The hypergeometric–zeta correspondence for the curves $\mathcal{C}_{z}^{(m,s)}$

Let $l$ and $q$ be odd primes such that $q\equiv 1 \pmod{l}$, and let $z\in\mathbb{F}_{q}$ with $z\neq 0,1$. For integers $1\leq m,s<l$ satisfying $m+s=l$, consider the smooth projective curve with affine equation

$$\mathcal{C}_{z}^{(m,s)}: y^l=t^m(1-t)^s(1-zt)^m.$$

Using the notation for $F_{i,q}(z)$ and $a_{i,q}(z)$ introduced earlier, the hypergeometric–zeta correspondence. After rearranging terms if necessary,

$$F_{i,q}(z)=-a_{i,q}(z)\qquad\text{for all }1\leq i\leq g.$$

This conjecture proposes an equality between finite-field hypergeometric values and the reciprocal-root contributions to the zeta function of $\mathcal{C}_{z}^{(m,s)}$. The paper states that it remains unproved in full generality, while establishing particular cases and advances toward the general case.