On the maximality of the $λ$-invariants of Mazur--Tate elements
Authors
Antonio Lei
Robert Pollack
Naman Pratap
Abstract
Let $E$ be an elliptic curve with good ordinary reduction at an odd prime $p$. Assuming that Greenberg's $μ=0$ conjecture holds, we show that the $λ$-invariants of the Mazur--Tate elements attached to $E$ either stabilise to the $λ$-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 $\ord_p\left(\frac{L(E',1)}{Ω_{E'}}\right)$ is negative for some $E'$ 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.
