Experimental study of the rank of families of elliptic curves over \(\mathbb{Q}\). (Étude expérimentale du rang de familles de courbes elliptiques sur \(\mathbb{Q}\).) (French) Zbl 0873.11033

The author studies 93 elliptic curves over \({\mathbb{Q}}(t)\). These curves have Mordell-Weil groups of rank \(r\leq 4\). The author studies the ranks of the Mordell-Weil groups of the specializations of these curves for many \(t\in{\mathbb{Z}}\). His calculation involves computing the ranks of the groups of rational points of 66918 elliptic curves over \({\mathbb{Q}}\). Lower bounds for the ranks are established by searching for rational points of small height. Upper bounds are deduced from a parity criterion and certain explicit formulas [J.-F. Mestre, Compos. Math. 58, 209-232 (1986; Zbl 0607.14012)]. Here the author assumes the truth of the Shimura-Taniyama-Weil conjecture and of the Birch-Swinnerton-Dyer conjecture. The author makes various interesting observations concerning this extensive numerical experiment. One of which is the fact that the proportion of specialized elliptic curves \(E_t\) that have a Mordell-Weil group of rank \(r+s\) seems to be independent of the rank \(r\) of the elliptic curve \(E\) over \({\mathbb{Q}}(T)\).
Reviewer: R.Schoof (Roma)


11G05 Elliptic curves over global fields
14H52 Elliptic curves
14Q05 Computational aspects of algebraic curves


Zbl 0607.14012


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