zeta functions of complexes from PGSp(4)

Open Access
Author:
Fang, Yang
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
March 04, 2011
Committee Members:
  • Winnie Li, Dissertation Advisor
  • Wen Ching Winnie Li, Committee Chair
  • Dale Brownawell, Committee Member
  • William C Waterhouse, Committee Member
  • Sencun Zhu, Committee Member
Keywords:
  • zeta
  • identity
  • PGSp(4)
Abstract:
In this thesis we study the zeta functions arising from PGSp$(4)$ over a nonarchimedean local field. In this case, the complexes have dimension two, like PGL$(3)$. However, the vertices are distinguished as special and nonspecial vertices, unlike the case of PGL$(3)$. We define the (edge) zeta function as the counting function of the number of tailless closed geodesics of all type-one or type-two edges, which has a closed form expression in terms of parahoric Hecke operators. The main result shows that the zeta function satisfies a zeta identity involving the Euler characteristic of the complex, the characteristic polynomial of the recurrence relations of the Hecke algebra, the Iwahori-Hecke operator and the number of special and nonspecial vertices. Moreover, we study the operators on nonspecial vertices.