Rationality of Motivic Zeta

2014 Fiscal Year Final Research Report

• Project/Area Number: 24654007
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: KIMURA Shun-ichi 広島大学, 理学(系)研究科(研究院), 教授 (10284150)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: モチビックゼータ

Outline of Final Research Achievements

When X is a finite set, the formal sum of its symmetric products is rational, and the degree of its denominator is the number of the elements of X. When Y is an algebraic variety over a finite field, the rationality of the formal sum of the number of elements of its symmetric products is the celebrated Weil Conjecture, which promoted the advance of algebraic geometry in 20th century. The aim of this research is study the rationality of the formal sum of symmetric products (which is called the motivic zeta) in many categories.
We studied the rationality of the Motivic Chow Series, which is a generalization of the motivic zeta in the category of motives of algebraic varieties, and found that the rationality depends on the varieties and the equivalence relations, which implies the subtlety of this business. We also defined the generating function (expedited to be promoted to a version of motivic zeta), conjectured that it is always rational, and proved the rationality in some cases.

Free Research Field

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