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arXiv · 2103.11001

Elliptic curves with exceptionally large analytic order of the Tate-Shafarevich groups

Andrzej Dąbrowski·Lucjan Szymaszkiewicz
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Abstract

We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|\sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\sza(E)| = 1029212^2 = 2^4\cdot 79^2 \cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $\sza$. For instance, $410536^2$ is the true order of $\sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.

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Andrzej Dąbrowski, Lucjan Szymaszkiewicz. 2021-03-19. Elliptic curves with exceptionally large analytic order of the Tate-Shafarevich groups. https://arxiv.org/abs/2103.11001

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