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

TL;DR

This paper provides a criterion to determine when a mod ll eigenvalue system from Katz cuspforms can be derived from lower level or weight, based on the intersection of kernels of Hecke operators.

Contribution

It introduces a new criterion involving kernel intersections of Hecke operators to identify minimal levels of mod ll eigenvalue systems.

Findings

The criterion applies to eigenvalue systems with ll not dividing the level.

It relates the existence of a prime dividing the level and ll to the origin of eigenvalue systems.

The result helps understand the minimal level and weight for mod ll eigenforms.

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 $1\leq k \leq \ell +1$, arises from lower level or weight if there exists a prime $r$ dividing $n\ell$ such that $$ \mathrm{dim}_{\overline{\mathbb{F}}_\ell} \bigcap_{p \neq r} \ker \left( T_p-f(T_p), S(n,k)_{\overline{\mathbb{F}}_\ell}\right)>1,$$ 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 $k$.