Paley-like graphs over finite fields from vector spaces

Extension and averaging operators in vector spaces over finite fields

We study the mapping properties of both extension operators and averaging operators associated with the algebraic variety S = {x ∈ Fq : Q(x) = 0} where Q(x) is a nondegenerate quadratic form over the finite field Fq. Carbery, Stones, and Wright [3] established a method to deduce L − L estimates for averaging operators over a general variety. As a core result in this paper, we give the sharp est...

Canonical Quantization of Symplectic Vector Spaces over Finite Fields

In this paper an affirmati5e answer is given to a question of D. Kazhdan on the existence of a canonical Hilbert spaces attached to symplectic vector spaces over finite fields. This result suggest a solution to a discrete analogue of a well known problem in geometric quantization where a naive canonical Hilbert space does not exist. As a result, a canonical model for the Weil representation of ...

Harmonic Analysis on Vector Spaces over Finite Fields

• (i) G = ZN = Z/NZ = {0, 1, 2, ....., N − 1} with addition modulo N . For 0 ≤ n ≤ N − 1 let γn : G → S, γn(m) = exp(2πimn/N). Then {γ0, ....., γN−1} is a complete list of the characters so that ZN is isomorphic to ZN . An example of a primitive N ’th root of unity is ω := exp 2πi/N . • (ii) G = T = R/Z; for n ∈ Z let γn : G→ S, γn(x) = exp(2πinx). Then G∗ = {γn : n ∈ Z} so that G∗ is isomorphi...

Extension Theorems in Vector Spaces over Finite Fields

Notes on quantization of symplectic vector spaces over finite fields

In these notes we construct a quantization functor, associating an Hilbert space H(V ) to a finite dimensional symplectic vector space V over a finite field Fq. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp (V ). The main technical result is a proof of a stronger form of the Stone-von Neumann theorem for the Heisenberg group over Fq. Our result ...