There is disagreement on this among the schools of thought.
According to the Hanafis it is makruh tahrimi for people following an Imam in congregational prayer to recite the Quran during the prayer. This includes Surah al-Fatiha and this includes both loud prayers (Fajr, Maghrib, Isha) and Silent prayers (Zuhr, Asr). However the prayer will be valid even if he violated this rule. Their evidence is the verse:
وإذا قرئ القرآن فاستمعوا له وأنصتوا لعلكم ترحمون
When the Qur’ān is recited, listen to it and be silent, so that you may be blessed.
— Quran 7:204
As it is recorded in its tafsir that this verse was revealed when some of the Sahaba recited the Quran during congregational prayers behind the Prophet ﷺ.
This stance is also supported by the ahadith:
إنما الإمام ليؤتم به فإذا كبر فكبروا وإذا قرأ فأنصتوا
The Imam is appointed to be followed, so when he says the takbir, say the takbir, and when he recites, be silent.
— Nasai , Muslim
من كان له إمام فإن قراءة الإمام له قراءة
Whoever has an Imam, the recitation of the Imam is his recitation
— Ibn Majah
According to the Malikis and Hanbalis, it is not obligatory on them to recite the Quran. Their evidence are the ahadith mentioned above. They also state that it is mustahab (recommended) for them to recite Surah al-Fatiha in silent prayers. Some of them have stated that it is obligatory in silent prayers.
According to the Shafi'is it is obligatory on them to recite Surah al-Fatiha - in both silent and loud prayers. And this can be done during the time when the Imam pauses if it is a loud prayer. This is based on interpreting the following hadith in general terms:
لا صلاة لمن لم يقرأ بفاتحة الكتاب
Whoever does not recite Al-Fatiha in his prayer, his prayer is invalid.
— Bukhari