phi4中的K3
A K3 in phi4
摘要：
受量子场论中费曼积分计算的启发，Kontsevich于1997年证明了有限域$\F_q$上图超曲面的点数是$q$中的（拟）多项式. Stembridge验证了所有图与$\leq12$边缘，但在2003年贝尔凯尔和Brosnan表明，计数函数是一般类型的大型图。本文给出了图具有多项式点数的一个充分组合判据，并构造了$ϕ^4$理论中Kontsevich猜想的几个显式反例.它们的计数函数由某个奇异K3曲面的模形式给出模$pq^2$（$q=p^n$）。
英文摘要：
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 $ϕ^4$ theory. Their counting functions are given modulo $pq^2$ ($q=p^n$) by a modular form arising from a certain singular K3 surface.
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