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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 $k$ ofcharacteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain,under suitable assumptions on the factorization of $f$, bounds for the numberof integral points on these curves. These bounds improve on a general result byHindry and Silverman. When $f$ has degree at most $6$, we give exactexpressions for the number of integral points of small height in terms ofcertain 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 byBremner, and gives new insight into Pillai's equation.

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