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It depends precisely what is meant by "co-regency". I will try to outline the widely accepted facts.
According to historian Beth Severy, Augustus in 4 CE announced his intention that Tiberius should inherent his title as emperor. Augustus adopted Tiberius as a son and give him a triumph and "a special grant of imperium". Over the following decade:
For the most part, Tiberius led the empire's military campaigns, while Augustus administered the city of Rome. The various problems which arose in this period were handled by both men in a way that reinforced the role of their family in the state.
In 12 CE, there was some sort of ceremony that implied recognition of Tiberius as "co-princeps". This is described by Suetonius.
Then, according to Wikipedia, after Augustus' death in 14 CE:
The Senate convened on 18 September, to validate Tiberius's position as Princeps and, as it had done with Augustus before, extend the powers of the position to him. These proceedings are fully accounted by Tacitus. Tiberius already had the administrative and political powers of the Princeps, all he lacked were the titlesβAugustus, Pater Patriae, and the Civic Crown (a crown made from laurel and oak, in honor of Augustus having saved the lives of Roman citizens).
Upvote:0
I don't recall which historian it was, but I believe it was Cassius Dio, who said that Tiberius shared authority with Augustus to carry out the census. As I recall, Tiberius had all the same authority as Augustus himself, minus some of the titles. But it was only during Augustus' final year, and only for the purpose of the census. He wasn't co-emperor or anything like that. He was granted authority to a specific purpose.