Iwasawa invariants of Mazur–Tate elements of elliptic curves and modular forms

Abstract

We investigate two questions regarding the lambda-invariants of Mazur-Tate elements of elliptic curves and modular forms 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. In addition, we give examples and a conjecture for the additive potentially supersingular case, supported by computational data from Sage in this setting. Using our methods for elliptic curves, we extend our results to lambda-invariants of Mazur-Tate elements of cuspidal Hecke eigenforms associated with potentially ordinary p-adic Galois representations. At good ordinary primes p dividing the denominator of the normalised central L-value of an elliptic curve E defined over the rationals, we prove that the lambda-invariant grows as p^n-1, which is the maximum value. Under mild hypotheses, we prove a converse result allowing us to characterise when lambda-invariants of the form p^n-1 arise for elliptic curves with good ordinary reduction at p. We relate this behaviour to the existence of congruences between the modular symbols of E and Eisenstein boundary symbols. In special cases, we show that the associated Hecke algebra satisfies the Gorenstein property and indicate how that can be related to the notion of mod p multiplicity one for modular symbols.