Rank growth of abelian varieties over certain finite Galois extensions

2024/10/22 by Seokhyun Choi, Bo-Hae Im, Choi, Seokhyun +2
Computer Science · Mathematics · #Abelian extension #Abelian group #Algebra over a field #Algebraic Geometry and Number Theory #Combinatorics #Elementary abelian group #Fundamental theorem of Galois theory #Galois group #Galois module #Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation #Pure mathematics #Rank (graph theory) #Rank of an abelian group #math.NT #msc:11G05 #msc:11G10

paper · pdf · doi:10.48550/arxiv.2410.16867

25 pages

openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · arxiv created 2026/08/03 · arxiv updated 2026/08/04 · openalex updated_date 2026/08/06

Abstract

Let A/K be an abelian variety over a number field K. We prove that a finite automorphism group G ⊆ Aut_K(X) of a smooth projective variety X/K such that X/G ≅ ℙ_K^d can force the rank growth of A over infinitely many mutually linearly disjoint G-extensions L_i/K. The proof is based on Hilbert irreducibility and Néron specialization. We then combine the theorem with finite group representations to obtain explicit lower bounds for rank growth. As applications, we obtain rank growth results for Jacobian varieties and construct explicit examples. We further prove arbitrarily large rank growth of abelian varieties over symmetric extensions. Finally, we study the connection with infinite rank conjectures.

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