Integral points on elliptic curves with $j$-invariant $0$ over $k(t)$

TL;DR

This paper establishes bounds on the number of integral points on elliptic curves with $j$-invariant 0 over function fields, improving previous results and providing explicit formulas for certain cases.

Contribution

It introduces new bounds for integral points on these elliptic curves using descent methods and offers explicit formulas for small degree cases, enhancing understanding of related Diophantine equations.

Findings

Bounds for integral points improve previous results

Explicit formulas for degree ≤ 6 cases

Connections to Picard groups and Pillai's equation

Abstract

We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable assumptions on the factorization of $f$, bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman. When $f$ has degree at most $6$, we give exact expressions for the number of integral points of small height in terms of certain subgroups of Picard groups of the $k$-curves corresponding to the $2$ and $3$-torsion of our curve. This allows us to recover explicit results by Bremner, and gives new insight into Pillai's equation.