The rank of Mazur’s Eisenstein ideal

2017/07/31 by Preston Wake, Carl Wang-Erickson · 14 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebra over a field #Algebraic Geometry and Number Theory #Eisenstein series #Hecke algebra #Ideal (ethics) #Prime (order theory) #Rank (graph theory) #math.NT

paper · pdf · doi:10.1215/00127094-2019-0039

published in Duke Mathematical Journal 169(1) (Duke University Press) · 63 pages. Final version. Improvements to exposition and minor corrections, added dedication. To appear in Duke Math J

openalex created_date 2017/07/14 · arxiv created 2019/07/10 · openalex publication_date 2019/11/21 · arxiv updated 2020/02/12 · openalex updated_date 2026/08/05

Abstract

We use pseudodeformation theory to study Mazur’s Eisenstein ideal. Given prime numbers N and p>3, we study the Eisenstein part of the p-adic Hecke algebra for Γ0(N). We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, thereby answering a question of Mazur and generalizing a result of Calegari and Emerton. We also give new proofs of Merel’s result on this rank and of Mazur’s results on the structure of the Hecke algebra.

Citations

• Cohomology of Number Fields
• Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings
• Matric Massey products
• Modular Elliptic Curves and Fermat's Last Theorem
• On the Galois structure of the class group of certain Kummer extensions