Characterizing Algebraic Curves With Infinitely Many Integral Points

Abstract

A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper we give necessary and sufficient conditions for C to have infinitely many S-integral points.

Integral points and orbits of endomorphisms on the projective plane

“…are in O k,S and F (P u ) = u. It follows that P u ∈ (P 2 \ D)(O k,S ), viewing P u in P 2 (k), and similar to the proof of Lemma 25, Theorem 1.1 of [1] tells us that for all but finitely many u, the member of the pencil containing P u contains infinitely many integral points with respect to D. Assuming that O * k,S is infinite, by varying u, we conclude that (P 2 \ D)(O k,S ) is Zariski-dense.…”

“…In case (3), each point in the orbit of [3 : 2 : 1] under φ 6 is (D, S)-integral for S = {∞, 2, 3}, and it is easy to see from the 2-adic and the 3-adic valuations that an algebraic curve can only contain finitely many points in this orbit. For cases (1) and (2), it follows from a recent work of Xie [35] that there exists an algebraic point whose orbit under φ 6 is Zariski-dense (if there exists ℓ ≥ 1 such that the two eigenvalues of the tangent map at a fixed point of φ 6ℓ are multiplicatively independent, we can invoke [2, Corollary 2.7] instead). By enlarging k and S so that all coordinates of this point are S-units and that all the coefficients of F and G lie in O k,S , we see that each point in this orbit is (D, S)-integral.…”

Markoff–Rosenberger triples in geometric progression

“…However, C(O K ) may be finite if g = 0 and |C ∞ | ≤ 2. These last cases where treated by Alvanos, Bilu and Poulakis [2], obtaining a complete characterization of the cases in which C(O K ) is finite. In particular, our genus zero curve G satisfies |G ∞ | = 2 and both points at infinity are defined over Q, therefore [2, Theorem 1.2] asserts that G(O K ) is finite if and only if K = Q or K is an imaginary quadratic field.…”

Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations

“…All these numbers satisfy recurrence relations. For example, if we set S wheres n k are certain associated Stirling numbers of the first kind, there is no known formula analogous to (1) in the literature for Stirling numbers of the first kind. From the above identities, we see that for varying n and fixed a the function S n a is an exponential polynomial, or the nth term of a linearly recurrent sequence whose roots are all simple and given by {1, .…”