  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.
