A K3 in phi4

Francis Brown; Oliver Schnetz

A K3 in phi4

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 1006.4064 v5 pith:MZ2D5XR3 submitted 2010-06-21 math.AG math-phmath.MP

Francis Brown , Oliver Schnetz This is my paper · ORCID

classification math.AGmath-phmath.MP

keywords countingfieldfunctionsgraphgraphskontsevichpolynomialtheory

verification ladder T0 review T1 audit T2 compute T3 formal

0 comments

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.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Graph integrals, Feynman periods, and single-valued multiple zeta values
math.QA 2026-07 conditional novelty 7.0 of 10

Canonical integrals of graphs with E=2V−2 equal RW integrals and evaluate to single-valued multiple zeta values, which are shown to lie in the space of Feynman periods.

Pith tools