Finiteness of the Tate–Shafarevich group for elliptic curves over number fields

Prove that the Tate–Shafarevich group Sha(E/K) is finite for every elliptic curve E over a number field K.

Background

The authors’ conditional algorithm for computing rational points on genus 1 curves over number fields assumes the finiteness of the Tate–Shafarevich group of elliptic curves, a classical conjecture in arithmetic geometry. This finiteness would ensure termination of their algorithm that combines Selmer computations with a search for rational points.

They explicitly define the conjecture in the text to fix terminology and assumptions used for their algorithmic consequences.

References

By the "finiteness-of-$\Sha(E/K)$" conjecture we mean the conjecture that all Tate-Shafarevich groups of elliptic curves over $K$ are finite.

— Conditional algorithmic Mordell  (2408.11653 - Alpöge et al., 2024) in Section “Proofs of Theorems …”, Theorem \ref{hilbert} context

For an elliptic curve $E/Q$, the Shafarevich--Tate group $(E/Q)$ is conjectured to be finite; and while this is known if $E$ has rank $0$ or $1$, it remains totally open for rank $\geq 2$, and to date there is not even a single known example of such a curve for which we unconditionally know finiteness of $(E/Q)$.

— Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves  (2609.08431 - Banwait, 8 Sep 2026) in Section 1, subsection “The problem”

Even the significantly weaker statement that $(E/Q)[p] = 0$ for merely infinitely many $p$ (as opposed to all but finitely many) remains open.

— Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves  (2609.08431 - Banwait, 8 Sep 2026) in Section 1, subsection “The problem”