what political and religious influence did the byzantine empire have on eastern europe?

I think mostly religious in the form of Eastern Orthodoxy, This ancient form of Christianity is dominate in eastern Europe certainly in Russia and Ukraine.

A HUGE effect. Far too much to simplify in this medium. I would suggest that You type "Byzantine Empire" into the area at the very top of the page where it says "search", press "web search", and take it from there. Happy hunting!