The theta-kernel weight divisibility conjecture

Let $p$ be a prime, let $n$, $j$, and $k$ be positive integers with $j<n$, and let $F\in M_k(\Gamma^{(n)}_{1}(N))_{\mathbb{Z}_{(p)}}$ with $k\in\mathbb{Z}_{\ge 1}$. Assume that $\Theta^{[j+1]}(F)\equiv 0\pmod p$ and $\Theta^{[j]}(F)\not\equiv 0\pmod p$. Suppose that $k$ is sufficiently small compared with $p$. Theta-kernel weight divisibility conjecture. Then

$$p\mid\bigl(2\omega_N(F)-j\bigr).$$

The preceding theorem establishes this divisibility only under explicit bounds and additional hypotheses; the stated general prediction concerns the broader regime in which the weight is sufficiently small relative to $p$.