A \(K3\) in \(\phi^{4}\). (English) Zbl 1253.14024
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.
The 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 \[ (\eta(\tau)\eta(7\tau))^3 \] which is attached to the singular \(K3\) surface of discriminant \(-7\).
The 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 \[ (\eta(\tau)\eta(7\tau))^3 \] which is attached to the singular \(K3\) surface of discriminant \(-7\).
Reviewer: Matthias Schütt (Hannover)
MSC:
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 05A15 | Exact enumeration problems, generating functions |
| 11G25 | Varieties over finite and local fields |
| 14M12 | Determinantal varieties |
| 81T18 | Feynman diagrams |
Software:
OEISOnline Encyclopedia of Integer Sequences:
Coefficients of L-series for elliptic curve ”49a1”: y^2 + x * y = x^3 - x^2 - 2 * x - 1.References:
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