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

TL;DR

This paper investigates the congruences between Ramanujan's tau function and Eisenstein series modulo small primes, and analyzes the Iwasawa invariants of associated Mazur--Tate elements, providing explicit formulas and confirming previous numerical observations.

Contribution

It establishes explicit congruences between Ramanujan's tau function and Eisenstein series modulo 3, 5, and 7, and derives formulas for the Iwasawa invariants of related Mazur--Tate elements.

Findings

Ramanujan's tau form is congruent to Eisenstein series modulo p for p in {3,5,7}.

Explicit formulas for Iwasawa invariants of Mazur--Tate elements are provided.

Numerical data on invariants are confirmed by theoretical results.

Abstract

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