A \(K3\) in \(\phi^{4}\) (Q442755)

The present paper concerns graph hypersurfaces and their number of points over finite fields. In 1997 Kontsevich conjectured that the number of points of graph hypersurfaces over a finite field \(\mathbb{F}_q\, (q=p^n, p \text{ a prime})\) is a polynomial or a quasi-polynomial in \(q\). This conjecture which was inspired by Feynman integral computations in quantum field theory, was verified by Stembridge for all graphs on at most 12 edges, but it tends to be false for large graphs by work of Belkale and Brosnan.NEWLINENEWLINEThe present paper provides a sufficient combinatorial criterion for a graph to have polynomial point-counts. It also constructs some explicit counterexamples to Kontsevich's conjecture which are actually arising from \(\phi^4\) theory. Their counting functions are related to the weight 3 Hecke eigenform NEWLINE\[NEWLINE (\eta(\tau)\eta(7\tau))^3 NEWLINE\]NEWLINE which is attached to the singular \(K3\) surface of discriminant \(-7\).