Global Igusa zeta function and $K$-equivalence

Abstract

Let $\mathfrak{X}$ and ${\mathfrak{X}}^{\prime }$ be two smooth projective varieties over the ring of integers of a $p$-adic field $\text{K}$ with generic fibers being $X$ and $X^{\prime }$. We introduce a (family of) good $s$-norms on the pluricanonical spaces of $X$ and $X^{\prime }$, which are called global Igusa zeta functions in $s$, and show that if the $r$-canonical maps send $X$ and $X^{\prime }$ birationally to their images respectively, then any isometry between $H^0(X,r{K}_X)$ and $H^0(X^{\prime },r{K}_{X^{\prime }})$ with respect to this $s$-norm (for some $s>0$ and $s\ne 1/r$) induces $K$-equivalence on the $\text{K}$-points between $X$ and $X^{\prime }$.