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We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y2=x3+n, n∈Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds of the canonical heights of the points. Using the result we show that any pair in the three points can always be a part of a basis of the free part of the Mordell–Weil...
We study the Mordell–Weil group MW(V) for cubic surfaces V over finite fields that are not necessarily irreducible and smooth. We construct a surjective map from MW(V) to a group that can be computed explicitly. For #MW(V), this yields a lower bound, which is (often but) not always trivial. To distinguish cases, we follow the classification of cubic surfaces, originally due to Schläfli and Cayley...
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