Let $k = \mathbb{F}_q$ be a finite field of characteristic $p$ and $C$ a projective, smooth curve over $k$. Denote by $\bar C$ the base change of $C$ to a separable closure $\bar k$ of $k$. Let $\ell$ be a prime $\ell \neq p$. Then we can consider the first étale homology by which I mean $$ H_1^{et}(\bar C, \mathbb{Z}_{\ell}) \stackrel{\mathrm{def}}{=} H^1_{et}(\bar C, \mathbb{Z}_{\ell}^{\vee})^{\vee} = H^1_{et}(\bar C, \mathbb{Q}_{\ell} / \mathbb{Z}_{\ell})^{\vee} $$ Here $(-)^{\vee}$ denotes the Pontryagin dual, i.e. $(-)^{\vee} = \mathrm{Hom}_{cts}(-, \mathbb{Q}/\mathbb{Z})$ (I think we might as well take $\mathrm{Hom}_{cts}(-, \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})$ in this setting).
Tensoring with $\mathbb{Q}_{\ell}$ yields $$ H_1^{et}(\bar C, \mathbb{Q}_{\ell}) \stackrel{\mathrm{def}}{=} H_{1}^{et}(\bar C, \mathbb{Z}_{\ell}) \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell} $$ Now by functoriality we have a Galois action on this $\mathbb{Q}_{\ell}$ vector space, i.e. a $\mathrm{Gal}_k = \mathrm{Gal}(\bar k / k)$ representation. From Weil / Deligne we know that the analogous action on étale cohomology $H^1_{et}(\bar C, \mathbb{Q}_{\ell})$ has no non-trivial fixed points, because the eigenvalues of the Frobenius all have absolute value $q^{1/2} \neq 1$.
I am assuming the same holds for $H_1^{et}(\bar C, \mathbb{Q}_{\ell})$, but can't deduce this from the above. My idea was to use Poincaré-Duality: For every $n$ we have a perfect pairing $$ H^1_{et}(\bar C, \mathbb{Z}/\ell^n) \times H^1_{et}(\bar C, \mu_{\ell^{n}}) \longrightarrow \mathbb{Z}/\ell^n $$ Also by definition we have a pairing $$ H^1_{et}(\bar C, \mathbb{Q}_{\ell} / \mathbb{Z}_{\ell}) \times H_1^{et}(\bar C, \mathbb{Z}_{\ell}) \longrightarrow \mathbb{Q}_{\ell} / \mathbb{Z}_{\ell} $$ By somehow taking limits and comparing the pairings, I am guessing that I obtain a pairing between étale cohomology and étale homology along the lines of $$ H^1_{et}(\bar C, \mathbb{Q}_{\ell}) \times H_1^{et}(\bar C, \mathbb{Q}_\ell) \longrightarrow \mathbb{Q}_{\ell}(?) $$ Where on the right hand side we twist by some factor. Then I could relate the Frobenius eigenvalues on $H^1_{et}(\bar C, \mathbb{Q}_{\ell})$ with those on $H_1^{et}(\bar C, \mathbb{Q}_{\ell})$.
Is this going in the right direction? Any hint on how to put these together or a reference is greatly appreciated!