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arXiv · 2103.06154

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

Anthony Doyon·Antonio Lei
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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.

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Anthony Doyon, Antonio Lei. 2021-03-10. Congruences between Ramanujan's tau function and elliptic curves, and Mazur--Tate elements at additive primes. https://doi.org/10.1007/s11139-021-00397-9

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