The Jiang Zemin and Hu Jintao periods were eventful, coinciding with and indeed facilitating China’s emergence on the world stage as a major player based on the spectacular economic growth over which they presided. 

【翻译】江泽民和胡锦涛时代是一个重要的历史时期。这一时期，恰逢并确实促进了中国作为主要参与者在世界舞台上的崛起，这基于他们领导的惊人的经济增长。

【分析】

1. The Jiang Zemin and Hu Jintao periods were eventful：主系表结构，是句子主干。

2. coinciding with and indeed facilitating China’s emergence on the world stage as a major player based on the spectacular economic growth over which they presided.  现在分词短语作结果状语。

3. coinciding with and indeed facilitating 是并列的现在分词，其逻辑注意是The Jiang Zemin and Hu Jintao periods。

4. China’s emergence on the world stage 是coinciding with and indeed facilitating 的宾语。

5. as a major player 方式状语。

6. based on the spectacular economic growth over which they presided.  过去分词短语作定语，修饰China’s emergence on the world stage，相当于which was based on the spectacular economic growth over which they presided。