It’s 10AD in Ancient Rome. My wife has just dropped dead suspiciously and all the locals think that I did it, they want justice. What kind of judicial process (if any) will there be and what rights of defence do I have?

There are a couple of questions to answer here. First, is killing your wife a crime? This seems like it should be obvious, but Romans husbands had at least some limited ability to kill their wives without penalty. A Roman patriarch, called the paterfamilias, theoretically had the power of life and death over his wife and children. Roman authors in the late Republic and early Empire often talked about this right as if it was an active power still used in their day. However, when they give examples of this, they are usually talking about the distant past, often quite nostalgically. When they give more recent examples, they are never completely arbitrary murders. The last known case in Roman sources of a husband judging his wife with the power of life and death comes Tacitus. He wrote that in 63 AD, a woman named Pomponia Graecina was charged with “foreign superstition” (possibly Christianity), and handed over to her husband for trial. It is implied that he could have killed her for this, but he chose to acquit her of the charge, and so it is unclear if he could have, or would have, killed her if he had judged her guilty. If Roman husbands ever had the right to kill their wives with impunity, it had been greatly reduced by the 1st century AD. However, it is still likely that killing your wife would still be viewed by many as a lesser crime then other types of murder, particularly if people believed you had a “reason” for doing so, such as catching her committing adultery. It is possible that even if people believed you had killed your wife, they would not blame you for doing so.

The second part of the question, what kind of judicial process there will be, would vary dramatically based on your status, and the status of your wife’s family. Only a Roman citizen could press charges in a Roman court. There was some kind of judicial system that served non-citizens in the imperial period, but it is poorly documented and not well understood, so I am going to assume for the sake of having something to talk about that the involved parties are Roman citizens. Unlike in the modern world, there were no state prosecutors in Rome. All legal cases, even murder cases, were brought by one person against another. The person bringing the case acted as the prosecutor for a criminal case, although they could hire someone to handle the actual legal work. Theoretically any Roman citizen could prosecute a murder charge against any other Roman, but in practice this duty fell to the family of the victim, usually their paterfamilias or another male relative. In your case, any of the suspicious neighbors theoretically could press charges against you if they were a Roman citizen, but it's very unlikely they would do so unless the family of your wife was on board with the prosecution. If your wife had no living family, it's possible a neighbor might take up prosecution if they believed you to have murdered her without cause, but I am not aware of any case where this ever happened. Most likely, if charges were brought, they would be brought by your wife’s father, or if he were dead, by another male relative of hers. Women had the right to bring criminal charges if the victim was a member of their immediate family, so it's possible that if her father was dead, her mother might bring charges instead of a male relative, but this is less likely.

Who would hear the case would depend on your status and the status of your wife’s family. In 10 AD, the Roman legal system was in flux due to the dramatic changes in government over the previous decades, and one of the major changes was the introduction of the emperor to the legal system. If your wife’s family was wealthy and politically connected, they could appeal directly to the emperor, who served as the supreme judge of the state. In 24 AD, the emperor Tiberius heard a case that is actually quite similar to yours. Plautius Silvanus, a praetor (a very high government official), was accused of murdering his wife by throwing her out a window. The father of Silvanus’s now deceased wife prosecuted the case, and as a wealthy and politically connected man himself, he could take the case directly to the emperor. Although he had the authority to judge the case himself, Tiberius referred the case to the senate, possibly to avoid political fallout from the decision. Silvanus claimed that his wife committed suicide, but the case was never tried because Silvanus himself committed suicide before it could occur. Trials of the Roman elite were often highly political affairs, as Silvanus’s case demonstrates, and were not representative of the legal experience of most Romans.

If the parties involved in the trial were not wealthy and politically connected, the legal procedures would be far more conventional, although in 10 AD the legal system was in such flux that defining what a “conventional trial” is difficult. In the late republic, most cases were heard before courts called quaestiones perpetuae (standing court). Procedure in the quaestiones perpetuae could vary a lot, but the most common method of holding a criminal trial was for a magistrate to summon a jury to hear the case. Only free male citizens over the age of 30 with a recorded wealth of at least 300,000 sesterces could sit on a jury. (For context, in order to be considered part of the senatorial class, 800,000 sesterces of wealth was required, making 300,000 a fairly high bar to meet). These juries could vary in size, sometimes including hundreds of members for major cases, although most were much smaller. Magistrates were not required to hear every case, but its unlikely a sensational murder trial like yours would be refused if a family member was prosecuting. The quaestiones perpetuae were gradually replaced by new institutions in 1st and 2nd century AD, but in 10 AD, your case would probably still be heard by a quaestiones perpetuae.

The first step in a trial was formally laying charges before the magistrate. Both the prosecutor and the accused had to be present for this. If the accused was absent, they may be assumed to have gone into exile rather than face charges. No imperial police force existed, so if the prosecutor wanted to continue to press charges, he would need to pursue the accused himself. After charges were filed before the magistrate, both parties were given a period of at least 10 days to prepare their case. If you were a citizen in good standing, you would be left free during this time to prepare your defense. If you were able, you would hire a team of lawyers to assist you (generally not just one, although I do not know why a team was preferred).