Explicit reciprocity laws and Iwasawa theory for modular forms

We prove that the Mazur-Tate elements of an eigenform $f$ sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic ${\mathbb{Z}}_p$-extension (up to scaling by a single constant). Our method begins with the construction of local cohomology classes built via the $p$-adic local Langlands correspondence. From these classes, we build algebraic analogues of the Mazur-Tate elements which we directly verify sit in the appropriate Fitting ideals. Using Kato’s Euler system and explicit reciprocity laws, we prove that these algebraic elements divide the corresponding Mazur-Tate elements, implying our theorem.