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Abstract

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 Zp-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.