On the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$

Keunyoung Jeong; Junyeong Park; Donggeon Yhee

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arXiv · 2104.03156

On the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$

Keunyoung Jeong·Junyeong Park·Donggeon Yhee

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Abstract

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$ 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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Keunyoung Jeong, Junyeong Park, Donggeon Yhee. 2021-04-07. On the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$. https://arxiv.org/abs/2104.03156

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