The Tate-Voloch Conjecture for Drinfeld modules

We study the $v$-adic distance from the torsion of a Drinfeld module to an affine variety

Towards the full Mordell-Lang conjecture for Drinfeld modules

Let $\phi$ be a Drinfeld module of generic characteristic, and let $X$ be a sufficiently generic affine subvariety of $\mathbb{G}_a^g$. We show that the intersection of $X$ with a finite rank $\phi$-submodule of $\mathbb{G}_a^g$ is finite

The Mordell-Lang Theorem for Drinfeld modules

We study the quasi-endomorphism ring of infinitely definable subgroups in separably closed fields. Based on the results we obtain, we are able to prove a Mordell-Lang theorem for Drinfeld modules of finite characteristic. Using specialization arguments we are able to prove also a Mordell-Lang theorem for Drinfeld modules of generic characteristic.Comment: 20 page

The Mordell-Lang Theorem for finitely generated subgroups of a semiabelian variety defined over a finite field

We determine the structure of the intersection of a finitely generated subgroup of a semiabelian variety $G$ defined over a finite field with a closed subvariety $X\subset G$

Elliptic curves over the perfect closure of a function field

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated

Integral points for Drinfeld modules

We prove that in the backward orbit of a non-preperiodic point under the action of a Drinfeld module of generic characteristic there exist at most finitely many points S-integral with respect to another nonpreperiodic point. This provides the answer (in positive characteristic) to a question raised by Sookdeo. We also prove that for each nontorsion point z, there exist at most finitely many torsion points which are S-integral with respect to z. This proves a question raised by Tucker and the author, and it gives the analogue of Ih's conjecture for Drinfeld modules

The Local Lehmer Inequality For Drinfeld Modules

We give a lower bound for the local height of a non-torsion element of a Drinfeld module.Comment: 20 page

A Bogomolov type statement for function fields

Let k be a an algebraically closed field of arbitrary characteristic, and we let h be the usual Weil height for the n-dimensional affine space corresponding to the function field k(t) (extended to its algebraic closure). We prove that for any affine variety V defined over the algebraic closure of k(t), there exists a positive real number c such that if P is an algebraic point of V and h(P)< c, then P has its coordinates in k

Equidistribution for torsion points of a Drinfeld module

We prove an equidistribution result for torsion points of Drinfeld modules of generic characteristic. We also show that similar equidistribution statements provide proofs for the Manin-Mumford and the Bogomolov conjectures for Drinfeld modules

The orbit intersection problem for linear spaces and semiabelian varieties

Let f_1 and f_2 be affine maps of the N-th dimensional affine space over the complex numbers, i.e., f_i(x):=A_i x + y_i (where each A_i is an N-by-N matrix and y_i is a given vector), and let x_1 and x_2 be vectors such that x_i is not preperiodic under the action of f_i for i=1,2. If none of the eigenvalues of the matrices A_i is a root of unity, then we prove that the set of pairs (n_1,n_2) of non-negative integers such that f_1^{n_1}(x_1)=f_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + \ell_1, m_2k + \ell_2) where m_1, m_2, \ell_1, \ell_2 are given non-negative integers, and k is varying among all non-negative integers. Using this result, we prove that for any two self-maps \Phi_i(x) := \Phi_{i,0}(x)+y_i on a semiabelian variety X defined over the complex numbers (where \Phi_{i,0} is an endomorphism of X and y_i is a given point of X), if none of the eigenvalues of the induced linear action D\Phi_{i,0} on the tangent space at the identity 0 of X is a root of unity (for i=1,2), then for any two non-preperiodic points x_1,x_2, the set of pairs (n_1,n_2) of non-negative integers such that \Phi_1^{n_1}(x_1) = \Phi_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + \ell_1, m_2k + \ell_2) where m_1,m_2,\ell_1,\ell_2 are given non-negative integers, and k is varying among all non-negative integers. We give examples to show that the above condition on eigenvalues is necessary and introduce certain geometric properties that imply such a condition. Our method involves an analysis of certain systems of polynomial-exponential equations and the p-adic exponential map for semiabelian varieties