On Larsen's conjecture on the ranks of Elliptic Curves

2024/11/21 by Hadavand, A.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.14097

Abstract

Let E be an elliptic curve over ℚ and G=⟨σ₁, …, σ_n⟩ be a finitely generated subgroup of Gal(ℚ/ ℚ). Larsen's conjecture claims that the rank of the Mordell-Weil group E(ℚ^G) is infinite where \mathbb Q^G is the G-fixed sub-field of \mathbb Q. In this paper we prove the conjecture for the case in which σ_i for each i=1, …, n is an element of some infinite families of elements of Gal(ℚ/ ℚ).

Related

• Larsen's conjecture for elliptic curves over ℚ with analytic rank at most 1
• Mordell-Weil groups and the rank of elliptic curves over large fields
• Ranks of elliptic curves in cyclic sextic extensions of ℚ
• Elliptic curves with large rank over function fields
• Heegner points and the rank of elliptic curves over large extensions of global fields