On the maximality of the $\lambda$-invariants of Mazur--Tate elements

Let $E$ be an elliptic curve with good ordinary reduction at an odd prime $p$. Assuming that Greenberg’s $\mu =0$ conjecture holds, we show that the $\lambda$-invariants of the Mazur–Tate elements attached to $E$ either stabilise to the $\lambda$-invariant of the $p$-adic $L$-function or they attain the largest possible value at all finite levels. We characterise the latter phenomenon:\ it occurs if and only if ${\text{ord}}_p\left(\frac{L(E^{\mathrm{\prime }},1)}{{\mathrm{\Omega }}_{E^{\mathrm{\prime }}}}\right)$ is negative for some $E^{\mathrm{\prime }}$ that is isogenous to $E$. Furthermore, we relate this condition to congruences with boundary symbols coming from Eisenstein series. We also study the extension of these results to Hecke eigenforms of weight two.