Heights and regulators of number fields and elliptic curves

Fabien Pazuki[1]

  • [1] Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark, and IMB, Université de Bordeaux, 351 cours de la Libération, 33405 Talence.

Publications mathématiques de Besançon (2014)

  • Issue: 2, page 47-62
  • ISSN: 1958-7236

Abstract

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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.

How to cite

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Pazuki, Fabien. "Heights and regulators of number fields and elliptic curves." Publications mathématiques de Besançon (2014): 47-62. <http://eudml.org/doc/275738>.

@article{Pazuki2014,
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.},
affiliation = {Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark, and IMB, Université de Bordeaux, 351 cours de la Libération, 33405 Talence.},
author = {Pazuki, Fabien},
journal = {Publications mathématiques de Besançon},
keywords = {Heights; abelian varieties; regulators; Mordell-Weil},
language = {eng},
number = {2},
pages = {47-62},
publisher = {Presses universitaires de Franche-Comté},
title = {Heights and regulators of number fields and elliptic curves},
url = {http://eudml.org/doc/275738},
year = {2014},
}

TY - JOUR
AU - Pazuki, Fabien
TI - Heights and regulators of number fields and elliptic curves
JO - Publications mathématiques de Besançon
PY - 2014
PB - Presses universitaires de Franche-Comté
IS - 2
SP - 47
EP - 62
AB - 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.
LA - eng
KW - Heights; abelian varieties; regulators; Mordell-Weil
UR - http://eudml.org/doc/275738
ER -

References

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