Mazur--Tate's refined exceptional zero conjecture

Let $p>2$, let $q_E$ be the Tate period of the elliptic curve $E$, and put

$$G_p=\left(\operatorname{\mathbb{Z}}/p\operatorname{\mathbb{Z}}\right)^\times/\langle-1\rangle.$$

Let $A\subseteq\operatorname{\mathbb{Q}}$ be a subring containing all modular symbols $\left[\frac{a}{p}\right]$ for $0\leq a\leq p-1$, together with $\frac{\left[\frac{0}{1}\right]}{2\operatorname{ord}_p(q_E)}$ and $\frac{1}{\#E(\operatorname{\mathbb{Q}})_{\mathrm{Tor}}}$. If $I(A,G_p)$ denotes the augmentation ideal of $A[G_p]$, then the asserted congruence is an equality in $I(A,G_p)/I(A,G_p)^2$.

Mazur--Tate's refined exceptional zero conjecture. One has

$$\sum_{1 \leq a \leq \frac{p-1}{2}}\left[\frac{a}{p}\right]a\in I(A,G_p)$$

and

$$\frac{1}{2}\sum_{1 \leq a \leq p-1}\left[\frac{a}{p}\right]a \equiv \frac{\left[\frac{0}{1}\right]}{2\operatorname{ord}_p(q_E)}(\widetilde q_E-1)\quad\text{in }I(A,G_p)/I(A,G_p)^2.$$

Here $\widetilde q_E$ is the image of the Tate period in the relevant group-ring construction.

This is the refined form of the exceptional-zero conjecture relating modular symbols to the Tate period through the augmentation quotient. The source explains that the factor $D$ used in the equivalent product formulation clears possible denominators; the conjecture is presented as open here, although it is attributed to Mazur and Tate and has partial results due to de Shalit.