The Eisenstein ideal of weight k and ranks of Hecke algebras

2021/06/08 by Shaunak V. Deo, Deo, Shaunak V.
Mathematics · #11F25 #11F33 (secondary) #11F80(primary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2106.04551

openalex publication_date 2021/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p and ℓ be primes such that p > 3 and p | ℓ-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 Γ₀(ℓ) at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the 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=\mathbbT theorems under certain hypothesis.

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