Muharram is also a religious month in Islam just like three others. There is a fast in this month called the fast of Ashura. There are many references in hadith I will mention some here.

Fasting in Muharram pre existed before Islam. After prophet Muhammad (PBUH) migrated to Madinah, He found that Jews were fasting on the same day. They gave the reason for that Sahih Albukahri 3397:

This is a great day on which Allah saved Moses and drowned the folk of Pharaoh. Moses observed the fast on this day, as a sign of gratitude to Allah.

That is why the prophet said to have two fasts on Muharram, a day before Ashura and the day of Ashura.

Fast on the day before and the day after Ashura.

In another Hadith of Sahih Muslim 1163, the Prophet Muhammad said:

The best of fasts after Ramadan is in the Month of Allah, which you call Muharram.

The day of Ashura is of great importance. There is another hadith of Sahih Muslims 1162 that explains that Allah forgives all sins of the previous year to have a fast on Ashura.

I hope from Allah that fasting on the Day of Arafah expiates for the sins of the year before it and the year after it.

It is important to know that fasting on Ashura is not compulsory but it is Sunnah. Reference from Sahih Al Bukhari:

Fast if you wish and those who do not, are not obliged to do so.