# GAUSS SUMS FOR U(2n + 1,$q^2$)

• Kim, Dae-San (Department of Mathematics Sogang University )
• Published : 1997.11.01

#### Abstract

For a lifted nontrivial additive character $\lambda'$ and a multiplicative character $\chi$ of the finite field with $q^2$ elements, the 'Gauss' sums $\Sigma\lambda'$(tr $\omega$) over $\omega$ $\in$ SU(2n + 1, $q^2$) and $\Sigma\chi$(det $\omega$)$\lambda'$(tr $\omega$) over $\omega$ $\in$ U(2n + 1, $q^2$) are considered. We show that the first sum is a polynomial in q with coefficients involving certain new exponential sums and that the second one is a polynomial in q with coefficients involving powers of the usual twisted Kloosterman sums and the average (over all multiplicative characters of order dividing q-1) of the usual Gauss sums. As a consequence we can determine certain 'generalized Kloosterman sum over nonsingular Hermitian matrices' which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.

#### References

1. Duke Math. J. v.22 Representations by Herinitian in a finite field L. Carlitz;J. H. Hodges
2. Math. Nachr v.17 Weighted partitions for hermitian matrices over a finite field J. H. Hodges
3. to appear in Monatsh. Math Gauss sums for symplectic groups over a finite field D. S. Kim
4. Gauss sums for O(2n+1,q) D. S. Kim
5. Acta Arith v.80 Gauss sums for $O^-$(2n, q) D. S. Kim
6. to appear in Glasgow Math. J. Gauss sums for U(2n, $Q^2$) D. S. Kim
7. Acta Arith v.78 Gauss sums for $O^+$(2n, q) D. S. Kim;I. S. Lee
8. Encyclopedia Math. Appl. v.20 R. Lidl;H. Niederreiter