Rigid local systems and a question of Wootters
Nicholas M. Katz · Communications in Number Theory and Physics · 2012
How it startedRecently, 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 F q of odd characteristic.[Thus F q 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.