More than once, New York Mob member "Lefty" (Al Pacino), brags to Johnny Depp (playing an undercover FBI agent) all he has done (hits) for the Mob:
"So when they open the books, they will find my name".
Who specifically in the MOB is "they"? Are these physical books? Are they metaphorical books? The Book of Mormon?
Having an actual book with Mob members written down would seem problematic given the line of work they are in. But Al Pacino mentions it several times, and he seems very proud of it.
At the time the movie was supposed to take place, the NYC "Mafia" operated under a commission. Individual families would propose associates who were good members to be "made", or become official members of the family. Being "made" came with benefits, as "made" members couldn't be randomly murdered; they needed approval and there needed to be good reason. Once you were "made", you were "on the books". You were known.
The commission would meet at set intervals. At that time, they would "open the books" and consider people proposed to be "made", and make a decision on each person proposed.
"So when they open the books, they will find my name".
It's a metaphor. There are no books, no official records of Lefty's killings.
He's just a low-level henchman, never promoted, the kind of guy you just need when you have some dirty work to do.
When he says "they" he means caporegime or underboss. The Italian mafia was organized like the Roman Empire army. Small units. Very specialized people. You would do the job until you get promoted because someone sees your good job and skills. Only then you could be promoted, and get more money and recognition.
Doing the jobs he was assigned, Lefty has always been craving for recognition. He expects his bosses to remember his good work ("read the account book and see the numbers").
Trivial, but, when killing people, you often put a case to an end, so to say, "you close the book".
