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

Let \(p\in \{3,5,7\}\) $p\in \{3,5,7\}$ and let \(\Delta \) $\mathrm{\Delta }$ denote the weight twelve modular form arising from Ramanujan’s tau function. We show that \(\Delta \) $\mathrm{\Delta }$ is congruent to an Eisenstein series \(E_{k,\chi , \psi }\) $E_{k,\chi ,\psi }$ modulo p for explicit choices of k and Dirichlet characters \(\chi \) $\chi$ and \(\psi \) $\psi$ . We then prove formulae describing the Iwasawa invariants of the Mazur–Tate elements attached to \(\Delta \) $\mathrm{\Delta }$ , confirming numerical data gathered by the authors in a previous work.