Arithmetic properties of Heilbronn sums

Abstract

For each odd prime q an integer NH q ( NH 3 = −1, NH 5 = −1, NH 7 = 97, NH 11 = −243, …) is defined as the norm from L to Q of the Heilbronn sum H q = Tr I Q (ζ) (ζ), where ζ is a primitive q 2th root of unity and L ⊃- Q (ζ) the subfield of degree q. Various properties are proved relating the congruence properties of H q and NH q modulo p ( p ≠ q prime) to the Fermat quotient (p q − 1 − 1) q ( mod q) ; in particular, it is shown that NH q is even iff 2 q − 1 ≡ 1 (mod q 2).