Global Igusa zeta function and $K$-equivalence

Abstract

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