1. Write a short letter. List any pertinant times, dates, and events in bullet points. If you want to go the extra mile, sign it & get it notarized (make sure to keep a copy). I called the police over a dozen times (my apartment neighbors caused serious noise problems, and I'm a pretty chill guy) & made sure I had my paperwork ready to present for backup.
2. Talk to management in person.
I dealt with this a few years ago. I complained endlessly to management (letters, calling, going down in person). They were unable to evict or do anything else about the noise-maker due to strict local laws...everyone else had complained about them too, but I was right above them & their speakers would literally make my bed vibrate at 3am in the morning. It got to the point where I was calling the cops on them at least twice a week. The guy even had a minor warrant out for his arrest, but it wasn't the "break down your door" kind of warrant, so if he didn't open, they couldn't go in. It got to the point where I was so sick from lack of sleep that I was unable to go to work, so I went down & sat in the management office for three days straight until they resolved it for me - they ended up giving me a new apartment down the hall. It was next to the laundry room & elevator, which was crap too, but it was better than what I had before. In short, I hate apartments & they can DIAF.