Recently, the electric network frequency (ENF), a natural signature embedded in many audio recordings, has been utilized as a criterion to examine the authenticity of audio recordings. ENF-based audio authentication system involves extraction of the ENF signal from a questioned audio recording, and matching it with the reference signal stored in an ENF database. This establishes a popular application of audio timestamp verification. In this paper, we explore another important application, i.e., ENF-based audio tampering detection, which has received less research attention. Specifically, we introduce the absolute-error-map (AEM) between the ENF signals obtained from the testing audio recording and the database. The AEM serves as an ensemble of the raw data associated with the ENF matching process. Through intensive analysis of the AEM, we propose two algorithms to jointly deal with timestamp verification and tampering detection, including insertion, deletion, and splicing attacks, respectively. The first algorithm is based on exhaustive point search and measurement, while the second algorithm leverages the image erosion technique to achieve fast detection of tampering type and tampered region, thus the second algorithm sacrifices some accuracy for speed. The authentication mechanism is that the system first determines if the testing data have been tampered with, and then outputs the timestamp information if no tampering is detected. Otherwise, it outputs the tampering type and tampered region. We demonstrate the effectiveness of the proposed solution via both synthetic and practical examples from our practically deployed audio authentication system.