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 $k\ge 2$ with $a_p=0$.

arXiv commentTo appear in Annales de l'Institut Fourier. 42 pages. Updates: main theorem has been strengthened so as not to assume that K/Q_p is unramified when k=p+2 mod(p^2-1). Includes results at p=2. Includes a new conjecture on the relation between signed lambda-invariants for p-congruent modular forms with a_p=0