107 research outputs found
The Tate-Voloch Conjecture for Drinfeld modules
We study the -adic distance from the torsion of a Drinfeld module to an
affine variety
Towards the full Mordell-Lang conjecture for Drinfeld modules
Let be a Drinfeld module of generic characteristic, and let be a
sufficiently generic affine subvariety of . We show that the
intersection of with a finite rank -submodule of 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 defined over a finite field with a closed
subvariety
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
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
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
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
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