Uniform Boundedness Conjecture Research Articles

Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}\in{\mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${\mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${\mathbb{Q}}$, then $\operatorname{Per}_{n}(0)$ is irreducible over ${\mathbb{C}}$. To do this, we exhibit a ${\mathbb{Q}}$-rational smooth point on a projective completion of $\operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\operatorname{Per}_{n}(0)$ itself has no ${\mathbb{Q}}$-rational points.

The 𝔭-primary uniform boundedness conjecture for Drinfeld modules

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the [Formula: see text]-primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa’s result on the [Formula: see text]-primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties.

Non-density of Points of Small Arithmetic Degrees

Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $\alpha_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree $\alpha_f(x)$ is strictly smaller than the first dynamical degree $\delta_f$ of $f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small arithmetic degree is not Zariski dense, and verify this sAND Conjecture for endomorphisms on projective varieties including surfaces, HyperK\"ahler varieties, abelian varieties, Mori dream spaces, simply connected smooth varieties admitting int-amplified endomorphisms, smooth threefolds admitting int-amplified endomorphisms, and some fibre spaces. We show the equivalence of the sAND Conjecture and another conjecture on the periodic subvarieties of small dynamical degree; we also show the close relations between the sAND Conjecture and the Uniform Boundedness Conjecture of Morton and Silverman on endomorphisms of projective spaces and another long standing conjecture on Uniform Boundedness of torsion points in abelian varieties.

Dynamical uniform boundedness and the abc-conjecture

We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the abc-conjecture. For unicritical polynomials of degree at least five, we require only the standard abc-conjecture.

Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

Fix$d\geqslant 2$and a field$k$such that$\operatorname{char}k\nmid d$. Assume that$k$contains the$d$th roots of$1$. Then the irreducible components of the curves over$k$parameterizing preperiodic points of polynomials of the form$z^{d}+c$are geometrically irreducible and have gonality tending to$\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of$z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

Preperiodic points for quadratic polynomials over cyclotomic quadratic fields

Given a number field $K$ and a polynomial $f(z) \in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $\alpha \to \beta$ if and only if $f(\alpha) = \beta$. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field $K$, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over $\mathbb{Q}$, while recent work of the author, Faber, and Krumm has provided a detailed study of this question for all quadratic extensions of $\mathbb{Q}$. In this article, we give a conjecturally complete classification like Poonen's, but over the cyclotomic quadratic fields $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. The main tools we use are dynamical modular curves and results concerning quadratic points on curves.

Torsion points and height jumping in higher-dimensional families of abelian varieties

In 1983, Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has “small height” is finite. We conjecture a generalization to higher-dimensional families, where we replace “finite” by “not Zariski dense.” We show that this conjecture would imply the uniform boundedness conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.

On the cyclic torsion of elliptic curves over cubic number fields (II)

Merel’s result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell–Weil groups of all elliptic curves defined over number fields of fixed degree d. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For N=49,40,25 or 22, we show that ℤ/Nℤ is not a subgroup of E(K) tor for any elliptic curve E over a cubic number field K.

Dynamical modular curves for quadratic polynomial maps

Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph G G , we construct a curve X 1 ( G ) X_1(G) whose points parametrize quadratic polynomial maps—which, up to equivalence, form a one-parameter family—together with a collection of marked preperiodic points that form a graph isomorphic to G G . Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials.

Uniform boundedness in terms of ramification

Let $$d\ge 1$$ be fixed. Let F be a number field of degree d, and let E / F be an elliptic curve. Let $$E(F)_{\text {tors}}$$ be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E / F, such that the size of $$E(F)_{\text {tors}}$$ is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of $$E(F)_{\text {tors}}$$ . In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in $$E(F)_{\text {tors}}$$ . It has been conjectured, however, that there is a bound for the size of $$E(F)_{\text {tors}}$$ that is polynomial in d. In this article we show that under certain hypotheses there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers F over (p).

Reduction of dynatomic curves

In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families ${\mathcal{F}}$ of polynomial maps, such as the family $f_{c}(x)=x^{m}+c$, where $m\geq 2$. We do this by making use of the dynatomic modular curves $Y_{1}(n)$ (respectively $Y_{0}(n)$) which parametrize maps $f$ in ${\mathcal{F}}$ together with a point (respectively orbit) of period $n$ for $f$. The key point in our strategy is to study the set of primes $p$ for which the reduction of $Y_{1}(n)$ modulo $p$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $n$, a discriminant $D_{n}$ whose list of prime factors contains all the primes of bad reduction for $Y_{1}(n)$. In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $p$ dividing $D_{n}$: one guarantees that $p$ is in fact a prime of bad reduction for $Y_{1}(n)$, yet this same criterion implies that $Y_{0}(n)$ is geometrically irreducible. The other guarantees that the reduction of $Y_{1}(n)$ modulo $p$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $Y_{1}(n)$ for several primes dividing $D_{n}$ when $n=7,8,11$, and $f_{c}(x)=x^{2}+c$. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.

Galois groups in a family of dynatomic polynomials

For every nonconstant polynomial f∈Q[x], let Φ4,f denote the fourth dynatomic polynomial of f. We determine here the structure of the Galois group and the degrees of the irreducible factors of Φ4,f for every quadratic polynomial f. As an application we prove new results related to a uniform boundedness conjecture of Morton and Silverman. In particular we show that if f is a quadratic polynomial, then, for more than 39% of all primes p, f does not have a point of period four in Qp.

Preperiodic points for quadratic polynomials with small cycles over quadratic fields

Given a number field K and a polynomial $$f(z) \in K[z]$$ , one can naturally construct a finite directed graph G(f, K) whose vertices are the K-rational preperiodic points of f, with an edge $$\alpha \rightarrow \beta $$ if and only if $$f(\alpha ) = \beta $$ . The dynamical uniform boundedness conjecture of Morton and Silverman suggests that, for fixed integers $$n \ge 1$$ and $$d \ge 2$$ , there are only finitely many isomorphism classes of directed graphs G(f, K) as one ranges over all number fields K of degree n and polynomials $$f(z) \in K[z]$$ of degree d. In the case $$(n,d) = (1,2)$$ , Poonen has given a complete classification of all directed graphs which may be realized as $$G(f,\mathbb {Q})$$ for some quadratic polynomial $$f(z) \in \mathbb {Q}[z]$$ , under the assumption that f does not admit rational points of large period. The purpose of the present article is to continue the work begun by the author, Faber, and Krumm on the case $$(n,d) = (2,2)$$ . By combining the results of the previous article with a number of new results, we arrive at a partial result toward a theorem like Poonen’s—with a similar assumption on points of large period—but over all quadratic extensions of $$\mathbb {Q}$$ .

Ramification in the division fields of elliptic curves with potential supersingular reduction

Let d≥1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F)tors be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E/F, such that the size of E(F)tors is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of E(F)tors. In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in E(F)tors. It has been conjectured, however, that there is a bound for the size of E(F)tors that is polynomial in d. In this article we show that if E/F has potential supersingular reduction at a prime ideal above p, then there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the ramification index of the prime of supersingular reduction. Mathematics Subject Classification: Primary: 11G05, Secondary: 14H52

Small Dynamical Heights for Quadratic Polynomials and Rational Functions

Let be a polynomial or rational function of degree 2. A special case of Morton and Silverman’s dynamical uniform boundedness conjecture states that the number of rational preperiodic points of φ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height of a nonpreperiodic rational point x is bounded below by a uniform multiple of the height of φ itself. We provide support for these conjectures by computing the set of preperiodic and small-height rational points for a set of degree-2 maps far beyond the range of previous searches.

Une base explicite de symboles modulaires sur les corps de fonctions

Abstract Les symboles modulaires pour le sous-groupe Γ 0 ⁢ ( 𝔫 ) $\Gamma_{0}(\mathfrak{n})$ de GL 2 ⁢ ( 𝔽 q ⁢ [ T ] ) $\mathrm{GL}_{2}(\mathbb{F}_{q}[T])$ , définis par Teitelbaum, possèdent une présentation par un nombre fini de générateurs et relations, dans un formalisme similaire à celui de Manin pour les symboles modulaires classiques. Nous résolvons complètement ces relations et donnons une base explicite de générateurs lorsque 𝔫 $\mathfrak{n}$ est un idéal premier de degré impair. Comme application, nous donnons un énoncé de non-annulation de fonctions L de certaines formes automorphes pour 𝔽 q ⁢ ( T ) $\mathbb{F}_{q}(T)$ . Le résultat principal est aussi un argument-clé pour un énoncé en direction de la conjecture de borne uniforme pour les modules de Drinfeld de rang 2. Modular symbols for the subgroup Γ 0 ⁢ ( 𝔫 ) $\Gamma_{0}(\mathfrak{n})$ of GL 2 ⁢ ( 𝔽 q ⁢ [ T ] ) $\mathrm{GL}_{2}(\mathbb{F}_{q}[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin’s for classical modular symbols. We completely solve the relations and get an explicit basis of generators when 𝔫 $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for L-functions of certain automorphic cusp forms for 𝔽 q ⁢ ( T ) $\mathbb{F}_{q}(T)$ . The main statement also provides a key step for a result towards the uniform boundedness conjecture for Drinfeld modules of rank 2.

Rational preimages in families of dynamical systems

Let $${\phi}$$ be a rational function of degree at least two defined over a number field k. Let $${a \in \mathbb{P}^1(k)}$$ and let K be a number field containing k. We study the cardinality of the set of rational iterated preimages Preim $${(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a {\rm for some} N \geq 1\}}$$ . We prove two new results (Theorems 2 and 4) bounding $${|{\rm Preim}(\phi, a, K)|}$$ as $${\phi}$$ varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim $${(\phi, a, K)}$$ and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.

On the torsion of Drinfeld modules of rank two

We prove that the curve Y0(𝔭) has no 𝔽2(T)-rational points where 𝔭 ⊲ 𝔽2[T] is a prime ideal of degree at least 3 and Y0(𝔭) is the affine Drinfeld modular curve parameterizing Drinfeld modules of rank two over 𝔽2[T] of generic characteristic with Hecke-type level 𝔭-structure. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over 𝔽2(T) implying the uniform boundedness conjecture in this particular case. We reach our results with a variant of the formal immersion method. Moreover we show that the group Aut(X0(𝔭)) has order two. As a further application of our methods we also determine the prime-to-p cuspidal torsion packet of X0(𝔭) where 𝔭 ⊲ 𝔽q[T] is a prime ideal of degree at least 3 and q is a power of the prime p.

Formal Drinfeld modules

The formal group of an elliptic curve at a finite prime of the field of definition has proven to be a useful tool in studying the elliptic curve. Moreover, these formal groups are interesting in themselves. In this paper we define and study formal Drinfeld modules in a general setting. We also define the formal Drinfeld module associated to a Drinfeld module at a finite prime. The results are applied to the uniform boundedness conjecture for Drinfeld modules.

On the uniform boundedness conjecture for Drinfeld modules

In analogy with the famous theorems of Mazur and Merel on the torsion subgroups of elliptic curves, one can formulate similar conjectures for the torsion points of Drinfeld modules. We prove some partial results for rank 2 Drinfeld 𝔽q[T]-modules, for example the uniform boundedness of the 𝔭-primary torsion.