Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
If you catch one rak`ah of the Prayer, either Jumu`ah or any other Prayer, then you have joined the congregation.
Responding to your question, Sheikh Hamed Al-Ali, instructor of Islamic Heritage at the Faculty of Education, Kuwait, and Imam of Dahiat As-Sabahiyya Mosque, states:
How to catch congregational prayer
As for Salatul-Jama`ah (congregational prayer), it is reached by joining the last ruku` (bowing). If one joins the jama`ah in the last ruku` that means one has caught up with the rak`ah and the jama`ah .
How to catch Friday prayer
As for Salatul-Jumu`ah (Friday Prayer), it is reached by joining one rak`ah, even by joining the Prayer while the imam is still performing the second ruku` (i.e. in the second rak`ah), then one has caught the Friday prayer. In this case, one is supposed to pray one more rak`ah and then finish his Prayer.
However, if one joins the Salah while the Imam has already risen from the second ruku`, then one has missed the Jumu`ah. In this case, he is supposed to assume the intention of praying Zhuhr and then join the Salah, then get up to pray four rak`ahs after the imam makes the tasleem and finishes the Salah.
In his book Fiqh-us-Sunnah, the late Azharite scholar Sheikh Sayyed Sabiq, may Allah bless his soul, adds,
Most scholars agree that if a person catches only one rak`ah of Salatul-Jumu`ah, then that rak`ah will be valid and the person needs only make up for the one rak`ah that he misses.
Ibn `Umar (may Allah be pleased with him) narrated that the Prophet (peace and blessings be upon him) says, “Whoever catches only one rak`ah of the Salah and then adds to it the other one, his prayer will be complete.” (An-Nasa’i, Ibn Majah, and Ad-Daraqutni)
Abu Hurairah (may Allah be pleased with him) quotes the Prophet (peace and blessings be upon him) as saying, “Whoever catches one rak`ah of the Friday prayer has indeed caught the whole prayer.” (Abu Dawud)
Whoever catches less than one rak`ah of the Salah has not caught the Jumu`ah and he is to pray four rak`ahs of the Zhuhr prayer according to the majority of scholars.
Ibn Mas`ud (may Allah be pleased with him) says, “Whoever catches one rak`ah from the Jumu`ah is only to add another one to it, but whoever misses both rak`ahs is to pray four rak`ahs.” (At-Tabarani)
Ibn `Umar (may Allah be pleased with him) says, “If one catches one rak`ah from the Jumu`ah, then he is to add another one to it. If he catches only the last sitting (at the end of the prayer, following the ruku`), then he is to pray four (rak`ahs).” (Al-Baihaqi)
Such is the opinion of the Shafi`i, Maliki and Hanbali Schools, and Muhammad Ibn Al-Hasan as well.
Abu Hanifah and Abu Yusuf say that if one catches the tashah-hud with the imam, then he has caught the Jumu`ah. He should pray two rak`ahs after the imam, makes the tasleem, and his Jumu`ah would be complete.
Thus, if you join the Jama`ah in the second ruku` in the Jumu`ah prayer, then your Jumu`ah is complete. But if you join the Jama`ah after the second ruku`, then it is not counted as Jumu`ah and you should intend to pray Zhuhr and add four rak`ahs.
Allah Almighty knows best.