conjectures on unlikely intersections in semiabelian schemes.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/676b0b8f59f30259d03dfe1025cf62a34a0c87a6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

O-minimality and certain atypical intersections (2014, result): “The analogous statement in the context of algebraic tori was proved by Bombieri and Zannier, cf. Theorem 1 [50].”
Ax–Schanuel for the $j$-function (2014, result): “Related functional transcendence results in connection with Zilber-Pink have also appeared in the work of Betrand-Masser-Pillay-Zannier [16, 4] on “torsion anomalous” points.”
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$ (2011, methodology): “Masser and Zannier [56] prove a spe- cial case of Pink’s relative Manin-Mumford conjecture, and Pila [68] gives new proofs of some simple results of André-Oort-Manin-Mumford type (including the X = C2 and X = C×E cases of Theorem 1.1).”
Current trends and open problems in arithmetic dynamics (2018, background): “The following theorem, which is a dynamical analogue of a theorem of Masser and Zannier for elliptic curves [165], illustrates the type of result that one seeks.”
Ax-Lindemann for $\mathcal{A}_g$ (2014, unknown): “It provides a geometric characterization of the exceptional set in the Counting Theorem and in this role is analogous to functional transcendence statements in the context of other Zilber-Pink type problems (e.g., [16], [23]).”
Counting Lattice Points and O-Minimal Structures (2012, methodology): “These results in turn have been applied to problems in Diophantine geometry (see [24], [22], [19], [20] and [16]).”
Torsion points on families of squares of elliptic curves (2012, methodology): “We conclude the proof over Q as in [17] by combining Silverman’s Specialization Theorem [24] with David’s result [7] on counting conjugates of torsion points to show by contrast that the number of rational points is of order at least n for some δ > 0.”