A new perspective on the rank of Mazur's Eisenstein Hecke algebra
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arXiv
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| Format: | Preprint |
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2026
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| author | Lang, Jaclyn Müller, Katharina Palvannan, Bharathwaj |
| author_facet | Lang, Jaclyn Müller, Katharina Palvannan, Bharathwaj |
| contents | Let $N, p \geq 5$ be primes such that $N \equiv 1 \bmod p$. We study the rank $r$ of the Hecke algebra that parametrizes modular forms of weight 2 and level $N$ that are Eisenstein modulo $p$. When $r$ is $2$ or $3$, we prove that $r-1$ equals the order of vanishing of the mod-$p$ reduction of a zeta element that interpolates Dirichlet $L$-values at $-1$, thereby recovering results of Merel and Lecouturier. This equality can fail in some cases when $r \geq 4$, and we provide a heuristic explanation of this failure. Our approach handles all of these cases uniformly by studying the analogous Hecke algebra in level $N^2$. When exactly one of $r-1$ or the order of vanishing equals $3$, we provide precise information about Galois orbits of cuspidal newforms in level $N^2$ that are Eisenstein modulo $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04195 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new perspective on the rank of Mazur's Eisenstein Hecke algebra Lang, Jaclyn Müller, Katharina Palvannan, Bharathwaj Number Theory Primary 11F33, Secondary 11F80, 11F85 Let $N, p \geq 5$ be primes such that $N \equiv 1 \bmod p$. We study the rank $r$ of the Hecke algebra that parametrizes modular forms of weight 2 and level $N$ that are Eisenstein modulo $p$. When $r$ is $2$ or $3$, we prove that $r-1$ equals the order of vanishing of the mod-$p$ reduction of a zeta element that interpolates Dirichlet $L$-values at $-1$, thereby recovering results of Merel and Lecouturier. This equality can fail in some cases when $r \geq 4$, and we provide a heuristic explanation of this failure. Our approach handles all of these cases uniformly by studying the analogous Hecke algebra in level $N^2$. When exactly one of $r-1$ or the order of vanishing equals $3$, we provide precise information about Galois orbits of cuspidal newforms in level $N^2$ that are Eisenstein modulo $p$. |
| title | A new perspective on the rank of Mazur's Eisenstein Hecke algebra |
| topic | Number Theory Primary 11F33, Secondary 11F80, 11F85 |
| url | https://arxiv.org/abs/2605.04195 |