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

2023/06/20 by Gillibert, Jean, Hallouin, Emmanuel, Levin, Aaron · 1 citation
#11D61 (Secondary) #14J20 #14J27 (Primary) 11G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.11353

Abstract

We consider elliptic curves defined by an equation of the form y²=x³+f(t), where f∈ 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.

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