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Dragos Ghioca

Publications and source records attributed to Dragos Ghioca.

At least 91 records · Page 5Linked to original sources
arXiv2012

Torsion points in families of Drinfeld modules

Let $Φ^ł$ be an algebraic family of Drinfeld modules defined over a field $K$ of characteristic $p$, and let $\bfa,\bfb\in K[ł]$. Assume that neither $\bfa(ł)$ nor $\bfb(ł)$ is a torsion point for $Φ^ł$ for all $ł$. If there exist infinitely many $ł\in\Kbar$ such that both $\bfa(ł)$ and $\bfb(ł)$ are torsion points for $Φ^ł$, then we show that for each $ł\in\Kbar$, we have that $\bfa(ł)$ is torsion for $Φ^ł$ if and only if $\bfb(ł)$ is torsion for $Φ^ł$. In the case $\bfa,\bfb\in K$, then we prove in addition that $\bfa$ and $\bfb$ must be $\Fpbar$-linearly dependent.

math.NT↗
arXiv2011

Periods of rational maps modulo primes

Let $K$ be a number field, let $ϕ\in K(t)$ be a rational map of degree at least 2, and let $α, β\in K$. We show that if $α$ is not in the forward orbit of $β$, then there is a positive proportion of primes ${\mathfrak p}$ of $K$ such that $α\mod {\mathfrak p}$ is not in the forward orbit of $β\mod {\mathfrak p}$. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $α$ by a hypersurface, such as the ramification locus of a morphism $ϕ: {\mathbb P}^{n} \to {\mathbb P}^{n}$.

math.AG↗
arXiv2011

Preperiodic points for families of polynomials

Let $a(λ)$ and $b(λ)$ be two polynomials with coefficients in complex numbers and let $f_{\lamb$ be a one-parameter family of polynomials indexed by all complex numbers $λ$. We study whether there exist infinitely many complex numbers $λ$ such that both $a(λ)$ and $b(λ)$ are preperiodic for $f_λ$.

math.DS↗
arXiv2010

Algebraic equations on the adelic closure of a Drinfeld module

Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\mathbf A}_K$ be a ring of adéles of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $Φ:{\mathbb F}[t] \to \operatorname{End}_K({\mathbb G}_a)$ over $K$ and a positive integer $g$ we regard both $K^g$ and ${\mathbf A}_K^g$ as $Φ({\mathbb F}_p[t])$-modules under the diagonal action induced by $Φ$. For $Γ\subseteq K^g$ a finitely generated $Φ(\F_p[t])$-submodule and an affine subvariety $X \subseteq \bG_a^g$ defined over $K$, we study the intersection of $X({\mathbf A}_K)$, the adèlic points of $X$, with $barΓ$, the closure of $Γ$ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than $X(K) \cap Γ$.

math.NT↗
arXiv2009

The Dynamical Mordell-Lang Conjecture

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $ϕ$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $ϕ$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(\bP^1)^g$ has only finite intersection with any curve contained in $(\bP^1)^g$. We also show that our result holds for indecomposable polynomials $ϕ$ with coefficients in $\bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $\bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(ϕ,ϕ)$ on $\bA^2$.

math.NT↗
arXiv2008

A gap principle for dynamics

Let $f_1,...,f_g\in {\mathbb C}(z)$ be rational functions, let $Φ=(f_1,...,f_g)$ denote their coordinatewise action on $({\mathbb P}^1)^g$, let $V\subset ({\mathbb P}^1)^g$ be a proper subvariety, and let $P=(x_1,...,x_g)\in ({\mathbb P}^1)^g({\mathbb C})$ be a nonpreperiodic point for $Φ$. We show that if $V$ does not contain any periodic subvarieties of positive dimension, then the set of $n$ such that $Φ^n(P) \in V({\mathbb C})$ must be very sparse. In particular, for any $k$ and any sufficiently large $N$, the number of $n \leq N$ such that $Φ^n(P) \in V({\mathbb C})$ is less than $\log^k N$, where $\log^k$ denotes the $k$-th iterate of the $\log$ function. This can be interpreted as an analog of the gap principle of Davenport-Roth and Mumford.

math.NT↗
arXiv2008

The dynamical Mordell-Lang problem for etale maps

We prove a dynamical version of the Mordell-Lang conjecture for etale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer positively a question of Keeler/Rogalski/Stafford for critically dense sequences of closed points of a Noetherian integral scheme.

math.NT↗
arXiv2008

Linear relations between polynomial orbits

We study the orbits of a polynomial f in C[X], namely the sets {e,f(e),f(f(e)),...} with e in C. We prove that if nonlinear complex polynomials f and g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C^d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell--Lang conjecture.

math.AG↗
arXiv2008

p-adic logarithms for polynomial dynamics

We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of the affine space A^g over a p-adic field, endowed with polynomial actions on each coordinate of A^g. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

math.NT↗
arXiv2008

Periodic points, linearizing maps, and the dynamical Mordell-Lang problem

We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism f:X --> X. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.

math.NT↗
arXiv2007

Siegel's theorem for Drinfeld modules

We prove an analog of Siegel's theorem for integral points in the context of Drinfeld modules. The result holds for finitely generated submodules of the additive group over a function field of transcendence dimension 1.

math.NT↗
arXiv2007

Equidistribution and integral points for Drinfeld modules

We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of Siegel's integral points theorem over number fields and to an analog of a theorem of Schinzel's regarding the order of a point modulo certain primes.

math.NT↗