Establish full rank of the symmetric-power Frobenius

Prove that the Frobenius operator \(\bar{\beta}_{k,a}\) has full rank on the first cohomology space \(H^1(\mathcal{S}_k,\nabla)\), thereby establishing equality in the degree bound for \(L({\rm Sym}^{k}Kl_{n,m}/\mathbb{F}_q,T)\).

Background

The paper constructs a symmetric-power cohomology space $H^1(\mathcal{S}_k,\nabla)$ and identifies the symmetric-power $L$-function with the characteristic polynomial of the induced Frobenius operator $\bar{\beta}_{k,a}$, under the condition $d_k(n,m,p)=0$. The main theorem gives only an upper bound for the degree of this polynomial.

The authors explain that the missing ingredient is a proof that $\bar{\beta}_{k,a}$ has full rank. Establishing this would show that the degree reaches the cohomological dimension $\frac{m}{mn+1}\binom{mn+k}{k}$, confirming equality in the degree bound.