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Sum over exponentiated bilinear form in finite-field vector space

This is a multidimensional Gauss sum, and can be handled by the same methods used to handle Gauss sums. $Z(A)=0$ if and only if $X^T A X$ is nonzero for some $X \in \ker (A + A^T)$, and, if $Z(A) \...
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Sum over exponentiated bilinear form in finite-field vector space

Denote $b(X)=X^TAX$, $c(X)=(-1)^{b(X)}$ for $A=[a_{is}]$, $X=[x_1,\dots,x_n]^T$. Replacing $a_{ii}$ and $a_{ji}$ with $a_{ij}+a_{ji}$ and $0$, we may assume $A$ is upper-triangular. The form $b$ ...
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Pointless groups

Perhaps I'm missing something, but it seems to me that $k=\mathbb{F}_2$, $G=\mathbb{G}_{m, k}$ works, where $H$ is the trivial subgroup.
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