Counting points on a family of degree one del Pezzo surfaces

Abstract

We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove asymptotics for a special subset of the rational points, specifically those that are integral with respect to the singularity. This method utilizes Mordell's parameterization of integral points on quadratic twists on elliptic curves, which is based on a syzygy for invariants of binary quartic forms. Let $F(A,B)$ denote the set of binary quartic forms with invariants $-4A$ and $-4B$ under the action of $\textrm{SL}_2(\mathbb{Z})$. We reduce the point-counting problem to the question of determining an asymptotic formula for the correlation sums of representation numbers of binary quadratic and binary quartic forms, where the quartic forms range in $F(A,B)$. These sums are then treated using a connection to modular forms.