Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups
Tom 166 / 2021
Colloquium Mathematicum 166 (2021), 217-225
MSC: Primary 11G05, 11G40, 11Y50.
DOI: 10.4064/cm8008-9-2020
Opublikowany online: 22 March 2021
Streszczenie
We exhibit examples of rank zero elliptic curves over the rationals with |{ш }(E)| \gt 63408^2, which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |{ш }(E)| = 1029212^2 = 2^4\cdot 79^2 \cdot 3257^2. We use deep results by Kolyvagin, Kato, Skinner–Urban and Skinner to prove that, in some cases, these orders are the true orders of {ш }. For instance, 410536^2 is the true order of {ш }(E) for E= E_4(21,-233) from the table in Section 2.3.
