. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...

Abstract. Given a global field K and a polynomial φ defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational preperiodic points of φ is bounded in terms of only the degree of K and the degree of φ. In 1997, for quadratic polynomials over K = Q, Call and Goldstine proved a bound which was exponential in s, the number of primes of bad reduction of φ. By careful analysis of the filled Julia sets at each prime, we present an improved bound on the order of s log s. Our bound applies to polynomials of any degree (at least two) over any global field K. Let K be a field, and let φ ∈ K(z) be a rational function. Let φ n denote the n th iterate of φ under composition; that is, φ 0 is the identity function, and for n ≥ 1, φ n = φ ◦ φ n−1. We will study the dynamics φ on the projective line P 1 (K). In particular, we say a point x is preperiodic under φ if there are integers n> m ≥ 0 such that φ m (x) = φ n (x). The point y = φ m (x) satisfies φ n−m (y) = y and is said to be periodic, as its iterates will forever cycle through the same finite sequence of values. Note that x ∈ P 1 (K) is preperiodic if and only if its orbit {φ n (x) : n ≥ 0} is finite. For example, let K = Q and φ(z) = z 2 −29/16. Then {5/4, −1/4, −7/4} forms a periodic

Abstract. We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x ↦ → x2 + c. This extends earlier results by Morton for N = 4 and Flynn, Poonen and Schaefer for N = 5. 1.

Abstract. We discuss the relationship between various additive problems concerning squares. 1. Squares in arithmetic progression Let σ(k) denote the maximum of the number of squares in a+b,...,a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [27] elegantly proved as follows: If there are more than δk squares amongst the integers a+b,..., a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such that each a + ijb is a square, by Szemerédi’s theorem. But then the a + ijb are four squares in arithmetic progression, contradicting a result of Fermat. This result can be extended to any given field L which is a finite extension of the rational numbers: From Faltings ’ theorem we know that there are only finitely many six term arithmetic progressions of squares in L, so from Szemerédi’s theorem we again deduce that there are oL(k) squares of elements of L in any k term arithmetic progression of numbers in L. (Xavier Xarles [28] recently proved that are never six squares in arithmetic progression in Z [ √ d] for any d.) In his seminal paper Trigonometric series with gaps [24] Rudin stated the following conjecture: Conjecture 1. σ(k) = O(k 1/2). It may be that the most squares appear √ in the arithmetic progression 49 + 24i, 1 ≤

The field of dynamical systems traces its roots to Poincaré’s qualitative study of solutions to differential equations in the late nineteenth century. The subfield of complex dynamics, which was initiated by Fatou and Julia in the late 1910s but which did not draw substantial attention until the 1980s, focuses mainly on the iteration of rational functions. In the 1990s, number theorists began to study such iterations as well, noting parallels to certain aspects of the theory of elliptic curves. Since then, number-theoretic dynamics has begun to emerge as a field in its own right, especially concerning the rationality properties of periodic points. 1. Rational preperiodic points A (discrete) dynamical system is a set X and a function φ: X → X; one considers the iterates φn: = φ ◦ φ ◦···◦φ for integers n ≥ 0. We say a point x ∈ X is periodic if φn (x) =x for some n ≥ 1; if n is the smallest such integer, the set {x, φ(x),...,φn−1 (x)} is called an n-cycle. More generally, given x ∈ X, ifthereis an integer m ≥ 0 such that φm (x) is periodic, we say x is preperiodic. Complex dynamics is devoted mostly to the case that X is the complex projective

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given k ≥ 4 and L ≥ 3 there are only finitely many arithmetic progressions of the form (x l0 0, xl1 1,..., xl k−1 k−1) with xi ∈ Z, gcd(x0, x1) = 1 and 2 ≤ li ≤ L for i = 0,1,..., k − 1. Furthermore, we show that, for L = 3, the progression (1, 1,..., 1) is the only such progression up to sign. Our proofs involve some well-known theorems of Faltings [F], Darmon and Granville [DG] as well as Chabauty’s method applied to superelliptic curves. 1.

Lower bounds on the canonical height associated to the morphism φ(z) = z d + c (Short title: Lower bounds on a certain canonical height)

Lower bounds on the canonical height associated to the morphism φ(z) = z d + c (Short title: Lower bounds on a certain canonical height)

Suppose that φ: P 1 → P 1 is a rational morphism, defined over a number field K. One may associate to φ, as a special case of a result of Call and Silverman [2], a canonical height function ˆ hφ: P 1 (K) → R satisfying the two properties ˆhφ(φ(α)) = deg(φ) ˆ hφ(α) and ˆ hφ(α) = h(α) + O(1), where h is the usual absolute logarithmic height. It is reasonably easy to show, from the two properties above, that ˆ hφ(α) = 0 just in case φ j (α) = φ i (α) for some j ̸ = i. If this is the case, we will say that α is a pre-periodic point for φ. It is natural to ask how small the value of ˆ hφ can be at points which are not pre-periodic, i.e., wandering points for φ. We will examine this question for morphisms for the form φ(z) = z d + c, for d ≥ 2. The canonical heights mentioned above are analogous to the canonical heights on elliptic curves (and more general abelian varieties) studied by Néron and Tate. The analogous question in this context, namely how small the canonical height of a non-torsion point on an elliptic curve may be, is the subject of a conjecture of Lang. Specifically, Lang conjectured that the height of such a point is bounded below by a constant multiple of max{h(jE),log |Norm K/Q D E/K|,1}, where jE and D E/K are the j-invariant and minimal discriminant of E/K respectively (see [5] for definitions of these terms). Silverman [4] has given a partial solution to this conjecture, proving that (for a non-torsion point P on an elliptic curve E) ˆh(P) ≥ C max{h(jE),log |Norm K/Q D E/K|,1}, where C depends on [K: Q], as well as the number of primes at which E has split multiplicative reduction.

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. Suppose that ϕc(z) = z2 + c, where c ∈ Q. We will say that α ∈ P1 (Q) is a periodic point with exact