When was Adi-Sankarāchārya born? Specifically, why is there so much of a variation in the dates given by Kānci, Dwāraka, Joshī Mutts and Śringerī Mutts?
Let's see whether he lived in 1st century BC(44 BC to 12 BC):
Sringeri mutt established by Shankara, records Shankara was born in 14th year from the era of Vikramadhitya. It is concluded that Vikram era started in 57 BC, so Adi Shankara born somewhere 13-14 years after 57 BC. But this mutt is in the south, Vikramaditya of Ujjain never ruled southern India, even if he was a historical figure. So, Karnataka was ruled by 2 Vikramadhityas of Chalukhyas. One of them lived in 7th century AD and another king ruled in 8th century AD. So,there is a huge possibility that Sringeri mutt might have recorded Chalukhya dynasty Vikramadhityas.
It is the only mutt claiming that Shankara born in 509 BC and died in 477 BC. But it was established after 1820 AD. It has nothing to do with ADI Shankara. SO, the claims of this mutt are irrelevant. This mutt is lying in the name of god.
This is the most important point. Hsuan Tsang, a Chinese traveler who visited India in the 7th century AD, said Dharmakirti was his contemporary. As he was a traveler, his antiquity cannot be shifted before 7th century AD. He also said Dharmakirti contributed a lot for Buddhism and praised him for this. He also mentions Bhartrhari, but not Shankara. As Adi Shankara defeated Buddhism in India, Hsuan Tsang definitely mentioned Shankara if he lived before him. This itself gives the clue that Shankara is post Dharmakirti and post Hsuan Tsang.
Adi Shankara is debated with Mandana Mishra which is confirmed by Shankara Digvijaya. This record is maintained by Sringeri mutt. We can fully trust Sringeri mutt, as it is the only one which has not experienced any destruction from Islamic rulers like other mutts. Mandana Mishra's historicity has been fixed as 8th century AD. In this debate, Adi Shankara mentioned Dharmakirti's work on Vijnanavada. As Dharmakirti' date has been fixed as 7th century AD by foreign accounts, Adi Shankara lived after 7th century AD.
So, the conclusion is, Adi Shankara lived somewhere between 750 to 850 CE.
This is from wikipedia- "509–477 BCE: This dating, is based on records of the heads of the Shankara's cardinal institutions Maṭhas at Dvaraka Pitha, the Govardhana matha and Badri and the Kanchi Peetham. This conforms to the chronology calculated based off the Hindu Puranas."
The timeline of 500 bc is not only based on kanchi mutt but the North indian mutts established by shankaracharya too, four places and purans wouldn't lie about the same thing simultaneously after all, the leaders of the mutts have said that historians can calculate the number of past mutt leaders of the four mutts and the 1st mutt leader would have timeline at some time of 500-400 bc. Which was the time they were established.
You may refer to the hyper link provided in my answer to another similar question.
The hyperlink gives you a document of the controversy behind Acharya's birth year and why Sankara Vijayam's give a more reliable data. The rest of the answer attempts to plot his horoscope based on shakara vijayams and validate his life events against the horoscope so as to show that sankara Vijayams are more reliable based on the data they provided.
The time of adi Shankara is generally given as 788 to 820 AD, As per Archaeological Survey of India first Hindu temple came up in Badami in 6th century AD, had no idle inside. First temple with Idle came up first in 7th century AD in Nagarjuna. Adi Shankara did is analysis of reason for decline of Hinduism sitting on to of a hillock in Srinagar, set up all the four Mats and his grand plan of revival of Hinduism in Badrinath. If you go there, you can read the division of responsibility is spelled out for all the mathadicaries,. Kedarnath temple is suspected to be Buddhist Monistry. There is enough evidence to conclude he died there. THe rest of the speculation id left to the Knowledgeable experts and historians, defiantly not religious experts.