Construction of elliptic curves over \(\mathbb{Q}(t)\) with high rank: A preview (Q1813898)

We have recently established a general method for constructing elliptic curves over the rational function field \(k(t)\) \((k\) any base field) having relatively high rank (up to 8) [Proc. Japan Acad., Ser. 65, No. 7, 268-271; No. 8, 296-299 and 300-303 (1989; Zbl 0715.14015-14017)]. Not only can we give the explicit equation of such an elliptic curve, but also we can write down explicit rational points generating the full Mordell-Weil group. --- As an illustration of this method, we give here an example of an elliptic curve \(E\) over \(\mathbb{Q}(t)\) with the Mordell-Weil group \(E(\mathbb{Q}(t))\) of rank 8, together with a set of explicit generators. The example is perhaps the first example of an elliptic curve over \(\mathbb{Q}(t)\) of rank 8, given with a set of explicit generators of the Mordell-Weil group. The detailed accounts of our method are given in J. Math. Soc. Japan 43, No. 4, 673-719 (1991; Zbl 0751.14018), where we treat also the case of rank 7 or 6, etc.