Kurihara’s strong Mazur–Tate conjecture for Selmer-group Fitting ideals

Let $E$ be an elliptic curve with good ordinary reduction at an odd prime $p$, let $\theta_n(f)$ be the Mazur–Tate element attached to the modular form $f$ associated with $E$ at $\mathbf{Q}^{(n)}$, and let $\Lambda_n=\mathbf{Z}_p[\operatorname{Gal}(\mathbf{Q}^{(n)}/\mathbf{Q})]$. Write $\operatorname{Sel}(\mathbf{Q}^{(n)},E[p^\infty])$ for the $p^\infty$-Selmer group. Strong Mazur–Tate conjecture. If $a_p\not\equiv 1\pmod p$ and $p$ does not divide the Tamagawa number of $E$, then

$$(\theta_n(f))=\operatorname{Fitt}_{\Lambda_n}\left(\operatorname{Hom}_{\mathbf{Z}_p}(\operatorname{Sel}(\mathbf{Q}^{(n)},E[p^\infty]),\mathbf{Q}_p/\mathbf{Z}_p)\right).$$

This strengthens the weak Mazur–Tate conjecture by asserting that the Mazur–Tate element generates the entire Fitting ideal. The source states that Kim and Kurihara proved this under additional hypotheses, including the Iwasawa main conjecture over $\mathbf{Q}_\infty$; the candidate itself is therefore recorded as solved.