Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree $L$-functions in $\mathbb{F}_q[t]$

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-one $L$-functions (i.e. situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair-correlation conjecture. Our calculations apply, for example, to elliptic curves defined over $\mathbb{F}_q[t]$.