Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

• 成果类型: Article
• 署名作者: Lecouturier, Emmanuel
• 署名单位: Tsinghua University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00996-1
• 发表日期: 2021
• 页码: 485-595
• 关键词: elliptic-curves; conjectures; congruences; HOMOLOGY; series; forms; birch

摘要

Let N and p be primes such that p divides the numerator of N-1/12. In this paper, we study the rank g(p) of the completion of the Hecke algebra acting on cuspidal modular forms of weight 2 and level Gamma(0)(N) at the p-maximal Eisenstein ideal. We give in particular an explicit criterion to know if g(p) >= 3, thus answering partially a question of Mazur. In order to study g(p), we develop the theory of higher Eisenstein elements, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler mass formula in characteristic p.