On the Jacobian of hyperelliptic curves y² = x⁵ + m²

2021/04/07 by Keunyoung Jeong, Jeong, Keunyoung, Junyeong Park +3
Computer Science · Mathematics · #11F27 #11G10 #11G30 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2104.03156

openalex publication_date 2021/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves y² = x⁵ + m² for integers m. Namely, we first provide a condition on m that gives a bound of the size of Selmer group and then we provide a condition on m that makes L-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch--Swinnerton-Dyer conjecture.

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