In The Rape of the Lock, is there an object that is associated with Ariel the sylph?
There isn't really a consistent object that is associated with Ariel the sylph in this brilliant mock-epic as opposed to an action or an ability. As Ariel is a supernatural creature, a fairly-like individual who is able to fly and also able to control and command a group of sylphs in order to try and protect Belinda, he is able to do other various supernatural things. The most important of these is his ability to see omens and read the future in the stars. It is this activity that is associated with his character if we look at his role in this poem. For example, Canto I describes him trying to warn Belinda about the impending disaster that he is able to see in her future:
Of these am I, who thy Protection claim,
A watchful Sprite, and Ariel is my name.
Late, as I rang'd the crystal Wilds of Air,
In the clear Mirror of thy ruling Star
I saw, alas! some dread Event impend,
Ere to the Main this morning's Sun descend,
But Heav'n reveals not what, or how, or where...
In the same way, his character is concerned as Belinda sets off on the boat ride down the Thames at the "black omens" that loom on the horizon and the sense of impending doom. Therefore, Ariel is a character who in this poem is associated with the ability to gain a sense or a feel for what is going to happen, even though he is not able to see the precise future clearly, as the above quote makes clear.
This is of course part of the massive irony of this mock-epic, as the impending doom Ariel sees for Belinda, his human charge, is nothing worse than the "rape" of one of her locks. This is one way in which Pope mocks and satirises the customs of court life.