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We compute the variances of sums in arithmetic progressions of arithmeticfunctions 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 byestablishing appropriate equidistribution results for the associated Frobeniusconjugacy classes. The variances are thus related to matrix integrals, whichmay be evaluated. Our results differ significantly from those that hold in thecase of degree-one $L$-functions (i.e. situations considered previously usingthis approach). They correspond to expressions found recently in the numberfield setting assuming a generalization of the pair-correlation conjecture. Ourcalculations apply, for example, to elliptic curves defined over$\mathbb{F}_q[t]$.

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