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.

THE EISENSTEIN IDEAL OF WEIGHT k AND RANKS OF HECKE ALGEBRAS

“…Combining this with Kummer's congruences, we get that ζ(1 − k) ∈ Z × (p) . Hence, the hypotheses of [26] are satisfied in our setup. • p is a regular prime.…”

“…In [26], Wake has proved that when k > 2, rank Zp (T 0 m ) = 1 if and only if the Eisenstein ideal of T 0 m is principal and a certain element ξ MT ∈ F p (that he defines in [26, Section 1.2.2]) is nonzero. See [26,Theorem 1.2.4] for more details. We don't use this result to prove part (2) of Theorem A, but we do need some other results from [26].…”

“…One can say that the approach of Merel and Lecouturier is on the "analytic side" and the approach of Calegari-Emerton and Wake-Wang-Erickson is on the "algebraic side". In [26], Wake studied the Hecke algebras T m and T 0 m and their Eisenstein ideals for weights k > 2 by unifying the two approaches mentioned above. In particular, he gave a necessary and sufficient condition (in terms of the Eisenstein ideal and the derivative of Mazur-Tate ζ-function that he defines) for the Z p -rank of T 0 m to be 1.…”

“…-valued ordinary pseudo-representations, analysis of pseudo-representations arising from actual representations and results from [26] and [19]. The results from [26] that we use are about the reducibility properties of the T m -valued pseudo-representation ([26, Theorem 5.1.1]) and the index of Eisenstein ideal in T 0 m ([26, Theorem 5.1.2]). Note that, in [28], Wake and Wang-Erickson work with pseudo-representations which are finite flat at p (a notion that they define and study in [27]).…”

Untitled

“…The novel idea here is that when we restrict our Galois representation to the Galois group of the splitting field of this cohomology class, then the representation becomes extra reducible, as expressed in (2). For an application of extra reducibility in another context, see [Wak22].…”