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

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