On the Shintani zeta function for the space of pairs of binary Hermitian forms

Abstract

Throughout this paper, k is a number field. We fix a quadratic extension k_1=k(a_0) of k, where a_0=sqrt(B_0) for a certain B_0 in k^\times/(k^\times)^2. In this paper, we consider the zeta function defined for the space of pairs of binary Hermitian forms. This is the prehomogeneous vector space we discussed in section 2 [1] and is a non-split form of the D_4 case in [4]. The purpose of this paper is to determine the principal part of the adjusted zeta function. Our main result is Theorem (8.15). Our case resembles the space of pairs of binary quadratic forms which we discussed in Chapter 5 [5]. However, the meaning of the problem, the pole structure of the adjusted zeta function, and the adjusting terms are different from those of the above case. The zeta function of our case is a counting function of the class number times the regulator of fields of the form k(sqrt(B_0),sqrt(B)) with B_0 fixed and B in k_, whereas the zeta function for the above case is essentially the same as the space of binary quadratic forms. Our case is a first example where we need two adjusting terms whereas the adjusting term of the above case was essentially the same as that of the space of binary quadratic forms. We will not prove the meromorphic continuation of the adjusting terms in this paper. For the sake of the density theorem, we still get the residue of the rightmost pole of the zeta function (without adjusting). If this space appears in a bigger prehomogeneous vector space. So the principal part formula of the adjusted zeta function should be enough for most applications. We have yet to obtain the corresponding density theorem because we have not carried out the local theory.