Exponential Sums for O+(2n,q) and Their Applications

For a nontrivial additive character λ of the finite field with q elements and each positive integer r, the exponential sums ∑ λ ( ( trw )r ) over w ∈ SO+(2n,q) and over w ∈ O+(2n,q) are considered. We show that both of them can be expressed as polynomials in q involving certain new exponential sums. Estimates on those new exponential sums are given. Also, we derive from these expressions the formulas for the number of elements w in SO+(2n,q) and O+(2n,q) with (trw)r = β, for each β in the finite field with q elements.