The second moment of the number of integral points on elliptic curves is bounded

In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219 \ldots$, the $s$-th moment of the number of integral points is bounded for many families of elliptic curves --- e.g., for the family of all integral short Weierstrass curves ordered by naive height, for the family of only minimal such Weierstrass curves, for the family of semistable curves, or for subfamilies thereof defined by finitely many congruence conditions. For certain other families of elliptic curves, such as those with a marked point or a marked $2$-torsion point, the same methods show that for $0 < s < \log_2 3 = 1.5850\ldots$, the $s$-th moment of the number of integral points is bounded. The main new ingredient in our proof is an upper bound on the number of integral points on an affine integral Weierstrass model of an elliptic curve depending only on the rank of the curve and the number of square divisors of the discriminant. We obtain the bound by studying a bijection first observed by Mordell between integral points on these curves and certain types of binary quartic forms. The theorems on moments then follow from H\"older's inequality, analytic techniques, and results on bounds on the average sizes of Selmer groups in the families.Comment: 14 pages, comments welcome

Orbit Parametrizations for K3 Surfaces

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose N\'eron-Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.Comment: 83 pages; to appear in Forum of Mathematics, Sigm