On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms

Abstract

Under some assumptions, we prove the rank-part of the Mazur–Tate refined conjecture of BSD type. More concretely, we prove that the rank of the Selmer group of an elliptic modular form is less than or equal to the order of zeros of Mazur–Tate elements, or modular elements, which are elements in certain group rings constructed from special values of the associated L-function. Our main result is regarded as a generalization of our previous work on elliptic curves.