Given is a verb and the past participle of give. It can also be used as an adjective meaning: a. Specified; fixed: We will meet at a given time and location. b. Granted as a supposition; acknowledged or assumed: Given the condition of the engine, it is a wonder that it even starts. 2. Having a tendency; inclined: My neighbor is given to lavish spending. 3. Bestowed as a gift; presented. 4. Law Issued on a specified date. Used of legal documents. As a noun given means something assumed or taken for granted. There is no such expression as 'so given' unless you say: So given the circumstances of the murder, who do you think was the killer?

Google shows no answer for it being an adverb. I guess it stems from something like 'so given that...', so, 'so' simlpy means 'so', lol. And 'given' is actaully a preposition here which means 'considering that... ', 'in view of...'.