On the maximality of the $λ$-invariants of Mazur--Tate elements
Antonio Lei, Robert Pollack, Naman Pratap
TL;DR
The article analyzes Mazur--Tate $p$-adic elements $ heta_n(E)$ for elliptic curves with good ordinary reduction at $p$, establishing a dichotomy for their Iwasawa invariants. It shows that, under Greenberg’s conjecture and along isogeny classes, either the $oldsymbol{ m ll}$-invariants stabilize to those of the $p$-adic $L$-function or they achieve the maximal value $oldsymbol{ m ll}( heta_n(E))=p^n-1$ at all levels, dictated by the negativity of $ ext{ord}_p(L(E,1)/ ext{Ω}_E)$. The work connects this behavior to Eisenstein congruences via boundary symbols, and extends the framework to Hecke eigenforms of weight two, providing explicit criteria and examples. By developing an abstract Iwasawa-invariant theory and linking it to congruences with boundary symbols, the paper clarifies when Mazur--Tate growth is maximal and how this relates to the reducibility of $E[p]$ and Eisenstein ideals. The results bridge $p$-adic $L$-functions, modular symbols, and Eisenstein theory, offering concrete criteria for predicting λ-behavior and highlighting the role of boundary-symbol congruences.
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.