The local zeta function for the non-trivial characters associated with the singular Jordan algebras

Abstract

Abstract. This paper investigates the local integralswhere O C represents the integers of a composition algebra over a non-archimedean local field K and χ is a non-trivial character on the units in the ring of integers of K extended to K * by setting χ(π) = 1. The local zeta function for the trivial character is known for all composition algebras C. In this paper, we show in the quaternion case that Z(t, χ) = 0 for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for χ equal to the quadratic character. In this latter case, Z(t, χ) = 0 for any character of order greater than 2.