Explicit reciprocity laws and Iwasawa theory for modular forms
OA Version
Citation
M. Emerton, R. Pollack, T. Weston. "Explicit reciprocity laws and Iwasawa theory for modular forms." https://doi.org/10.48550/arXiv.2210.02013
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.