We show that the Burkhardt quartic threefold is rational over any field of characteristic distinct from 3. We compute its zeta function over finite fields. We realize one of its moduli interpretations explicitly by determining a model for the universal genus 2 curve over it, as a double cover of the projective line. We show that the j-planes in the Burkhardt quartic mark the order 3 subgroups on the Abelian varieties it parametrizes, and that the Hesse pencil on a j-plane gives rise to the universal curve as a discriminant of a cubic genus 1 cover.

In this paper we consider the density of rational points on the "stacky" curve X(P 1 ; 0, 2; 1, 2; ∞, 2) which is P 1 with three half points, with respect to the so-called Ellenberg-Satriano-Zuerick-Brown height. In particular, we prove a conjecture of Ellenberg.

In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is nonpositive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space P 1 , is equivalent to the abc-conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.

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Let f : X → X be a surjective endomorphism of a normal projective variety defined over a number field. The dynamics of f may be studied through the dynamics of the linear action f * : Pic(X) R → Pic(X) R , which are governed by the spectral theory of f * . Let λ 1 (f ) be the spectral radius of f * . We study Q-divisors D with f * D = λ 1 (f )D and κ(D) = 0 where κ(D) is the Iitaka dimension of the divisor D. We analyze the base locus of such divisors and interpret the set of small eigenvalues in terms of the canonical heights of Jordan blocks described by Kawaguchi and Silverman. Finally we identify a linear algebraic condition on surjective morphisms that may be useful in proving instances of the Kawaguchi-Silverman conjecture.