The Secure Sockets Layer (SSL) and Transport Layer Security (TLS) protocols are the foundation of network security. The certificate verification in SSL/TLS implementations is vital and may become the weak link in the whole network ecosystem. In previous works, some research focused on the automated testing of certificate verification, and the main approaches rely on generating massive certificates through randomly combining parts of seed certificates for fuzzing. Although the generated certificates could meet the semantic constraints, the cost is quite heavy, and the performance is limited due to the randomness. To fill this gap, in this paper, we propose DRLGENCERT, the first framework of applying deep reinforcement learning to the automated testing of certificate verification in SSL/TLS implementations. DRLGENCERT accepts ordinary certificates as input and outputs newly generated certificates which could trigger discrepancies with high efficiency. Benefited by the deep reinforcement learning, when generating certificates, our framework could choose the best next action according to the result of a previous modification, instead of simple random combinations. At the same time, we developed a set of new techniques to support the overall design, like new feature extraction method for X.509 certificates, fine-grained differential testing, and so forth. Also, we implemented a prototype of DRLGENCERT and carried out a series of real-world experiments. The results show DRLGENCERT is quite efficient, and we obtained 84,661 discrepancy-triggering certificates from 181,900 certificate seeds, say around 46.5% effectiveness. Also, we evaluated six popular SSL/TLS implementations, including GnuTLS, MatrixSSL, MbedTLS, NSS, OpenSSL, and wolfSSL. DRLGENCERT successfully discovered 23 serious certificate verification flaws, and most of them were previously unknown.