The rank of hyperelliptic Jacobians in families of quadratic twists

Sebastian Petersen[1]

  • [1] Universität der Bundeswehr München Institut für Theoretische Informatik und Mathematik D-85577 Neubiberg

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 3, page 653-676
  • ISSN: 1246-7405

Abstract

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The variation of the rank of elliptic curves over ℚ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over ℚ admits infinitely many quadratic twists of rank ≥ 1 . Most elliptic curves have even infinitely many twists of rank ≥ 2 and examples of elliptic curves with infinitely many twists of rank ≥ 4 are known. There are also certain density results. This paper studies the variation of the rank of hyperelliptic Jacobian varieties in families of quadratic twists in an analogous way.

How to cite

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Petersen, Sebastian. "The rank of hyperelliptic Jacobians in families of quadratic twists." Journal de Théorie des Nombres de Bordeaux 18.3 (2006): 653-676. <http://eudml.org/doc/249638>.

@article{Petersen2006,
abstract = {The variation of the rank of elliptic curves over $\{\mathbb\{Q\}\}$ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over $\{\mathbb\{Q\}\}$ admits infinitely many quadratic twists of rank $\ge 1$. Most elliptic curves have even infinitely many twists of rank $\ge 2$ and examples of elliptic curves with infinitely many twists of rank $\ge 4$ are known. There are also certain density results. This paper studies the variation of the rank of hyperelliptic Jacobian varieties in families of quadratic twists in an analogous way.},
affiliation = {Universität der Bundeswehr München Institut für Theoretische Informatik und Mathematik D-85577 Neubiberg},
author = {Petersen, Sebastian},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {rank; hyperelliptic curve; quadratic twist; density},
language = {eng},
number = {3},
pages = {653-676},
publisher = {Université Bordeaux 1},
title = {The rank of hyperelliptic Jacobians in families of quadratic twists},
url = {http://eudml.org/doc/249638},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Petersen, Sebastian
TI - The rank of hyperelliptic Jacobians in families of quadratic twists
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 3
SP - 653
EP - 676
AB - The variation of the rank of elliptic curves over ${\mathbb{Q}}$ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over ${\mathbb{Q}}$ admits infinitely many quadratic twists of rank $\ge 1$. Most elliptic curves have even infinitely many twists of rank $\ge 2$ and examples of elliptic curves with infinitely many twists of rank $\ge 4$ are known. There are also certain density results. This paper studies the variation of the rank of hyperelliptic Jacobian varieties in families of quadratic twists in an analogous way.
LA - eng
KW - rank; hyperelliptic curve; quadratic twist; density
UR - http://eudml.org/doc/249638
ER -

References

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