我目前研究主題是關於函數體上Eisenstein級數的算術。在各數學領域中，Eisenstein 級數扮演一個很主要的角色。在SL(2,R)的表現理論裡，這些級數生成了在非歐Laplacian的譜分解裡連續的部分。另外Eisenstein級數也出現在自守L-函數的積分表示裡，在Langlands 計畫中是不可或缺的。透過Kronecker極限公式，我們可以將在SL(2,R) 上實解析Eisenstein 級數的“第二項”特殊值利用模單元來表示。這公式自然地連接模Galois 表現上的Euler 系統以及所對應的L-函數。除此之外，Beilinson（利用模單元）在模曲線的K-群裡構造了一些特殊元素，並透過這個極限公式證明這些元素的“regulator” 等同於所對應的L-特殊值。透過我最近在函數體上關於Kronecker 極限公式的工作， 這個計畫包含兩個方向：(一) 研究在Drinfeld 模簇的K-群上的Beilinson regulator；(二) 透過研究Bruhat-Tits building 上的Green 函數來尋找mirabolic Eisenstein 級數的幾何解釋。本研究目標在深入探討總函數體上的算術現象，並得到相較代數體上目前發展更遠的結果。 ;The main research topic in my current pursuit is on arithmetic of Eisenstein series in the function field context. Eisenstein series play a major role in various topics of mathematics. In the representation theory of SL(2,R), these series generates the continuous part in the spectral decomposition of the non-Euclidean Laplacian. Also, Eisenstein series appears as the “kernel functions” in the integral representations of automorphic L-functions, which are fairly essential in Langlands program. The celebrated Kronecker limit formula expresses the “second-term” special values of non-holomorphic Eisenstein series on SL(2,R) in terms of modular units. This formula naturally bridges Euler systems associated to modular Galois representations and the corresponding L-functions. Moreover, Beilinson constructed special elements (viamodular units) in the K-groups of modular curves, and applied the formula to show that their “regulator images” are essentially the corresponding special L-values. From my recent work on the Kronecker limit formula over function fields, there are two directions in this project:(1) Study Beilinson’s regulator on the K-groups of Drinfeld modular varieties;(2) Seek geometric interpretations of mirabolic Eisenstein series via “Green’s functions” on Bruhat-Titsbuildings.Our goal is to explore arithmetic phenomena over global function fields, and prove results which go farbeyond the classical developments over number fields.
