Background: The eigenvalues of the Hecke opeator $T_p$ on the space of cusp forms $S_2(\Gamma_0(N))$ (of dimension $g=\text{genus}(X_0(N))$) are analysed using reduction mod $p$ and the use of the Eichler-Shimura relation on the Tate module of the Jacobian of the reduction mod $p$ of $X_0(N)$.
I will denote the Tate module of the Jacobian by $H^1$ (to stand for etale cohomology), although I don't yet feel comfortable with etale cohomology.
Now, working mod $p$, $\text{Frob}+\text{Ver}$ acts on $H^1$ which is of dimension $2g$. But all the eigenvalues are real and each appears twice.
Question: Given a smooth projective curve $C$ over $\mathbb{F}_p$, let the $2g$ eigenvalues of $\text{Frob}$ on $H^1$ be $\alpha_1,\dotsc,\alpha_g$ and their conjugates $\overline{\alpha_1},\dotsc,\overline{\alpha_g}$. Is there a natural way to get a $g$-dimensional vector space $V$ and an operator $T:V\rightarrow V$ with eigenvalues $\alpha_1+\overline{\alpha_1},\dotsc,\alpha_g+\overline{\alpha_g}$?
Remark: Of course, we can always define such an operator. My question is whether there are natural constructions of such pairs $V,T$ from the curve $C$. Specific answers for specific curves would still be very nice. One example is of course: If $C$ is the reduction mod $p$ of the modular curve $X_0(N)$ (assume $p\nmid N$), then we can take $T_p$ to be the Hecke operator on $V=S_2(\Gamma_0(N))$. So, in a sense I'm looking for an analogue of $T_p$ and $S_2(\Gamma_0(N))$ for a smooth projective curve $C$ over $\mathbb{F}_p$ which is not necessarily the reduction of $X_0(N)$ mod $p$.
Some ideas: Maybe it's always (or sometimes) possible to lift $C$ to a curve defined over $\mathbb{Q}$, and also to lift $\text{Frob}+\text{Ver}$ and look at the action of the lift on the space of holomorphic differentials on the lifted curves (viewed as a complex curve)? I'm pretty much guessing here.