Rigid local systems and a question of Wootters

Recently, we learned from Ron Evans of some fascinatingquestions raised by Wootters \cite{S-S-W}. These questions,which concern exponential sums, arose from hisinvestigations of a particular quantum state with specialproperties, where the underlying vector space is the spaceof functions on the finite field $\F_p:=\Z/p\Z$, $p$ aprime which is $3$ mod $4$. Due to our ignorance of theunderlying physics, we concentrate on the exponential sumsthemselves. In our approach, it costs us nothing to workover an arbitrary finite field $\F_q$ of oddcharacteristic. [Thus $\F_q$ is “the” finite field of $q$elements, $q$ a power of some odd prime $p$.] We alsointroduce a parameter $a \in \F_q^{\times}$. In the Wootterssetup, where $q=p$ is $3$ mod $4$, the parameter $a$ issimply $a= -1$. Ultimately, we end up proving identitiesamong exponential sums, but not at all in a straightforwardway; we need to invoke the theory of Kloosterman sheavesand their rigidity properties, as well as the fundamentalresults of \cite{De-Weil II} and \cite{BBD}. It would beinteresting to find direct proofs of these identities.

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Published 19 October 2012