Larsen’s Conjecture on Infinite Rank over Fixed Subfields

Establish that for every abelian variety A defined over a finitely generated infinite field k and every topologically finitely generated subgroup G of the absolute Galois group Gal(k^sep/k), the Mordell–Weil rank of A over the fixed subfield (k^sep)^G is infinite.

Background

The paper studies ranks of elliptic curves over infinite extensions of global function fields, focusing on whether infinite rank persists after descending to fixed subfields of large Galois extensions. A central guiding statement is Larsen’s conjecture, which predicts that for any abelian variety over a finitely generated infinite field, the rank remains infinite over the fixed field of any topologically finitely generated subgroup of the absolute Galois group.

Prior work has established the conjecture in several cases, including for procyclic subgroups and under various additional hypotheses (e.g., Heegner point constructions or assumptions related to the Birch–Swinnerton-Dyer conjecture). The present paper proves new cases for non-isotrivial elliptic curves over global function fields of characteristic p>3 with analytic rank 1, showing infinite rank over fixed subfields of certain ring class field towers, thus providing evidence toward Larsen’s conjecture in the function field setting.