A K3 in phi4

TL;DR

This paper investigates the polynomial nature of point counts of graph hypersurfaces over finite fields, providing criteria for when this occurs and presenting counterexamples related to phi^4 theory involving modular forms from K3 surfaces.

Contribution

It offers a combinatorial criterion for polynomial point-counts and constructs explicit counterexamples to Kontsevich's conjecture within phi^4 theory.

Findings

Counterexamples with non-polynomial point counts are constructed.

Counting functions relate to modular forms from K3 surfaces.

Provides criteria distinguishing polynomial from non-polynomial cases.

Abstract

Inspired by Feynman integral computations in quantum field theory, Kontsevich conjectured in 1997 that the number of points of graph hypersurfaces over a finite field $\F_q$ is a (quasi-) polynomial in $q$. Stembridge verified this for all graphs with $\leq12$ edges, but in 2003 Belkale and Brosnan showed that the counting functions are of general type for large graphs. In this paper we give a sufficient combinatorial criterion for a graph to have polynomial point-counts, and construct some explicit counter-examples to Kontsevich's conjecture which are in $\phi^4$ theory. Their counting functions are given modulo $pq^2$ ($q=p^n$) by a modular form arising from a certain singular K3 surface.