Elliptic curves and finitely generated Galois groups

2025/10/01 by Bo-Hae Im, Michael Larsen, Im, Bo-Hae +1
#math.NT

paper · pdf · doi:10.48550/arxiv.2510.00750

Abstract

Let K be an extension of ℚ and A/K an elliptic curve. If Gal( 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 K₀ is finitely generated over ℚ, the points of an abelian variety A₀/K₀ over the compositum of all bounded-degree Galois extensions of K₀ form a virtually free abelian group. This, in turn, follows from a second Mordellic result, which asserts that the group of A₀ over the extension of K₀ defined by the torsion of A₀( K₀) is free modulo torsion.

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