Upper bound for the moment of shifted values of cubic $L$-functions over function fields

In this paper, we study correlations of shifted values of cubic $L$-functions over function fields and derive an upper bound for moments of these shifted values in the limit where the genus of the corresponding cubic characters tends to infinity over a fixed finite field ${\mathbb{F}}_q$. Our results apply to the non-Kummer case when $q\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$. The Kummer case, when $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$, can be treated similarly.