The Khazars where a people who built an empire centred North of the Caucasus Range, between the 7th and 11th centuries. They are not at all mainstream (at least in Europe), even though, defeating the Arabs, they prevented their expansion in present day Southern Russia.
They were of Turkic origin and (apparently) converted in mass to Judaism.
The meaning of my question is broad: I'm interested in genetic as well as cultural heritage, which includes religion - but please do abstain from discussing the legitimacy of modern day Israel here. I believe there's plenty of space elsewhere on the internet for such an interesting but different topic. Thank you. If you do not understand the above disclaimer, ignore it.
EDIT: A new study has been published by Oxford UP that revives the Ashkenazim hypotesis. I do not have the means to judge it, but must be serious if it's been published here