半模函式與Shimura對應

Abstract

1973年，G.Shimura 利用theta-函數來定義半模函式，並且寫出Hecke算子作用在半模函式上的一般項。他發現Hecke算子作用在半模函式上的特徵值與整模函式的特徵值有對應關係，這就是所謂的Shimura對應。 另一方面，eta-函數是一個 (1/2) weight 的模函式在Shimura的定義之下。本篇論文當中主要探討用eta-函數所定義出來的半模函式空間，這也許會使我們能夠證明出一些分析函數的同於式。歷史上這種由eta-函數所定義出來的半模函式空間的研究開使於Li Guo 和 Ken Ono在“The partition function and the arithmetic of certain modular L-functions”中，並且證明了在某些例子中這種子空間是同構於一個整模函式的子空間，而這個整模函式的子空間是一些算子的不變子空間。我們現在把他們的結果更一般化，並且算出對應空間的維度。不過由於時間的關係，我們仍無法算出對於一般Hecke算子的trace formula，也許在將來的日子裡會有機會把他算出來。

In 1973, G.Shimura defined modular forms of half-integral weight by using theta-function. He showed that there are Hecke operators on half-integral weight modular forms, and he found that there is a correspondence between each eigenvalue for Hecke operator for integral weight modular form and half-integral weight modular form. And it is the so-called Shimura correspondence. On the other hand, eta-function is a modular form of weight (1/2) in Shimura′s sense. In this paper, we study the space of half-integral weight modular forms defined by eta-function, so that we may find some congruence of partition functions. Historically, these spaces were first studied by Li Guo and Ken Ono in their paper“The partition function and the arithmetic of certain modular L-functions”. They proved that in some case these space are isomorphic to space of integral-weight modular forms which is eigenspace of some operators. Now we make more general results, and we compute the dimensions in our cases. For the isomorphism, we try to prove it by using trace formula, but it is so complicated that we have not figured it out yet.