Orbital Exponential Sums for Some Quadratic Prehomogeneous Vector Spaces

Let (G, V ) be a prehomogeneous vector space over a finite field of odd characteristic.Taniguchi and Thorne [5] developed a method to calculate explicit formulas of the Fourier transforms of any G-invariant functions over V .By means of their method, we calculate the Fourier transform of any G-invariant function for several "quadratic" prehomogeneous vector spaces, parametrizing quadratic fields.We use PARI/GP [8] for explicit matrix calculations.Here Tr F q /F p : F q F p is the trace map.Let F G V be the set of all G-invariant maps from V to C, i.e.,V is a finite dimensional vector space over C. We can easily see that if is a G-invariant function, is also G-invariant.In fact, the Fourier transform map) be all the distinct G-orbits in V , and for each i let e i be the indicator function of O i .The functions e 1 , . . .