Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$

TL;DR

The paper proves Larsen's conjecture for elliptic curves $E/\,\mathbb{Q}$ with analytic rank at most $1$. It develops a Heegner-point framework extended to ring class fields, constructing an infinite family of inert primes and leveraging Serre open image, Chebotarev density, and trace relations to produce non-torsion Heegner traces whose $E$-rational points generate infinite rank over the fixed field $\overline{\mathbb{Q}}^{G}$. The argument treats separately the cases when $G$ fixes a chosen imaginary quadratic field $K$ and when it does not, using towers of class fields and an involution argument to secure unbounded rank in all circumstances. Consequently, Larsen's conjecture holds for $E/\mathbb{Q}$ under the stated rank hypothesis, with implications tied to Goldfeld's conjecture for quadratic twists and potential extensions to totally real fields or CM settings.

Abstract

We prove Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. Specifically, let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$. If $E/\mathbb{Q}$ has analytic rank at most $1$, then we prove that for any topologically finitely generated subgroup $G$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the rank of $E$ over the fixed subfield $\overline{\mathbb{Q}}^G$ of $\overline{\mathbb{Q}}$ under $G$ is infinite.

Theorems & Definitions (20)

• Conjecture 1
• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1