研究期間：10108~10207;Let F be a nonarchimedean local field. Recently, zeta functions of finite complexes arising from the Bruhat-Tits building associated to the symplectic group Sp4(F) have been introduced by Fang-Li-Wang. They showed that these zeta functions have a closed form expression related to the degree 4 spin L-functions of GSp4(F). The notion of Ramanujan complexes in the symplectic case was also defined in the same paper, with a classification of Ramanujan complexes given in terms of the behavior of zeta functions. In this project, we propose to study the related questions further. Specifically, since there is another important L-functions associated to GSp4(F), known as the degree 5 standard L-function, it is natural to seek for another family of zeta functions for finite complexes arising from the building associated to Sp4(F), which have a closed form expression revealing these degree 5 standard L-functions. Also, as the construction of Ramanujan complexes in the case of GLn involves deep results from representation theory including the Jacquet-Langlands correspondence and Ramanujan conjectures, it would be interesting to understand the nature of Ramanujan complexes, including their existence and explicit construction, in the setting of symplectic groups.
