Do you really need all those backticks? I doubt that "Abstract_CoAuthors" is a reserved word in MySQL....
I see that you're using Co_Authors twice; as the error message suggests, you should use aliases so that MySQL can tell which use is which.
SELECT Abstract_CoAuthors.Abstract_ID , Abstract_CoAuthors.Co_Author_ID , co_author1.Last_Name
FROM Abstract_CoAuthors , Co_Authors as co_author1
INNER JOIN Abstract_Information ON Abstract_Information.Abstract_ID = Abstract_CoAuthors.Abstract_ID
INNER JOIN Co_Authors as co_author2 ON co_author2.Co_Author_ID = Abstract_CoAuthors.Co_Author_ID
LIMIT 0 , 30
Unless there's a more sensible choice of names than "co_author1" and "co_author2".
That would fix the alias problem - whether it gives you the result you want is another matter.
As you've already found, to get just the columns you want, refer to them by name.