A note on the minimal level of realization for a mod $\ell$ eigenvalue system

Abstract

In this article we give a criterion for a mod $\ell$ eigenvalue system attached to a mod $\ell$ Katz cuspform to arise from lower level or weight. Namely, we prove the following: the eigenvalue system associated to a ring homomorphism $f:\mathbb{T}\to {\overline{\mathbb{F}}}_{\ell }$ from the Hecke algebra of level ${\Gamma }_1(n)$ and weight $k$ to ${\overline{\mathbb{F}}}_{\ell }$, where $\ell$ is a prime not dividing $n$ and $1\le k\le \ell +1$, arises from lower level or weight if there exists a prime $r$ dividing $n\ell$ such that $${\dim }_{{\overline{\mathbb{F}}}_{\ell }}\bigcap _{p\ne r}\ker \left(T_p-f(T_p),S(n,k{)}_{{\overline{\mathbb{F}}}_{\ell }}\right)>1,$$ where $T_p$ is the $p$-th Hecke operator and $S(n,k{)}_{{\overline{\mathbb{F}}}_{\ell }}$ is the space of mod $\ell$ Katz cuspforms of level ${\Gamma }_1(n)$ and weight $k$.

arXiv comment12 pages, comments are welcome