Research
My research is in number theory, and in particular diophantine geometry.
My specific interests lie in the arithmetic of elliptic curves and
surfaces, and in the theory of dynamical systems over global fields.
Publications
These are listed in approximate reverse chronological order. Please note that draft versions on the arXiv may differ from published versions, and only the journal versions are "official."
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Solutions to difference equations have few defects
draft available on the
arxiv -
Effective finiteness of solutions to certain differential and difference equations
draft available on the
arxiv -
Degree gaps for multipliers and the dynamical Andre-Oort conjecture
draft available on the
arxiv -
Minimally critical regular endomorphisms of \(\mathbb{A}^N\)
draft available on the
arxiv -
Minimally critical endomorphisms of \(\mathbb{P}^N\)
draft available on the
arxiv -
Post-Critically Finite Maps on \(\mathbb{P}^n\) for \(n\geq 2\) are Sparse
draft available on the
arxiv -
Current Trends and Open Problems in Arithmetic Dynamics
draft available on the
arxiv -
Critical orbits of polynomials with a periodic point of specified multiplier
draft available on the
arxiv -
The critical height is a moduli height
draft available on the
arXiv
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p-adic uniformization and the action of Galois on certain affine correspondences
draft available on the
arXiv
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Canonical heights and preperiodic points for weighted homogeneous families of polynomials
draft available on the
arXiv
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Finite ramification for preimage fields of postcritically finite morphisms
draft available on the
arXiv
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Critical dynamics of variable-separated affine correspondences
draft available on the
arXiv
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Canonical heights for correspondences
draft available on the
arXiv
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Rigidity and height bounds for certain post-critically finite endomorphisms of projective space
draft available on the
arXiv
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The torsion submodule of an abelian t-module
in preparation
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Variation of the canonical height for polynomials in several variables
draft available on the
arXiv
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The filled Julia set of a Drinfeld module and uniform bounds for torsion
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A lower bound for the canonical height associated to a Drinfeld module
draft available on the
arXiv
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Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps
draft available on the
arXiv
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Canonical heights for Hénon maps
draft available on the
arXiv
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Algebraic divisibility sequences over function fields
draft available on the
arXiv
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Arboreal Galois representations and uniformization of polynomial dynamics
draft available on the
arXiv
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A finiteness result for post-critically finite polynomials
draft available on the
arXiv
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Variation of the canonical height for a family of polynomials
draft available on the
arXiv
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Cubic polynomials with periodic cycles of a specified multiplier
draft available on the
arXiv
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Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group
draft available on the
arXiv
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On Poonen's conjecture concerning rational preperiodic points of quadratic maps
draft available on the
arXiv
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Uniform bounds on pre-images under quadratic dynamical systems
draft available on the
arXiv
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A quantitative primitive divisor result for elliptic divisibility sequences
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Primitive divisors on twists of the Fermat cubic
draft available on the
arXiv
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Multiples of integral points on elliptic curves
draft available on the
arXiv
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The uniform primality conjecture for elliptic curves
Acta Arithmetica 134 (2008), pp. 157-181
draft available on the
arXiv
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Lower bounds on the canonical height associated to the morphism \(\phi(z)= z^d + c\)
draft available on the
arXiv
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Primitive divisors in arithmetic dynamics
draft available on the
arXiv
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Uniform estimates for primitive divisors in elliptic divisibility sequences
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Approximating algebraic numbers by j-invariants of elliptic curves
Acta Arithmetica 131 (2008), pp. 57-68
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Elliptic divisibility sequences over certain curves
Received one of twenty "Top Cited Article 2007 - 2011" awards from JNT
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Diophantine analysis and torsion on elliptic curves
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On k-th power numerical centres
Comptes rendus mathematiques de l'Academie des sciences (2006)
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Torsion subgroups of elliptic curves in short Weierstrass form
