Evaluating symplectic Gauss sums and Jacobi Symbols

Abstract

Stark [9] has explicitly evaluated some symplectic Gauss sums when the “denominator” matrix has odd prime level. This result is useful in computing the exact tranformation formulas of multivariable theta functions (see Stark [10], Friedberg [3] and Styer [11]). It is particularly useful when considering theta functions with quadratic forms having an odd number of variables, often a troublesome case (see Eichler [2] and Andrianov-Maloletkin [1]).