The Eisenstein ideal for weight $k$ and a Bloch–Kato conjecture for tame families

Abstract

We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of extra reducibility: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur–Tate $L$-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we recall Kato’s formulation of this main conjecture in the case of a family of motives given by twists by characters of conductor $N$ and $p$-power order and its relation to other formulations of the equivariant main conjecture.