Elliptic surfaces over $\mathbb{P}^1$ and large class groups of number fields

Jean Gillibert; Aaron Levin

Elliptic surfaces over $\mathbb{P}^1$ and large class groups of number fields

Number Theory 2019-05-20 v2

Authors: Jean Gillibert , Aaron Levin

Abstract

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$ , infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$ -torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{11}$ .

Keywords

elliptic curves class number of quadratic fields isogeny of elliptic curves

Cite

@article{arxiv.1811.08166,
  title  = {Elliptic surfaces over $\mathbb{P}^1$ and large class groups of number fields},
  author = {Jean Gillibert and Aaron Levin},
  journal= {arXiv preprint arXiv:1811.08166},
  year   = {2019}
}

Comments

10 pages, LaTeX. Minor improvements following the referee's suggestions. To appear in Int. J. Number Theory

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