F-signature functions of diagonal hypersurfaces

Abstract

Let $f$ be a diagonal hypersurface in $A_p={\mathbb{F}}_p[[x_1,\dots ,x_n]]$. We study the behavior of the function ${\varphi }_{f,p}(a/p^e)=p^{-ne}{\dim }_{{\mathbb{F}}_p}(A_p/(x_1^{p^e},\dots ,x_n^{p^e},f^a))$ which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when $p$ goes to infinity ${\varphi }_{f,p}$ converges to a piecewise polynomial function ${\varphi }_f$ and the left and right derivatives of ${\varphi }_{f,p}$ converge to ${\varphi }_f^{\prime }$. We use this fact to prove the existence of the limit F-signature and limit Hilbert-Kunz multiplicity for diagonal hypersurfaces. When $f$ is a Fermat hypersurface, we investigate the shape of the F-signature function of $f$ and provide an explicit formula for the limit F-signature and, in some cases, also for the F-signature for fixed $p$. This allows us to answer negatively to a question of Watanabe and Yoshida.

are welcome