從p-adic 可約群誘導出的複體之Zeta 函數和上同調群

Abstract

Ihara zeta 函數已經廣受研究的題目，它是一個在一維複體上的幾何算術函數。 Ihara 的定理透過 trace formula 給了局部的 L 函數和 zeta 函數之間的關聯。 也就是給了幾何與質譜的關聯性。近幾年對於二維複統體上的 zeta 函數，使用 組合以及表現論的方法，可以得到Ihara 定理的推廣。但是對於更一般的複體， 這些方法都有它的困難性。 在第一年，我們計畫研究從一般可約群來的複體的上的 zeta 函數。透過在 lattices 上定義特殊的上同調理論以及偽Laplacian 算子，可以得到這些 zeta 函數 交錯乘積與局部 L‐函數的關聯性。 在第二年，我們將研究偽Laplacian 算子的特性。對於一維與二維的情形，我們 已經知道他和古典的Laplcian 一般，會有 Hodge decomposition 的性質。對於一 般的情形，透過群表現理論，我們可能計算偽Laplacian 的特徵值來證明這這個 性質。

Ihara zeta function has been widely investigated for years, which can be regarded as a geometric counting function on a 1-complex arising from PGL2. Ihara theorem can be reformulated in terms of the trace formula, which describes the relation between Ihara zeta function and the local L-factor. Zeta functions of 2-complexes arising from PGL3 and PGSP4 are studied recently by combinatoric and representation-theoretic approaches. However, each approach has its own difficulty to apply to general p-adic Lie groups. In the first year, we plan to investigate the zeta functions on complexes from general Lie groups and theirs relation to local L-factors using cohomology theory on lattices. We have shown that this method can be applied to PGLn for any n and we expect it can be applied to other families of Lie groups. Moreover, we would also like to study if there exists different kind of zeta identity related to the local L-factors associated to other representations. In the second year, we plan to study the relation between the cohomology theory we used and the usual cohomology theory on complexes. Especially, we expect that the pseudo -Laplacian we defined has Hodge decomposition as the usual Laplacian. It may be proved by computing the eigenvalues of the pseudo-Laplacian.