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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 nonzero 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 2adic 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(x1)(xt), 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 xcoordinates 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 NeronTate height elliptic surfaces, together with complexdynamical 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 Bogomolovtype 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 'nonspecial' 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

Creator  
Publisher 
University of British Columbia

Date Issued 
2018

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 nonzero 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 2adic 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(x1)(xt), 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 xcoordinates 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 NeronTate height elliptic surfaces, together with complexdynamical 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 Bogomolovtype 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 'nonspecial' section P of a family of abelian varieties A → B that split as a product of elliptic curves.

Genre  
Type  
Language 
eng

Date Available 
20180411

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0365550

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201805

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

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Rights
AttributionNonCommercialNoDerivatives 4.0 International