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

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Abstract: We consider elliptic curves defined by an equation of the form

y^{2} = x^{3} + f (t)

, where

f i n 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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