On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes

TL;DR

This paper studies the growth of Iwasawa lambda-invariants of Mazur--Tate elements for elliptic curves at additive primes, exploring their relation to other invariants and extending results to modular forms.

Contribution

It introduces new insights into the behavior of Iwasawa invariants at additive primes and proposes a conjecture supported by computational evidence.

Findings

Growth patterns of lambda-invariants at additive primes

Relation between invariants and reduction types

Conjecture for additive potentially supersingular case

Abstract

We investigate the $\lambda$-invariants of Mazur--Tate elements of elliptic curves defined over the field of rational numbers at primes of additive reduction. We explain their growth and how these invariants relate to other better understood invariants depending on the potential reduction type. We give examples and a conjecture for the additive potentially supersingular case, supported by computational data from Sage in this setting. Further, we extend our results to $\lambda$-invariants of Mazur--Tate elements of cuspidal Hecke eigenforms associated with potentially ordinary $p$-adic Galois representations.