Praise be to Allah.
Do you have to recite a surah after al-Fatihah?
Reciting Quran after al-Fatihah in prayer is not obligatory, either in an obligatory prayer or a supererogatory prayer, whether the prayer is one in which Quran is recited out loud or one in which it is recited silently, whether one is joining the prayer late or otherwise.
‘Ata said: Abu Hurayrah said: In every prayer there is recitation. What we heard the Prophet (peace and blessings of Allah be upon him) say, we tell you, and what he did not tell us about we cannot tell you. Whoever recites Umm al-Kitab (i.e., Surat al-Fatihah) has done enough, and whoever does more than that, that is better. (Narrated by al-Bukhari, 738; Muslim, 396)
“The phrase, “Whoever recites Umm al-Kitab (i.e., Surat al-Fatihah) has done enough, and whoever does more than that, that is better” indicates that reciting al-Fatihah is obligatory , and that nothing else will do.
It also indicates that reciting a surah after al-Fatihah is mustahabb. There is scholarly consensus on this point with regard to Fajr, Jumu`ah and the first two rak`ahs of every prayer, and this is Sunnah according to all scholars. Al-Qadi ‘Iyad (may Allah have mercy on him) narrated that some of the companions of Malik said that reciting a second surah is obligatory, but this is an odd view which is to be rejected.
Do you have to recite a surah in the third and fourth rak`ahs?
With regard to reciting an additional surah in the third and fourth rak`ahs, there was a difference of opinion among the scholars as to whether it is mustahabb or not. Malik (may Allah have mercy on him) regarded it as makruh. Al-Shafi`i (may Allah be pleased with him) regarded it as mustahabb in his new opinion but not in his old opinion; the old opinion in this case is more correct.” (Sharh Muslim, 4/105, 106)
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said:
“According to the majority of scholars, reciting another surah (after al-Fatihah) is Sunnah and is not obligatory, because it is obligatory to recite only al-Fatihah.” (Al-Sharh al-Mumti’, 3/103)
If you stand up to complete your prayer after the imam has said the taslim, then the correct scholarly view is that the rak`ahs in which you caught up with the imam form the first part of your prayer.
If what remains of your prayer is the third and fourth rak`ahs, then you should recite al-Fatihah only; if what remains is the second and subsequent rak`ahs, then in the second you should recite al-Fatihah plus another surah, and in the remaining rak`ahs you should recite al-Fatihah only.
It is permissible for a worshipper to recite another surah after al-Fatihah in the third and fourth rak`ahs, but he should do that only sometimes; it was narrated that the Prophet (peace and blessings of Allah be upon him) did this according to a hadith narrated by Muslim (452).
And Allah knows best.