Do you have any source?
I did not not find any reference to Genoa in dfferent dictionaries (in English and Italian).
https://www.etymonline.com/search?q=lottery
https://www.merriam-webster.com/dictionary/lottery
https://www.treccani.it/vocabolario/lotteria/?search=lotter%...
https://www.treccani.it/vocabolario/lotto/?search=l%C3%B2tto...