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@WillSawin Hypergeometric sums are previously defined by nontrivial multiplicative characters of $\mathbb{F}_q$, then we deduce there exists a geometrically irreducible $\ell$-adic middle-extension sheaf on $\mathbb{A}^1_{\mathbb{F}_q}$. Next, we define a hypergeometric sum but over $E$, a finite extension of $\mathbb{F}_{q}$. Whether there will lead to the similar result or not?
