Local zeta functions and Meuser's invariant functions
Let \(K_1\) be a fixed finite field extension of \(\mathbb{Q}_p\), the field of \(p\)-adic numbers. For a positive integer \(d\), let \(K_d\) denote the unique unramified extension of \(K_1\) of degree \(d\) with ring of integer \(\mathcal O_{K_d}\). Let \(h(x)\) be a polynomial in \(m\) variables \(x=(x_1,x_2,\ldots,x_m)\) with coefficients in \(\mathcal O_{K_1}\). Igusa's local zeta function of \(h(x)\) on the field \(K_d\) is defined as \[ Z(s,K_d)=\int_{\mathcal O^m_{K_d}} | h(x)|^s_d\, | dx|_d\text{ for } s\in\mathbb{R},\quad s>0, \] where \(| dx|_d\) is the Haar measure on \(K_d^m\) such that \(\mathcal O^m_{K_d}\) has measure \(1\). The author shows that a generalization of Igusa's local zeta function to integrals over sets defined in a first order language of valued fields is an invariant function (in the sense of Meuser) of the class of unramified extensions of a \(p\)-adic field. As a corollary the author proves the invariance of certain Poincaré series which was conjectured by Fried and Jarden.
scientific article; zbMATH DE number 4214159