Dragos Ghioca

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - Nov 2019)
Unlikely intersections and equidistribution with a dynamical perspective (2018)

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 parameterst = 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 andb 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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Master's Student Supervision (2010 - 2018)
Two Special Cases of the Dynamical Mordell-Lang Conjecture in Positive Characteristic (2017)

We prove the positive characteristic version of the Dynamical Mordell-Lang Conjecture in two novel cases. Let p be a prime and K a field of characteristic p>0. Let k ∈ ℕ, and let G denote the multiplicative group of K, of dimension k. Let α be an element of G, and V a variety contained in G. Let φ: G →G be a group endomorphism defined over K. We knowφ(x₁,x₂,...,xk)=(x₁^a1,1 x₂^a1,2 ··· xk^a1k , ... , x₁^ak1 x₂^ak2 ··· xk^akk),for some integer exponents aij. In the case where the matrix of exponents, ( aij ) is similar to a single Jordan block, we show that the set S = { n ∈ ℕ double : φ^n(α) ∈ V } is a finite union of arithmetic progressions. When the dimension k = 3, we show S is a finite union of arithmetic progressions for any group endomorphism φ.

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Points of small height on plane curves (2014)

Let K be an algebraically closed field, and let C be an irreducible plane curve, defined over the algebraic closure of K(t), which is not defined over K. We show that there exists a positive real number c₀ such that if P is any point on the curve C whose Weil height is bounded above by c₀, then the coordinates of P belong to K.

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