On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes
Antonio Lei, Robert Pollack, Naman Pratap
TL;DR
This work investigates how the Iwasawa invariants $\mu$ and $\lambda$ of Mazur--Tate elements behave at primes of additive reduction for elliptic curves over $\mathbb{Q}$, linking their growth to the reduction type via potential semistability and twists to good reduction. It delivers explicit formulae for $\lambda(\theta_{n,i}(E))$ in several additive scenarios, notably showing $\lambda$ grows like $\frac{p-1}{2}p^{n-1}$ plus a term coming from a quadratic twist when $E$ attains good ordinary or multiplicative reduction over a quadratic field $F$, and a BSD-guided formula in the potentially good ordinary case. The paper also discusses potential supersingular cases, connecting plus/minus $p$-adic $L$-functions to conjectural growth patterns, provides CM-curve and modular-form generalizations, and includes numerical evidence to support these conjectures. Overall, the results illuminate how period ratios, twists, and BSD-type conjectures shape the cyclotomic growth of Mazur--Tate invariants, contributing to a coherent framework tying $p$-adic $L$-functions, Selmer groups, and Iwasawa theory at additive primes.
Abstract
We investigate the $λ$-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 $λ$-invariants of Mazur--Tate elements of cuspidal Hecke eigenforms associated with potentially ordinary $p$-adic Galois representations.