秩序、體嵌入和埃奇勒類數公式

Abstract

此篇論文主要分為三個部分;在第一個部分裡，我們考慮對於一個對於全球域F的有限維度的中心簡單代數A和一個對於此域F的有限擴張域K(可分擴張或者是不可分擴張)，其中F的次數必須整除A的次數。我們給出了域F可以K-線性嵌入中心簡單代數A的充分必要條件，以及對於某一個K上的位v，局部域F_v可以K_v-線性嵌入中心簡單代數A_v的充分必要條件，然後給出了數值上對於一組$(K,A)$，哈瑟原則是否成立的判斷方法。 第二部分主要是探討對於一個局部域上有限維度的中心簡單代數上的秩序，並且討論在哪些條件之下，一個單項秩序會是一個巴斯秩序或者是一個古瑞斯丹秩序。事實上，我們發現如果是在一個上三角的單秩序條件下，埃奇勒秩序和古瑞斯丹秩序是一樣的。在一般條件之下，一個單項秩序是巴斯秩序充分必要條件是它是西利地特瑞秩序或者是周期為二的埃奇勒秩序。 最後一部分主要的目的是計算對應於p-韋伊數為根號p的簡單同源類的阿貝耳簇之同構類數目有多少，其中p為一個質數。此部分主要的工具是用到本田-塔特理論以及推廣後的埃奇勒類數公式。

This thesis is separated into three parts. In the first part, we consider a finite-dimensional central simple algebra A over a global field F and a finite (separable or not) field extension K of F whose degree divides the degree of A over K. We give the necessary and sufficient condition for which the field K (resp. K_v) can be F-linearly (resp. F_v-linearly) embedded into A (resp. A_v). Here v is a place of F, and F_v denotes the completion of F at v, K_v:=Kotimes_F F_v and A_v:=A otimes_F F_v. This yields a more numerical criterion for a pair (K,A) for which Hasse principle holds or not. Secondly, we study a class of orders called monomial orders in a central simple algebra over a non-Archimedean local field and determine which monomial orders are Gorenstein or Bass orders. In fact, we can show that for upper triangular monomial orders, the sets of Gorenstein orders and Eichler orders are the same. For general case, a monomial order is Bass if and only if it is either a hereditary or an Eichler order of period two. The goal of the third part is to compute the number of mathbb{F}_p-isomorphism classes of abelian varieties in the simple isogeny class corresponding to the p-Weil number pi =sqrt{p}, where p is a prime integer. Main tools are the Honda-Tate theory and extended methods for Eichler's class number formula.