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

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 $(\mathbb{Z}/2\mathbb{Z}{)}^{11}$.

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

References in corpus (3)

• Three lectures on elliptic surfaces and curves of high rank
• Selmer groups and class groups
• Descent on elliptic surfaces and arithmetic bounds for the Mordell-Weil rank