Poles of a two-variable $P$-adic complex power

Abstract:

For almost all $P$-adic completions of an algebraic number field, if $s \in {\mathbf {C}}$ is a pole of ${f^s} = \int _{}^{} {\int _{}^{} {|f(x,y){|^s}|dx{|_{{K_p}}}|dy{|_{{K_p}}}} }$ , where $f$ is a polynomial whose only singular point is the origin, $f(0,0) = 0$, and $f$ is irreducible in $\overline K [[x,y]]$, then $\operatorname {Re} (s)$ is $- 1$ or one of an explicitly given set of rational numbers, whose cardinality is the number of characteristic exponents of $f = 0$.