Asymptotic formulae for Iwasawa Invariants of Mazur--Tate elements of elliptic curves

Abstract: We investigate two related questions regarding the $\lambda$-invariants of Mazur--Tate elements of elliptic curves defined over the field of rational numbers. At additive primes, we explain their growth and how these invariants relate to other better understood invariants depending on the potential reduction type. At good ordinary primes dividing the denominator of the normalised $L$-value of the elliptic curve, we prove that the $\lambda$-invariant grows as $p^n-1$, which is the maximum value. In addition, we give examples and a conjecture for the additive potentially supersingular case, supported by computational data from Sage in this setting.