Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series

Abstract

Let $p\in \{3,5,7\}$ and let $\hat{I}”$ denote the weight twelve modular form arising from Ramanujan's tau function. We show that $\hat{I}”$ is congruent to an Eisenstein series $E_{k,\ddot{I}‡,\ddot{I}ˆ}$ modulo $p$ for explicit choices of $k$ and Dirichlet characters $\ddot{I}‡$ and $\ddot{I}ˆ$. We then prove formulae describing the Iwasawa invariants of the Mazur--Tate elements attached to $\hat{I}”$, confirming numerical data gathered by the authors in a previous work.

Added calculations on mu-invariants and made several corrections. To appear in Research in Number Theory