The historical study of Pure Land Buddhism turns on a small number of hard data and a large number of translation questions, and it is a good illustration of how early Mahayana history actually gets done.
The hardest datum is an inscription. A pedestal from Govindnagar, now in the Government Museum at Mathura, is dated by Gregory Schopen to the twenty-sixth year of the Great King Huveshka, which is 104 of the common era. It records the dedication of an image of the Blessed One, the Buddha Amitabha, and the donor's hope that all living things obtain the unexcelled knowledge of a buddha. This is the earliest known epigraphic mention of him, and it puts him in India in the early second century, considerably earlier than a casual account of Pure Land as an East Asian development would suggest.
The earliest sutra mention is in the Pratyutpanna Samadhi Sutra, translated into Chinese by the Kushan monk Lokaksema, and dated between the first century before the common era and the second after it. Compilation of the Larger Sukhavativyuha has been proposed for Kushan-era Gandhara, but that origin is disputed.
The translator problem. Five Chinese translations of the Larger Sutra survive. The one catalogued as T. 362 is traditionally attributed to Zhi Qian but is now thought likely to be the work of Lokaksema or his lineage; T. 361 is probably Zhi Qian's revision of it. T. 360, made between 359 and 429 by Buddhabhadra and Baoyun, became the standard version in China and is the one East Asian Pure Land actually reads.
This matters because doctrine follows the text a tradition uses. Jan Nattier's work, including her Guide to the Earliest Chinese Buddhist Translations of 2008, has revised a number of these attributions, and early Chinese versions show traces of having been translated from Gandhari rather than Sanskrit.