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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.
Item Metadata
Title |
Unlikely intersections and equidistribution with a dynamical perspective
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2018
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Description |
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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Genre | |
Type | |
Language |
eng
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Date Available |
2018-04-11
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0365550
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2018-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
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DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International