THE EISENSTEIN IDEAL OF WEIGHT k AND RANKS OF HECKE ALGEBRAS

Abstract

Let p and $\ell $ be primes such that $p> 3$ and $p \mid \ell -1$ and k be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight k and level $\Gamma _0(\ell )$ at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the $\mathbb {Z}_p$ -rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb {T}$ theorems under certain hypotheses.

GEOMETRIC WEIGHT-SHIFTING OPERATORS ON HILBERT MODULAR FORMS IN CHARACTERISTIC p

“…One can also define cruder partial Θ-operators by composing with multiplication by (products of) partial Hasse invariants. For example, the operator [8], and for any j = 1, . .…”

“…Here we instead exploit the observations and techniques introduced in [12], applying them directly to the special fibre of the Pappas-Rapoport model to construct and relate partial Θ and Frobenius operators. In particular, this eliminates extraneous multiples of partial Hasse invariants that appear in [8] and yields results whose implications for minimal weights are motivated by the forthcoming generalisation to the ramified case of the geometric Serre weight conjectures of [12]. The main contributions of this article may be summarised as follows:…”

“…Since then, however, a smooth integral model was constructed by Pappas and Rapoport [29], and the theory of partial Hasse invariants was further developed in this context by Reduzzi and Xiao in [32]. The theory of partial Θ-operators was revisited in that light by Deo, Dimitrov and Wiese in [8], where they closely follow [1]. Here we instead exploit the observations and techniques introduced in [12], applying them directly to the special fibre of the Pappas-Rapoport model to construct and relate partial Θ and Frobenius operators.…”