Plaintiffs' attorneys made the filing Thursday night, while Arpaio's attorneys filed their brief a few hours earlier.
Arpaio is accused of ordering some patrols not based on reports of crime but rather on letters from people who complained about people with dark skin congregating in an area or speaking Spanish.
The sheriff's lawyers deny the allegation and said Arpaio placed no value on the letters.
Deputies are accused of pulling over some vehicles only to make immigration checks. Arpaio denies the allegation.
The judge hasn't revealed when he'll deliver his verdict.