Deitmar schemes, graphs and zeta functions

JOURNAL OF GEOMETRY AND PHYSICS

卷:117

Article

Merida-Angulo, Manuel¹  Thas, Koen¹ 

[1] Univ Ghent, Dept Math, Krijgslaan 281,S25, B-9000 Ghent, Belgium

关键词: Field with one element;    Deitmar scheme;    Loose graph;    Zeta function;    Ihara zeta function;   

DOI  :  10.1016/j.geomphys.2017.01.027

来源: Elsevier

【 摘 要 】

In Thas (2014) it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, F-1) to a so-called loose graph (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over F1 such as combinatorial F-1-projective and Fraffine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of loc. cit., and show that Deitmar schemes which are defined by finite trees (with possible end points) are defined over F-1 in Kurokawa's sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated F-q-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Ihara's zeta function (which is trivial in this case). Using a process called surgery, we show that one can determine the zeta function of a general loose graph and its associated IDeitmar, Grothendieck)-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the Grothendieck ring of F-1 -schemes along the way in order to perform the calculations. Finally, we include a computer program for performing more tedious calculations, and compare the new zeta function to Ihara's zeta function for graphs in a number of examples. (C) 2017 Published by Elsevier B.V.

【 授权许可】

Free