Mazur–Tate's vanishing-order conjecture

Let $\mathbb{Q}_n$ be a finite cyclotomic layer, let $\chi$ be a character of $\operatorname{Gal}(\mathbb{Q}_n/\mathbb{Q})$, and let $I_\chi$ be the kernel of the induced map from the group ring to $\mathbb{Z}_p[\chi]$. The order of vanishing of an element at $\chi$ is measured by membership in powers of $I_\chi$. Mazur–Tate's vanishing-order conjecture. The order of vanishing of $\theta_n(f)$ at $\chi$ is greater than or equal to the dimension of the $\chi$-part of the Mordell–Weil group of $E(\mathbb{Q}_n)$. This refines the rank aspect of the Birch and Swinnerton-Dyer conjecture at finite layers; the supplied source gives no general resolution.