Vanishing of the p-primary Shafarevich–Tate group for rank-two CM elliptic curves

The problem asks whether the $p$-primary Shafarevich–Tate group vanishes for rank-two CM elliptic curves. Existing work established substantial finite computations but explicitly left the case $p=577$ unresolved.

Known results

• A 2010 preprint proved finiteness and near-complete vanishing for five CM curves at good-reduction primes $p<30{,}000$, leaving $p=577$ for one curve among the exceptions.
• C. Wuthrich computed relevant $p$-adic heights and confirmed vanishing for the previously exceptional cases $p=17,29,277$.
• A general conditional estimate gives $t_{E/\mathbb{Q},p}\leq (1/2+\epsilon)p-g_{E/\mathbb{Q}}$ for sufficiently large good ordinary $p$, far weaker than universal vanishing.
• A 2009 preprint reported smaller-range computations for two CM curves.

September 2026 computational criterion

A new arXiv preprint reports a $p$-adic criterion applied to good ordinary primes below $30{,}000$ for five curves, including $p=577$, thereby claiming to resolve that previously open computational case. This is a claimed advance, not a verified solution of the broader conjecture.

Current status (as of September 2026): the finite computational case $p=577$ is claimed resolved by a new criterion, while universal vanishing for rank-two CM elliptic curves remains open.