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.
Acknowledgments
Authors thank to Dohyeong Kim for the useful suggestions, and the referee for the careful reading and introducing the reference [1] to us. K. Jeong is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1C1C1004264 and 2020R1A4A1016649). D. Yhee is supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIT) (2017R1A5A1015626).
Citation
Keunyoung Jeong. Junyeong Park. Donggeon Yhee. "On the Jacobian of a family of hyperelliptic curves." Osaka J. Math. 60 (1) 43 - 60, January 2023.
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