Regulators of rank one quadratic twists

Delaunay, Christophe; Roblot, Xavier-François

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 601-624.

Nous étudions les régulateurs des courbes elliptiques de rang 1 appartenant à des familles de tordues quadratiques d’une courbe fixée. En particulier, nous formulons des conjectures sur la taille moyenne de ces régulateurs. Nous décrivons également un algorithme performant pour calculer explicitement les invariants des tordues quadratiques de rang 1 d’une courbe elliptique (régulateur, ordre du groupe de Tate-Shafarevich, etc.) et nous comparons les données numériques obtenues avec les prédictions.

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.

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