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Unlikely intersections and equidistribution with a dynamical perspective Mavraki, Niki Myrto

Abstract

In this thesis we investigate generalizations of a theorem by Masser and Zannier concerning torsion specializations of sections in a fibered product of two elliptic surfaces. We consider the Weierstrass family of elliptic curves Et : y² = x³ + t and points Pt(a) = (a, √a³ + t) ∊ Et parametrized by non-zero t ∊ ℚ₂, where a ∊ ℚ₂. Given α,β ∊ ℚ₂ such that α/β ∊ ℚ, we provide an explicit description for the set of parameters t = λ, such that Pλ(α) and Pλ(β) are simultaneously torsion for Eλ. In particular, we prove that the aforementioned set is empty unless α/β ∊ {-2, -1/2}. Furthermore, we show that this set is empty even when α/β ∉ ℚ provided that a and b have distinct 2-adic absolute values and the ramification index e(ℚ₂(α/β) | ℚ₂) is coprime with 6. Our methods are dynamical. Using our techniques, we derive a recent result of Stoll concerning the Legendre family of elliptic curves Et : y² = x(x-1)(x-t), which itself strengthened earlier work of Masser and Zannier by establishing, as a special case, that there is no parameter t = λ ∊ ℂ \ {0,1} such that the points with x-coordinates a and b are both torsion Eλ, provided a,b have distinct reduction modulo 2. We also consider an extension of Masser and Zannier’s theorem in the spirit of Bogomolov's conjecture. Let π : E → B be an elliptic surface defined over a number field K, where B is a smooth projective curve, and let P : B → E be a section defined over K with canonical height ĥE(P) ≠ 0. We use Silverman's results concerning the variation of the Neron-Tate height elliptic surfaces, together with complex-dynamical arguments to show that the function t ↦ ĥE₁ (Pt) satisfies the hypothesis of Thuillier and Yuan’s equidistribution theorems. Thus, we obtain the equidistribution of points t ∊ B(K) where Pt is torsion. Finally, combined with Masser and Zannier’s theorems, we prove the Bogomolov-type extension of their theorem. More precisely, we show that there is a positive lower bound on the height ĥAt(Pt), after excluding finitely many points t ∊ B, for any 'non-special' section P of a family of abelian varieties A → B that split as a product of elliptic curves.

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