They based their ideas on the prior works of Joachim of Fiore (c. 1135 – 1202), though rejecting the Church of their day more strongly than he had.
[4] Joachim believed that the Jews were the elect people of God during the Old Testament, he believed that during the "first seal" of the Old Testament the Jews endured oppression by the Egyptians, in the "second seal" they battled against the Canaanites and established their royal power and priesthood in Jerusalem.
[7] The coming of Jesus resulted in the replacement of God's chosen people, Joachim believed that the blindness of Tobit (Tob 2:7-10) represented the blindness of the Hebrews, he believed that the Hebrews were too "carnal" in observing the law and did not have the "inner sight" to see the "light of Jesus".
Elizabeth's pregnancy revealed that the Synagogue would give birth to Jesus, Zachary did not originally believe the angel Gabriel's proclamation that Elizabeth would have a child and became mute, which represents the duality of Jewish priesthood.
[7] Joachim drew connections between the rise of Islam and errors of the Greek church, he especially criticized the Orthodox rejection of filioque as a heresy.
The rise of the Franks and their union with the papacy was a fascinating thing for Joachim, first the event showed that the power of the Byzantines had been destroyed, even though it wasn't completely destroyed, it also secondly inaugurated the "fifth seal", which in his interpretation was an era of peace and stability for the Western Church, which paralleled the protection given to Hezekiah.
[7] Joachim believed that he saw that the Latin church had clear indications of entering into the days of lamentations predicted by Jeremiah.
Joachim saw that pope Leo IX aligned with King Josiah of Judah who reformed religious life in his kingdom, but was destroyed by Pharaoh, after the failure of Leo at the battle of Civitate Joachim declared that the Pope had erred by trusting in "material arms" rather than spiritual weapons.
Joachim also attacked the views of Peter Lombard concerning the Trinity, in his book "Psaltery of Ten Strings".
[1] Further, his admirers came to believe the beginning of this New Age would be ushered in by the coming of a virtuous Pope from the Franciscan order.