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

TL;DR

This paper investigates the behavior of $mbda$-invariants of Mazur--Tate elements for elliptic curves, showing they either stabilize to the $p$-adic $L$-function's invariant or reach a maximum, linked to isogenies and Eisenstein series.

Contribution

It characterizes the maximality of $mbda$-invariants in terms of isogenies and boundary symbol congruences, extending results to weight two Hecke eigenforms.

Findings

$mbda$-invariants either stabilize or attain maximum at finite levels.

Maximality occurs iff $ ext{ord}_p(rac{L(E',1)}{\u03a9_{E'}})$ is negative for some isogenous $E'$.

Relation established between maximality and congruences with Eisenstein series boundary symbols.

Abstract

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 $\ord_p\left(\frac{L(E',1)}{\Omega_{E'}}\right)$ 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.