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Let $m, n$ be non-negative integers. Assume that $\boldsymbol{\chi} = \left( \chi_i \right)_{1 \leq i \leq m}$ and $\boldsymbol{\eta} = \left( \eta_j \right)_{1 \leq j \leq n}$ are two collections of nontrivial multiplicative characters in $\widehat{\mathbb{F}_{q}^{\times}}$. N. Katz introduced the following hypergeometric sum: $$H(t,q;\boldsymbol{\chi}, \boldsymbol{\eta}) := \frac{\left( -1 \right)^{m+n-1}}{q^{(m+n-1)/2}} \mathop{\sum \sum}_{\substack{\mathbf{x} \in \left( \mathbb{F}_{q}^{\times} \right)^{m},\ \mathbf{y} \in \left( \mathbb{F}_{q}^{\times} \right)^{n}\\N(\mathbf{x}) = tN(\mathbf{y})}} \boldsymbol{\chi}(\mathbf{x}) \overline{\boldsymbol{\eta}(\mathbf{y})} \psi \left( T(\mathbf{x}) - T(\mathbf{y}) \right),$$ where $t \in \mathbb{F}_{q}^{\times}$, and for $\mathbf{x} = \left( x_1, x_2, \ldots, x_m \right) \in \left( \mathbb{F}_{q}^{\times} \right)^{m}$ (similarly for $\mathbf{y}$), we define: \begin{align*} \boldsymbol{\chi}(\mathbf{x}) &= \prod_{i=1}^{m} \chi_i(x_i), \\ T(\mathbf{x}) &= x_1 + x_2 + \cdots + x_m, \\ N(\mathbf{x}) &= x_1 x_2 \cdots x_m. \end{align*} With the above notation, if $\boldsymbol{\chi}$ and $\boldsymbol{\eta}$ are disjoint (i.e. $\chi_i\neq\eta_j,\forall i,j$), then for any $\ell \neq p$, there exists a geometrically irreducible $\ell$-adic middle-extension sheaf $\mathcal{H}(\boldsymbol{\chi}, \boldsymbol{\eta})$ on $\mathbb{A}^{1}_{\mathbb{F}_q}$ with Frobenius algebraic trace function given by $t \mapsto H(t,q; \boldsymbol{\chi}, \boldsymbol{\eta})$, such that it is

  1. pointwise pure of weight $0$ and of rank $\max \{m, n\}$;
  2. lisse on $\mathbb{G}_{m, \mathbb{F}_q}$ if $m \neq n$;
  3. lisse on $\mathbb{G}_{m, \mathbb{F}_q} - \{1\}$ and of rank $m$ if $m = n$.

The above result is Theorem 8.4.2 in the book "Exponential Sums and Differential Equations" by N. Katz.

Question: Does there exist a geometrically irreducible $\ell$-adic middle-extension sheaf on $\mathbb{A}^{1}_{E}$ with Frobenius algebraic trace function defined by $a \mapsto \mathrm{Hyp}\left( \psi; \boldsymbol{\chi}, \boldsymbol{\rho} \right)\left( a, E \right)$ (this hypergeometric sum is constructed as in the attached image) and satisfying the properties mentioned above, where $E/\mathbb{F}_{q}$ is a finite field extension? Fig.1. Fig.2.

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    $\begingroup$ The formula appearing in Fig. 1 you link should be basically the same as the formula in Theorem 8.4.2, with slightly different notation. Can you explain what discrepancy you see between them that needs to be dealt with to deduce the existence of the sheaves associated to the $(a,E)$ sum from the Theorem 8.4.2 sum? $\endgroup$
    – Will Sawin
    Commented Oct 6 at 13:51
  • $\begingroup$ @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? $\endgroup$
    – Takatoshi Kashiwara
    Commented Oct 6 at 14:50
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    $\begingroup$ $E$ is of the form $\mathbb F_{q'}$ for some power $q'$ of $q$, and one just has to plug $q'$ instead of $q$ into the prior result. $\endgroup$
    – Will Sawin
    Commented Oct 6 at 15:49

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