The Skolem–Mahler–Lech theorem states that if f(n) is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that f(m) is equal to 0 is the union of a finite number of arithmetic progressions in m � 0 and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y,andσ is an automorphism of Y, then the set of m such that σ m (q) lies in X is a union of a finite number of complete doubly-infinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem. 1.

We give a survey of Skolem’s problem for linear recurrence sequences. We cover the known decidable cases for recurrence depths of at most 4, and give detailed proofs for these cases. Moreover, we shall prove that the problem is decidable for linear recurrences of depth 5.

Abstract. Using the Skolem-Mahler-Lech theorem, we prove a dynamical Mordell-Lang conjecture for semiabelian varieties. 1.

Abstract. We study the orbits of a polynomial f ∈ C[X], namely the sets {α, f(α), f(f(α)),...} with α ∈ C. We prove that if two nonlinear complex polynomials f, 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. 1.

Abstract. This is an introduction to the method of Chabauty and Coleman, a p-adic method that attempts to determine the set of rational points on a given curve of genus g ≥ 2. We present the method, give a few examples of its implementation in practice, and discuss its effectiveness. An appendix treats the case in which the curve has bad reduction. 1. Rational points on curves of genus ≥ 2 We will work over the field Q of rational numbers, although everything we say admits an appropriate generalization to a number field. Let Q be an algebraic closure of Q. For each finite prime p, let Qp be the field of p-adic numbers (see [Kob84] for the definition). Curves will be assumed to be smooth, projective, and geometrically integral. Let X be a curve over Q of genus g ≥ 2. We suppose that X is presented as the zero set in some P n of an explicit finite set of homogeneous polynomials. We may give instead an equation for a singular (but still geometrically integral) curve in A 2; in this case, it is understood that X is the smooth projective curve birational to this singular curve. Rational points on X can be specified by giving their coordinates. (A little more data may be required if a singular model for X is used.) Let X(Q) be the set of rational points on X. Faltings ’ theorem [Fal83] states that X(Q) is finite. Thus we have the following welldefined problem: Given X of genus ≥ 2 presented as above, compute X(Q). Faltings ’ proof is ineffective in the sense that it does not provide an algorithm for solving this problem, even in principle. In fact, it is not known whether any algorithm is guaranteed to solve the problem. Even the case g = 2 seems hard. Nevertheless there are a few techniques that can be applied: see [Poo02] for a survey. On individual curves these seem to solve the problem often, perhaps even always when used together, though it seems very difficult to prove that they always work. One of the methods used is the method of Chabauty and Coleman.

Abstract. 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 (P 1) g has only finite intersection with any curve contained in (P 1) g. Our proof uses results from p-adic dynamics together with an integrality argument. 1.

Abstract. 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 the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang conjecture remains true in this setting if either (1) the semiabelian variety is simple, (2) the semiabelian variety is A 2, where A is a one-dimensional semiabelian variety, (3) the subvariety is a connected one-dimensional algebraic subgroup, or (4) each endomorphism has diagonalizable Jacobian at the origin. We also give examples showing that the conclusion fails if we make slight modifications to any of these hypotheses. 1.

This paper grew out of the following question. Given an ordinary elliptic curve E/Fq over a finite field Fq of characteristic p, consider the sequence of integers A(n), n ≥ 1, defined by #E(Fq n) = qn + 1 − A(n).

www.elsevier.com/locate/jnt Periodic points, linearizing maps, and the dynamical Mordell–Lang problem

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 (P 1) g.Ifthecoefficientsofϕ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (P 1) g has only finite intersection with any curve contained in (P 1) g.Wealsoshowthatourresultholdsfor indecomposable polynomials ϕ with coefficients in C. Ourproofusesresultsfrom p-adic dynamics together with an integrality argument. The extension to polynomials defined over C uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (ϕ, ϕ) on A 2.