Rigid local systems and a question of Wootters

Recently, we learned from Ron Evans of some fascinating questions raised by Wootters [1]. These questions, which concern exponential sums, arose from his investigations of a particular quantum state with special properties, where the underlying vector space is the space of functions on the finite field F_p:= Z/pZ, p a prime which is 3 mod 4. Due to our ignorance of the underlying physics, we concentrate on the exponential sums themselves. In our approach, it costs us nothing to work over an arbitrary finite field Fq of odd characteristic. [Thus Fq is "the" finite field of q elements, q a power of some odd prime p.] We also introduce a parameter a ∈ F ×/q. In the Wootters setup, where q = p is 3 mod 4, the parameter a is simply a = -1. Ultimately, we end up proving identities among exponential sums, but not at all in a straightforward way; we need to invoke the theory of Kloosterman sheaves and their rigidity properties, as well as the fundamental results of [6] and [3]. It would be interesting to find direct proofs of these identities.