Patrick Ingram

Primitive divisors in arithmetic dynamics

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.

On Poonen's conjecture concerning rational preperiodic points of quadratic maps

The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are at most 9 periodic points defined over the rational numbers for any map in the family x^2 + c for c rational. We verify this conjecture for c values up to height 10^8. For quadratic number fields, we provide evidence that the upper bound on the exact period of Q-rational periodic point is 6.

Arboreal Galois representations and uniformization of polynomial dynamics

Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) to the multiplicative action z->z^d. The relation between this construction and the Bottcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves, and Tates p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory.

Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatous classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

Uniform estimates for primitive divisors in elliptic divisibility sequences

Let P be a nontorsion rational point on an elliptic curve E, given by a minimal Weierstrass equation, and write the first coordinate of nP as A n ∕ D n 2, a fraction in lowest terms. The sequence of values D n is the elliptic divisibility sequence (EDS) associated to P. A prime p is a primitive divisor of D n if p divides D n , and p does not divide any earlier term in the sequence. The Zsigmondy set for P is the set of n such that D n has no primitive divisors. It is known that Z is finite. In the first part of the paper we prove various uniform bounds for the size of the Zsigmondy set, including (1) if the j-invariant of E is integral, then the size of the Zsigmondy set is bounded independently of E and P, and (2) if the abc Conjecture is true, then the size of the Zsigmondy set is bounded independently of E and P for all curves and points. In the second part of the paper, we derive upper bounds for the maximum element in the Zsigmondy set for points on twists of a fixed elliptic curve.

Primitive Divisors on Twists of Fermat's Cubic

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m , with m cube-free, all the terms beyond the first have a primitive divisor.

The uniform primality conjecture for elliptic curves

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Langs conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed.

Torsion subgroups of elliptic curves in short Weierstrass form

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves E/Q in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any e > 0, all but finitely many curves E A,B : y 2 = x 3 + Ax + B, where A and B are integers satisfying A > |B| 1+e > 0, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to |A| > |B| 2+e > 0, then the E A,B now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterle.

Canonical heights for Hénon maps

We consider the arithmetic of Henon maps f(x, y)=(ay, x+f(y)) defined over number fields and function fields, usually with the restriction that a=1. We prove a result on the variation of Kawaguchis canonical height in families of Henon maps, and derive from this a specialization theorem, showing that the set of parameters above which a given non-periodic point becomes periodic is a set of bounded height. Proving this involves showing that the only points of canonical height zero for a Henon map over a function field are those which are periodic (in the non-isotrivial case). In the case of quadratic Henon maps f(x, y)=(y, x+y^2+b), we obtain a stronger result, bounding the canonical height below by a quantity which grows linearly in the height of b, once the number of places of bad reduction is fixed. Finally, we propose a conjecture regarding rational periodic points for quadratic Henon maps defined over the rational numbers, namely that they can only have period 1, 2, 3, 4, 6, or 8. We check this conjecture for the first million values of the parameter b, ordered by height.

Algebraic Divisibility Sequences Over Function Fields

In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.