According to the Theravada tradition, the Abhidhamma was taught by the Buddha.
From "The Abhidhamma in Practice" by N.K.G. Mendis:
Theravaada tradition holds that the Buddha conceived the Abhidhamma in the fourth week after his enlightenment, while still sitting in the vicinity of the Bodhi tree. Tradition also has it that he first preached the Abhidhamma to the assembly of deities in the Taavatimsa heaven; his mother, reborn as a deity, was present in the assembly. This can be taken to mean that the Buddha, by intense concentration, transcended the earth-bound mentality and rose mentally to the world of the deities, a feat made possible by his attainment of higher powers (abhiññaa) through utmost perfection in mental concentration. Having preached the Abhidhamma to the deities, he returned to earth, that is, to normal human consciousness, and preached it to the venerable Saariputta, the arahant disciple most advanced in wisdom.
However, in this YouTube video, Ajahn Brahm stated that the Theravada Abhidhamma was not taught by the Buddha and that the complexity of the Abhidhamma is not needed. He said that the Buddha's original teachings in the Pali suttas is simpler and completely sufficient.
One of the evidences he cited is as follows - only the Suttas (Dhamma) and the Vinaya were recited at the First Buddhist Council and there was no mention of the Abhidhamma.
I quote from Harvey, P (2013), "An Introduction to Buddhism: Teachings, History and Practices" (2nd Edition), Cambridge University Press, p. 88:
Just after the Buddha’s passing away (c. 404 BCE), a 'communal recitation' (council) of 500 Arahats was held at Rājagaha (Skt Rājagraha, Vin.ii.284–7) to agree the contents of the Dhamma and Vinaya which the Buddha had left as ‘teacher’ (D.ii.154). Ānanda, the Buddha’s faithful attendant monk, recited the Suttas, such that each begins: ‘Thus have I heard’. The monk Upāli recited the Vinaya.
This wikipedia page on Theravada Abhidhamma states that according to scholars, the Theravada Abhidhamma was probably composed in 3rd century BCE, which is after the Buddha's lifetime. I also call it the Theravada Abhidhamma specifically, because there seems to be another Sarvastivada Abhidharma.
So, my question is: Was the Theravada Abhidhamma really taught by the Buddha? Or is that an exaggeration?