Elliptic curves and finitely generated Galois groups
Bo-Hae Im, Michael Larsen
TL;DR
The work addresses whether the Mordell–Weil group of an abelian variety over a field with finitely generated Galois group has infinite rank, specifically proving that for an elliptic curve $A_0$ over a finitely generated field $K_0$ of characteristic zero, the rank over the invariant field fixed by a finitely generated Galois subgroup is infinite. The authors combine a Kummer-theoretic analysis of abelian varieties over torsion extensions, a Silverman–Néron–Lang framework for extensions of bounded degree, a Hales–Jewett combinatorial construction to generate many potential rational points, and Chebotarev density for schemes to certify density and independence of reductions. The main contributions include establishing that the torsion-free part of $A(K_{tor})$ is free, showing $A(K(d))$ is virtually free for finitely generated $K$, and proving the genus-1 case of the Junker–Koenigsmann conjecture as a corollary. These results provide a robust bridge between arithmetic geometry, Galois representations, and model-theoretic conjectures, with potential implications for understanding the distribution and density of rational points on abelian varieties in arithmetic families.
Abstract
Let $K$ be an extension of $\mathbb{Q}$ and $A/K$ an elliptic curve. If $\mathrm{Gal}(\bar K/K)$ is finitely generated, then $A$ is of infinite rank over $K$. In particular, this implies the $g=1$ case of the Junker-Koenigsmann conjecture. This "anti-Mordellic'' result follows from a new "Mordellic'' theorem, which asserts that if $K_0$ is finitely generated over $\mathbb{Q}$, the points of an abelian variety $A_0/K_0$ over the compositum of all bounded-degree Galois extensions of $K_0$ form a virtually free abelian group. This, in turn, follows from a second Mordellic result, which asserts that the group of $A_0$ over the extension of $K_0$ defined by the torsion of $A_0(\bar K_0)$ is free modulo torsion.