This paper considers challenges resulting from the use of advanced artificial judicial intelligence (AAJI). We argue that these challenges should be considered through the lens of value alignment. Instead of discussing why specific goals and values, such as fairness and nondiscrimination, ought to be implemented, we consider the question of how AAJI can be aligned with goals and values more generally, in order to be reliably integrated into legal and judicial systems. This value alignment framing draws on AI safety and alignment literature to introduce two otherwise neglected considerations for AAJI safety: specification and assurance. We outline diverse research directions and suggest the adoption of assurance and specification mechanisms as the use of AI in the judiciary progresses. While we focus on specification and assurance to illustrate the value of the AI safety and alignment literature, we encourage researchers in law and philosophy to consider what other lessons may be drawn.