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."

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 AndreOort 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

PostCritically 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

padic uniformization and the action of Galois on certain affine correspondences
draft available on the
arXiv

Canonical heights and preperiodic points for weighted homogeneous families of polynomials
draft available on the
arXiv

Finite ramification for preimage fields of postcritically finite morphisms
draft available on the
arXiv

Critical dynamics of variableseparated affine correspondences
draft available on the
arXiv

Canonical heights for correspondences
draft available on the
arXiv

Rigidity and height bounds for certain postcritically finite endomorphisms of projective space
draft available on the
arXiv

The torsion submodule of an abelian tmodule
in preparation

Variation of the canonical height for polynomials in several variables
draft available on the
arXiv

The filled Julia set of a Drinfeld module and uniform bounds for torsion

A lower bound for the canonical height associated to a Drinfeld module
draft available on the
arXiv

Attracting cycles in padic dynamics and height bounds for postcritically finite maps
draft available on the
arXiv

Canonical heights for Hénon maps
draft available on the
arXiv

Algebraic divisibility sequences over function fields
draft available on the
arXiv

Arboreal Galois representations and uniformization of polynomial dynamics
draft available on the
arXiv

A finiteness result for postcritically finite polynomials
draft available on the
arXiv

Variation of the canonical height for a family of polynomials
draft available on the
arXiv

Cubic polynomials with periodic cycles of a specified multiplier
draft available on the
arXiv

Specializations of elliptic surfaces, and divisibility in the MordellWeil group
draft available on the
arXiv

On Poonen's conjecture concerning rational preperiodic points of quadratic maps
draft available on the
arXiv

Uniform bounds on preimages under quadratic dynamical systems
draft available on the
arXiv

A quantitative primitive divisor result for elliptic divisibility sequences

Primitive divisors on twists of the Fermat cubic
draft available on the
arXiv

Multiples of integral points on elliptic curves
draft available on the
arXiv

The uniform primality conjecture for elliptic curves
Acta Arithmetica 134 (2008), pp. 157181
draft available on the
arXiv

Lower bounds on the canonical height associated to the morphism \(\phi(z)= z^d + c\)
draft available on the
arXiv

Primitive divisors in arithmetic dynamics
draft available on the
arXiv

Uniform estimates for primitive divisors in elliptic divisibility sequences

Approximating algebraic numbers by jinvariants of elliptic curves
Acta Arithmetica 131 (2008), pp. 5768

Elliptic divisibility sequences over certain curves
Received one of twenty "Top Cited Article 2007  2011" awards from JNT

Diophantine analysis and torsion on elliptic curves

On kth power numerical centres
Comptes rendus mathematiques de l'Academie des sciences (2006)

Torsion subgroups of elliptic curves in short Weierstrass form