What does the “pinnacle of the temple” refer to?

Matthew 4:5 (ESV)

5 Then the devil took him to the holy city and set him on the pinnacle of the temple

In short

There are two possible meanings for the pinnacle of the temple:

The Greek word pterugion refers to the edge of a home or of a roof, here meaning the roof over a side entrance to the temple, probably on the southern side, looking over a steep precipice.
The expression used is vague, but it probably refers to the roof of the arcade of Solomon, from which you could see into the deepest ravine beside the temple mount.

Interpretation 1:

The Greek word pterugion literally means the edge of a home or the edge of a roof. Here we must think in terms of the roof over one of the temple’s side entrances, most likely near the royal portico on the southern side, where there was a steep precipice.¹

Interpretation 2:

The vague expression precludes us from identifying the exact place on the roof of the temple or the battlements of the temple. The reference here is probably to the roof of the arcade of Solomon. From it you could look into the deepest ravine beside the temple mount.²