Thomas J Tucker

University of Rochester | UR · Department of Mathematics

Let K K be the function field of a smooth irreducible curve defined over Q ¯ \overline {Q} . Let f ∈ K [ x ] f\in K[x] be of the form f ( x ) = x q + c f(x)=x^q+c , where q = p r , r ≥ 1 , q = p^{r}, r \ge 1, is a power of the prime number p p , and let β ∈ K ¯ \beta \in \overline {K} . For all n ∈ N ∪ { ∞ } n\in \mathbb {N}\cup \{\infty \} , the G...

A QUESTION FOR ITERATED GALOIS GROUPS IN ARITHMETIC DYNAMICS - ANDREW BRIDY, JOHN R. DOYLE, DRAGOS GHIOCA, LIANG-CHUNG HSIA, THOMAS J. TUCKER

We advance a new conjecture in the spirit of the dynamical Manin–Mumford conjecture. We show that our conjecture holds for all polarisable endomorphisms of abelian varieties and for all polarisable endomorphisms of $(\mathbb{P}^{1})^{N}$ . Furthermore, we show various examples which highlight the restrictions one would need to consider in formulati...

We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry and partly from p p -adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some...

Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $\beta\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(\beta)=\text{Gal}(K(f^{-n}(\beta))/K)$ embed into $\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major prob...

Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $\beta\in \overline{K}$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(\beta)=\mathop{\rm{Gal}}(K(f^{-n}(\beta))/K(\be...

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some...

Using methods of p-adic analysis, along with the powerful result of Medvedev-Scanlon (Annals of Mathematics, 2014) for the classification of periodic subvarieties of (P^1)^n, we bound the length of the orbit of a periodic subvariety Y of (P^1)^n under the action of a dominant endomorphism.

Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $\beta\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(\beta)=\text{Gal}(K(f^{-n}(\beta))/K)$ embed into $\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major prob...

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general...

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general...

Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such that for some integers m_{1,t} and m_{2,t}, we have that [m_{i,t}](P_i)_t = (Q_i)_t on E_i (for i = 1,2), then a...

Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such that for some integers m_{1,t} and m_{2,t}, we have that [m_{i,t}](P_i)_t = (Q_i)_t on E_i (for i = 1,2), then a...

Let L be a field of characteristic p, and let a, b, c, d ε L(T). Assume that a and b are algebraically independent over Fp. Then for each fixed positive integer n, we prove that there exist at most finitely many λ ε L satisfying f(a(λ)) = c(λ) and g(b(λ)) = d(λ) for some polynomials f, g ε Fpn[Z] such that f(a) ≠ c and g(b) ≠ d. Our result is a cha...

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n - 1, b^n - 1) \mid h\] We prove a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$...

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n - 1, b^n - 1) \mid h\] We prove a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$...

Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes where f has bad reduction. After excluding all the natural counter-examples, we define a subset A(f,a) of pair...

Let K be a number field or a function field of characteristic 0. If K is a number field, assume the abc-conjecture for K. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in that are not postcritically finite. For example, suppose K is a number field and is not postcritically finite, and let be...

Let X be a Noetherian space, let Φ : X → X be a continuous function, let Y ⊆ X be a closed set, and let x ∈ X. We show that the set S := {n ∈ N: Φn(x) ∈ Y} is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety X, any rational map Φ : X →...

Let $c_1, c_2, c_3$ be distinct complex numbers, and let $d\ge 3$ be an integer. We show that the set of all pairs $(a,b)\in \mathbb{C}\times \mathbb{C}$ such that each $c_i$ is preperiodic for the action of the polynomial $x^d+ax+b$ is not Zariski dense in the affine plane.

Let $\varphi: {\mathbb P}^1 \longrightarrow {\mathbb P}^1$ be a rational map of degree greater than one defined over a number field $k$. For each prime ${\mathfrak p}$ of good reduction for $\varphi$, we let $\varphi_{\mathfrak p}$ denote the reduction of $\varphi$ modulo ${\mathfrak p}$. A random map heuristic suggests that for large ${\mathfrak p...

Let K be a function field over an algebraically closed field k of characteristic 0, let ϕ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which ϕ is totally ramified and let α ∈ K. We show that for all but finitely many pairs (m, n) ∈ \$\mathbb{Z}\$≥0 × \$\mathbb{N}\$ there exists a place \$\mathfrak{p}\$...

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any f(z) in K(z) of degree d at least 2 which is not a d-th power in \bar{K}(z), Siegel's theorem implies that the image set f(K) contains only finitely many S-units. We conjecture that the number of such S-units is bounded by a function...

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given...

Using methods of p-adic analysis we give a different proof of Burnside's problem for automorphisms of quasiprojective varieties X defined over a field of characteristic 0. More precisely, we show that any finitely generated torsion subgroup of Aut(X) is finite. In particular this yields effective bounds for the size of torsion of any semiabelian va...

The Mordell-Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which...

Using methods of p-adic analysis we give a different proof of Burnside's problem for automorphisms of quasiprojective varieties X defined over a field of characteristic 0. More precisely, we show that any finitely generated torsion subgroup of Aut(X) is finite. In particular this yields effective bounds for the size of torsion of any semiabelian va...

We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on (\mathbb P1)g{(\mathbb P^1)^g}. If the coefficients of φ are algebraic, we show that the orbit o...

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such...

Let K be a number field, let phi(x)is an element of K(x) be a rational function of degree d > 1, and let alpha is an element of K be a wandering point such that phi(n)(alpha)not equal 0 for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that v(p)(phi(n)(al...

Let K be a number field, let f: P_1 --> P_1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2 such that f^m(u) is S-integral relative to f^n(w) is finite and effectively computable. This may be thought of as...

Let $K$ be a number field, let $\phi \in K(t)$ be a rational map of degree at least 2, and let $\alpha, \beta \in K$. We show that if $\alpha$ is not in the forward orbit of $\beta$, then there is a positive proportion of primes ${\mathfrak p}$ of $K$ such that $\alpha \mod {\mathfrak p}$ is not in the forward orbit of $\beta \mod {\mathfrak p}$. M...

We provide a family of counterexamples to a first formulation of the dynamical Manin–Mumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of Abelian varieties under the action of group endomorphisms and for lines under the action of diagonal endomorphisms of .

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

Given two rational maps $\varphi$ and $\psi$ on $\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<\varphi,\psi>$ which is closely related to the canonical height functions $h_\varphi$ and $h_\psi$ associated to these maps. Our main results show a strong connection between the value of $<\varphi,\psi>$ and the c...

Given two rational maps $\varphi$ and $\psi$ on $\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<\varphi,\psi>$ which is closely related to the canonical height functions $h_\varphi$ and $h_\psi$ associated to these maps. Our main results show a strong connection between the value of $<\varphi,\psi>$ and the c...

Under suitable hypotheses, we prove a dynamical version of the Mordell–Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism Φ:X→X. We also prove a version of the Mordell–Lang conjecture that holds for any endomorphism of a semiabelian variety. We use an analytic method based on the technique of Skol...

Let $f_1,...,f_g\in {\mathbb C}(z)$ be rational functions, let $\Phi=(f_1,...,f_g)$ 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 $\Phi$. We show that if $V$ does not contain any periodic su...

We prove a dynamical version of the Bogomolov conjecture in the special case of lines in affine space A^m under the action of a map (f_1,...,f_m) where each f_i is a polynomial in Q-bar[X] of the same degree.

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...

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...

Let K be a number field with algebraic closure K-bar, let S be a finite set of places of K containing the archimedean places, and let f be a Chebyshev polynomial. We prove that if a in K-bar is not preperiodic, then there are only finitely many preperiodic points b in K-bar which are S-integral with respect to a.

For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K:Q], and B, and is valid for all b outside an explicit finite set. We also sh...

We prove a dynamical version of the Mordell–Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to those employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(P^1...

Using the Skolem-Mahler-Lech theorem, we prove a dynamical Mordell-Lang conjecture for semiabelian varieties.

We prove a Siegel type statement for finitely generated f\phi-submodules of \mathbbGa\mathbb{G}_a under the action of a Drinfeld module f\phi. This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of \mathbbP1\mathbb{P}^1 o...

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.

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture. Comment: 19 pages; to appear in Inventiones Mathematicae; in this version we modified the exposition in...

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.

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...

The Mahler measure formula expresses the height of an algebraic number as the integral of the log of the absolute value of its minimal polynomial on the unit circle. The height is in fact the canonical height associated to the monomial maps xn. We show in this work that for any rational map ϕ(x) the canonical height of an algebraic number with resp...

Using an alternative notion of good reduction, an analog of the Shafarevich theorem for elliptic curves is proved for morphisms of the projective line over number fields.

We give a geometric proof that one may compute a particular generalized Mahler integral using equidistribution of preperiodic points of a dynamical system on the sphere. The dynamical system is associated to the multiplication by 2 map on an elliptic curve over a number field K with Weierstrass equation y^2 = P(x) (a Lattes dynamical system).

We show that if f: X → Y is a finite, separable morphism of smooth curves defined over a finite field 𝔽q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(𝔽q) surjectively onto Y(𝔽q) if and only if f maps X(𝔽q) injectively into Y(𝔽q). Surprisingly, the bounds on q for these two implicat...

If K is a number field and \(\varphi: {\mathcal{P}}_{K}^{1}\rightarrow {\mathcal{P}}_{K}^{1}\) is a rational map of degree d > 1, then at each place v of K, one can associate to φ a generalized Mahler measure for polynomials F ∈ K[t]. These Mahler measures give rise to a formula for the canonical height h φ(β) of an element \(\beta\in \overline{K}\...

This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k, there exist infinitely many k-rational points on the projective line such that the fiber of f over P is irreducible over k. In this paper, we consid...

Introduction Let f : X Gamma! Y be a finite separable morphism of (smooth, projective, geometrically irreducible) curves defined over the finite field F q . Then f is called an exceptional cover if the diagonal is the only geometrically irreducible component of the fiber product X Theta Y X which is defined over F q . The primary interest of except...

If one cannot solve a problem, then – according to a heuristic of George Pólya – there must be an easier problem that one cannot solve. The authors apply this principle to number theory.

This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k, there exist infinitely many k-rational points on the projective line such that the fiber of f over P is irreducible over k. In this paper, we consid...

The abc-conjecture is applied to various questions involving the number of distinct fields Q( p f(n)), as we vary over integers n.

this article that g(X F;h ) 2

This paper addresses questions involving the sharpness of Vojtas inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D,-) in such a way that Vojtas conjecture is sharp when the approximation term is not included (that is, when D=0). In Theorem 3.7, it is demonstrated that Vojtas co...

This paper addresses questions involving the sharpness of Vojta's conjecture and Vojta's inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D,-) in such a way that Vojta's inequality is sharp in Theorem 2.3. Partial results are obtained for the more difficult problem of showing th...

this paper. We begin with some definitions and explanations of notation in Section 2. Then, in Section 3, we show that an analogue of Hilbert's irreducibility theorem holds for elliptic curves. It follows immediately from a result of [A-H] that one hold as well for any curve possessing infinitely many points of degree 3 or less. The question for el...

this paper. For example, it is probably possible to find explicit constants in the case of genus 0 algebraic approximations of bounded degree (recall that [V 3] is only for rational approximations). These constants would most likely be very large, at least relative to [V 3], in light of the fact that we apparently must use Roth's lemma rather than...

This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pullbacks of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent a...

If the finite group G acts on the finite non-empty set X (i.e., G is represented as a group of permutations of X), then ....

Thesis (Ph. D.)--University of Rochester. Dept. of Mathematics, 2009. [Abstract would not render]--Submitter.

Thesis (Ph. D.)--University of Rochester. Dept. of Mathematics, 2009. We investigate an assortment of questions about variations of heights in a number field K and a function field Fp(T), and the effect these variations have on the distribution of points. A theorem of Y. Bilu describes the equidistribution of points (and their conjugates) with smal...