Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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Fabien Pazuki
Heights and regulators of number fields and elliptic curves
Publications mathématiques de Besançon no. 2 (2014), p. 47-62, doi: 10.5802/pmb.8
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Class. Math.: 11G50, 14G40
Keywords: Heights, abelian varieties, regulators, Mordell-Weil

Abstract

We compare general inequalities between invariants of number fields and invariants of elliptic curves over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded regulator. On the elliptic curve side, assuming the height conjecture of Lang and Silverman, we obtain a Northcott property for the regulator on the set of elliptic curves with dense rational points over a number field. This amounts to say that the arithmetic of CM fields is similar, with respect to the invariants considered here, to the arithmetic of elliptic curves over a number field having a non Zariski dense Mordell-Weil group, i.e. with rank zero.

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