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Congruences between Ramanujan's tau function and elliptic curves, and Mazur--Tate elements at additive primes
Anthony Doyon, Antonio Lei
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We show that if E/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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