Iwasawa invariants of modular forms with $a_p=0$

TL;DR

This paper establishes a method to compute Iwasawa invariants of Mazur-Tate elements for modular forms with specific properties, linking them to signed p-adic L-functions and enabling analysis across various weights.

Contribution

It introduces a way to determine Iwasawa invariants of Mazur-Tate elements using signed p-adic L-functions, applicable to modular forms of any weight with a_p=0.

Findings

Computed Iwasawa invariants of Mazur-Tate elements in terms of signed p-adic L-functions.

Determined p-adic valuations of critical L-values for modular forms.

Established relations between invariants of congruent modular forms of different weights.

Abstract

Fix a prime $p$ and a cuspidal newform $f$ of level coprime to $p$ with $a_p=0$. Attached to $f$ are signed $p$-adic $L$-functions $L_p^\pm(f)$ and Mazur-Tate elements $\theta_n(f)$, both of which encode arithmetic data about $f$ along the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$. We compute the Iwasawa invariants of Mazur-Tate elements in terms of the corresponding invariants of the signed $p$-adic $L$-functions. As corollaries, we determine the $p$-adic valuation of critical values of the $L$-function of $f$, and describe a relation between the Iwasawa invariants of congruent modular forms of weights 2 and $p+1$. Our results provide an asymptotic method for computing the signed Iwasawa invariants attached to newforms of any weight $k\geq 2$ with $a_p=0$.