Congruences between Ramanujan's tau function and elliptic curves, and Mazur--Tate elements at additive primes

TL;DR

This paper explores congruences between Ramanujan's tau function and elliptic curves with rational p-torsion, using these to compute Iwasawa invariants of Mazur--Tate elements at additive primes.

Contribution

It establishes new congruences linking Ramanujan's tau function and elliptic curves with rational p-torsion, and applies these to compute Iwasawa invariants of Mazur--Tate elements.

Findings

Congruences between tau function and elliptic curves with rational p-torsion for p=2,3.

Computed Iwasawa invariants of 2-adic and 3-adic Mazur--Tate elements.

Numerical investigation of Iwasawa invariants at additive primes.

Abstract

We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.