Duke Mathematical Journal· 2012Q1
A K3 in ϕ4
- 123citations
- Q1SCImago
- 2012year
Short summary
A new combinatorial criterion identifies graphs whose point counts over finite fields are polynomial, and constructs counterexamples to Kontsevich's conjecture using ϕ4 theory and a singular K3 surface.
AI-generated from the title and abstract; the full text is not read.
Key points
- Introduces a sufficient combinatorial criterion for graph point counts over finite fields to be polynomial.
- Constructs explicit counterexamples to Kontsevich's conjecture using ϕ4 theory.
- The counting functions of these counterexamples are modular forms related to a singular K3 surface.
- Addresses a conjecture by Kontsevich regarding graph hypersurfaces in quantum field theory.
AI-generated from the title and abstract; the full text is not read.
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 Fq is a (quasi-) polynomial in q. Stembridge verified this for all graphs with at most twelve 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 counterexamples to Kontsevich’s conjecture which are in ϕ4 theory. Their counting functions are given modulo pq2 (q=pn) by a modular form arising from a certain singular K3 surface.
The authors' abstract, as published at the source. Duke Mathematical Journal, 2012 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics