Counting points on a family of degree one del Pezzo surfaces

TL;DR

This paper investigates rational points on a specific elliptic surface defined by a cubic equation with quadratic form coefficients, providing asymptotic counts for integral points related to binary quartic forms and modular forms.

Contribution

It introduces a novel approach to counting rational points on a family of degree one del Pezzo surfaces using invariants of binary quartic forms and modular forms.

Findings

Established asymptotics for integral rational points on the surface.

Connected point-counting to correlation sums of binary quadratic and quartic forms.

Utilized modular forms to analyze the sums and derive results.

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.