Graham Everest

S-integer dynamical systems: periodic points.

We associate via duality a dynamical system to each pair (RS,x), where RS is the ring of S-integers in an A-field k, and x is an element of RS\{0}. These dynamical systems include the circle doubling map, certain solenoidal and toral endomorphisms, full one- and two-sided shifts on prime power alphabets, and certain algebraic cellular automata. In the arithmetic case, we show that for S finite the systems have properties close to hyperbolic systems: the growth rate of periodic points exists and the periodic points are uniformly distributed with respect to Haar measure. The dynamical zeta function is in general irrational however. For S infinite the systems exhibit a wide range of behaviour. Using Heath-Browns work on the Artin conjecture, we exhibit examples in which S is infinite but the upper growth rate of periodic points is positive.

Primes in elliptic divisibility sequences.

Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.

PRIMES GENERATED BY ELLIPTIC CURVES

For a rational elliptic curve in Weierstrass form, Chud- novsky and Chudnovsky considered the likelihood that the denom- inators of the x-coordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a ratio- nal point under an isogeny, we use Siegels Theorem to prove that only nitely many primes will arise. The same question is consid- ered for elliptic curves in homogeneous form prompting a visit to Ramanujans famous taxi-cab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.

Primitive divisors of elliptic divisibility sequences

Silverman proved the analogue of Zsigmondys Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.

Prime powers in elliptic divisibility sequences

Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.

Primes Generated by Recurrence Sequences

We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of terms with a primitive divisor has a natural density. We discuss two heuristic arguments to suggest a value for that density, one using recent advances made about the distribution of roots of polynomial congruences.

Descent on elliptic curves and Hilbert’s tenth problem

. Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilberts Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers.

The Elliptic Mahler Measure

This chapter has three themes. In Section 6.1 a very short introduction to the classical theory of elliptic functions is given. In Sections 6.2, 6.3 and 6.5 some recent work on the elliptic Mahler measure is presented. Sections 6.4 and 6.6 contain evidence for a possible family of dynamical systems associated to elliptic curves whose dynamical properties are linked to the elliptic Mahler measure in a manner analogous to the connection between Chapters 1 and 3 and Chapters 2 and 4. This third theme is more speculative in nature.

Primes in sequences associated to polynomials (after Lehmer)

In a paper of 1933, D.H. Lehmer continued Pierces study of integral sequences associated to polynomials, generalizing the Mersenne sequence. He developed divisibility criteria, and suggested that prime apparition in these sequences -- or in closely related sequences -- would be denser if the polynomials were close to cyclotomic, using a natural measure of closeness. We review briefly some of the main developments since Lehmers paper, and report on further computational work on these sequences. In particular, we use Mossinghoffs collection of polynomials with smallest known measure to assemble evidence for the distribution of primes in these sequences predicted by standard heuristic arguments. The calculations lend weight to standard conjectures about Mersenne primes, and the use of polynomials with small measure permits much larger numbers of primes to be generated than in the Mersenne case.

Dirichlet series for finite combinatorial rank dynamics

We introduce a class of group endomorphisms - those of finite combinatorial rank - exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.