Timeline for Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants

9 events

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Nov 3, 2023 at 2:57	history	edited	Duality	CC BY-SA 4.0	added 2 characters in body
Nov 3, 2023 at 2:35	history	edited	Duality	CC BY-SA 4.0	added 103 characters in body
Nov 3, 2023 at 2:29	history	edited	Duality	CC BY-SA 4.0	added 373 characters in body
Nov 3, 2023 at 2:00	history	edited	Duality	CC BY-SA 4.0	deleted 323 characters in body
Nov 3, 2023 at 1:34	comment	added	Jeremy Rouse		I think you are confusing $\dim_{\mathbb{F}_{2}} {\rm Sha}(E/\mathbb{Q})[2]$ and the $2$-adic valuation of the order of ${\rm Sha}$. For the $n=16$, $p=48$ curve mentioned in your post, a $2$-descent says that the $2$-Selmer group is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{4}$.
Nov 2, 2023 at 21:51	history	edited	Michael Hardy	CC BY-SA 4.0	added 78 characters in body; edited title
Nov 2, 2023 at 17:15	comment	added	Chris Wuthrich		For $E$ without rational $2$-torsion, I would look for examples among those curves for which the $2$-torsion of the class number of $\mathbb{Q}(E[2])$ is large.
Nov 2, 2023 at 16:19	comment	added	Stanley Yao Xiao		Partial answer: the answer is no if $E$ has a rational 2-torsion point, which is basically just an application of genus theory. In this case the 2-Selmer rank is bounded explicitly by $2^{\omega(\Delta_E)}$ (up to some fudge factor).
Nov 2, 2023 at 15:34	history	asked	Duality	CC BY-SA 4.0