Iwasawa invariants of modular forms with $a_p=0$

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 $a_p=0$.