are ashkenazi jews really jews though?
the y-dna says that, yes, they are. the mt-dna suggests some admixture with indo-european types, although that's also true of mizrahi jews, due to the massive influence of the persian empire on hebrew ethnogenesis.
the science on this advanced rapidly about 15 years ago or so, when the technology finally allowed for it, and it's not really an open question. but, it's never really been clear how so many jews got so far north, and i'll admit having looked into the khazar thing...
there really does appear to have been a jewish empire parked roughly around the ukraine in the dark ages, and the story as to how they became jewish has not come down to us in any form other than myth. the story is that the khazar king converted to judaism to act as a buffer between the recently converted christian russians (they didn't actually convert for a long time after that) and the more seriously muslim caliphate to the south of the caucasus. i can't prove that didn't happen, but nobody really takes the legend all that seriously. so, nobody really knew for sure until they did the testing...
the thing is that eastern european jews don't actually look turkish, either. the turks are a central asian people that really have strongly eastern phenotypes. so, while the jewish empire was there, and for several centuries, it was actually observationally obvious that all these eastern european jews couldn't really be all that turkish, either.
they must have just migrated north sometime around the year 700. that would have been about the same time that the caliphate came in, and introduced coercive rules to stimulate conversion.
i have a similar hypothesis with the kurds, who show up out of nowhere at the same time, and the curiously close genetic relationship we've uncovered between kurds and jews is maybe kind of illuminating in that respect. both populations seem to have become migrant at about the same time, and apparently as refugees of the muslim takeover, after the conclusion of the lengthy roman-persian wars that ended the classical period.
but, yes - they are genetically jewish, even if they really don't look like it. and, it's just that curve tending to the limit, over time.
i used the example because it was the most obvious i could find...