J. Korean Math. Soc. 2023; 60(5): 1087-1107
Online first article August 18, 2023 Printed September 1, 2023
https://doi.org/10.4134/JKMS.j230085
Copyright © The Korean Mathematical Society.
Dongho Byeon, Taekyung Kim, Donggeon Yhee
Seoul National University; Cryptolab; Seoul National University
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$, $c$ the Manin constant of $E$, and $m$ the product of Tamagawa numbers of $E$ at prime divisors of $N$. Let $K$ be an imaginary quadratic field where all prime divisors of $N$ split in $K$, $P_K$ the Heegner point in $E(K)$, and ${\rm III}(E/K)$ the Shafarevich-Tate group of $E$ over $K$. Let $2u_K$ be the number of roots of unity contained in $K$. Gross and Zagier conjectured that if $P_K$ has infinite order in $E(K)$, then the integer $ c \cdot m \cdot u_K \cdot |{\rm III}(E/K)|^{\frac{1}{2}}$ is divisible by $|E(\mathbb{Q})_{\rm{tor}} |$. In this paper, we prove that this conjecture is true if $E(\mathbb{Q})_{\rm{tor}} \cong \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/4\mathbb{Z}$ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein--Watkins conjecture.
Keywords: Elliptic curve, Manin constant, Tamagawa number, Shafarevich-Tate group
MSC numbers: 11G05
Supported by: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2023R1A2C1002612).
2021; 58(1): 123-132
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