oistos Abarios ὀιστός Ἀβάριδος [?] [Gk];
["abaris" means "without weight"]

Flying arrow upon which Abaris the Hyperborean (Ἄβαρις Υπερβορέος) ^*1 [circum]navigated the globe while fasting all the while, according to alleged stories that Herodotus^*2 has heard being told.

A summary of the alleged deeds of the arrow is found in Brewer's Dictionary of Phrase & Fable article on : Ab’aris:

"The dart of Abaris. Abaris, the Scythian, was a priest of Apollo; and the god gave him a golden arrow on which to ride through the air. This dart rendered him invisible; it also cured diseases, and gave oracles. Abaris gave it to Pythagoras".

The above definition nearly checks out fact-for-fact (golden arrow, diseases), except I have not been able to track down the source or passage about invisibility. And although Abaris did apparently gave oracles or predictions, I'm not certain this was done through the faculty of the arrow.
According to Iamblichus, he talked to the arrow while performing purifications, and driving out pestilence and wind. Thus "calming storms" was another feat that the arrow can be credited with.

Certain astrological treatises (the Catasterism) comment that Abaris' arrow was identified ⇒Apollo's arrow, that is to say, the arrow which the god Apollo used to kill off the Cyclops in vengeance for the death of his son Aesclepius. In earlier tradition, it was merely an emblem (badge) carried by the priest, but the story later developed into that of an arrow of great size which Abaris rides "like a witch's broom".

§ Iamblichus (ca. 250-ca. 330), De vita Pythagorica

Iamblichus's biography of Pythagorus, De vita Pythagorica^*1 (Vita Pyth. ch. 19; sect.90-94) states that Abaris the Scythian^*2, a priest of Apollo, brought his collected contributionsto Hellas (Greece), and in transit he met Pythagoras (in real life they were not contemporaries).

Abaris thought he recognized the god Apollo in Pythagoras (who later on confirms his divinity by exhibiting his golden thigh ^*3 to Abaris).

In obeisance, Abaris presents his magic arrow:

Πυθαγόρᾳ ἀπέδωκεν ὀιστόν, ὃν ἔχων ἀπό τοῦ ἱεροῦ ἐξῆλθε, χπήσιμον αὐτῷ ἐσόμενον πρὸς τὰ συμπίπτοντα δυσμήχανα κατὰ τὴν τοσαύτην ἄλην. Ἐποχούμενος γὰρ αὐτῷ καὶ τὰ ἄβατα διέβαινεν, οἷον ποταμοὑς καὶ λίμνας καὶ τέλματα καὶ ὄρη καὶ τὰ τοιαῦτα, καὶ προσλαλῶν, ὡς λόγος, καθαρμούς τε ἐπετέλει καὶ λοιμοὺς ἀπεδίωκε καὶ ἀνέμους ἀπὸ τῶν είς τοῦτο ἀξιουσῶν πόλεων βοηθὸν αὐτὸν γενέσθαι. [92] Λακεδίμονα γοῦν παρειλήφαμεν μετὰ τὸν ὑπ᾽ ἐκείνου γενόμενον αὐτῇ καθαρμὸν μηκέτι λοιμῶξαι, πολλάκις πρότερον τούτῳ τῷ παθήματι περιπεσοῦσαν διά τὴν δυστραπελίαν τοῦ τόπου, καθ᾿ ὃυ ᾤκισται, τῶν Ταϋγέτων ὀρῶν πνῖγος ἀξιόλογν αὐτῇ παρεχόντων διά τὸ ὑπερκεῖσθαι, καὶ Κρήτης Κνωσσόν. καὶ ἄλλα τοιαῦτα τεκμήρια ἱστορεῖται τῆς τοῦ ᾿Αβάριδος δυνάμεωσ. —Vita Pyth. ch. 19, §91-92

.. he gave to Pythagoras an arrow which he had when he left his temple, and which would be useful to him in the many difficulties encountered on a very long journey. For riding on it he crossed impassible places; for example, rivers, lakes, swamps, mountains, and the like. And talking to the arrow, so goes the story, he performed purifications and drove off plagues and winds from the cities which asked for his assitance. At any rate, we have ascertained that Lacedaemon after being purified by him never suffered from plague again; formerly it was often afflicted by this misfortune because of the unhealthiness of the area in which it was situated (since the Taygetus range, towering over the city, provides a notable degree of stifling heat); and also he purifed Knossos in Crete. And other such signs of Abaris' power are on record. —tr. John Dillon and Jackson Hershbell pp.114-7

Lacedaemon, which Abaris saves from plague, is here described as a city over which the "Taygetus range" towers over, but it is equivalent to Laconia, the part of Peloponnesos where Sparta was located.

It is in a later chapter where Iamblichus reiterates the story that the arrow is definitively elucidated as being "golden arrow" (ὀιστὸν χρυσῆς [Gk., acc. case])^*4.

§ epitome of the pseudo-Eratosthenes Katasterismoi (Catasterism) (1st/2nd century AD)

Astronomical treatises identifies Apollo's arrow with Abaris' arrow.

Eratosthenes's Katasterismoi ^*1, "Chapter XXIX: On the Arrow" [i.e., on the constellation Sagitta] says^*2:

    This constellation is the arrow of Apollo, with which he is said to have killed the Kyklopes who fashioned the thunderbolt for Zeus, on account of Asklepios. Apollo hid the arrow among the Hyperboreans, where the feather temple is also. He is said to have returned later, when Zeus freed him from service to Admetos, about which Euripedes speaks in the Alkestis. It is believed that the arrow was then borne through the air with fruit-bearing Demeter. (p.147)

Aesclepius (cf. ⇒«Asclepius' staff»), physician extraordinaire, had such medical skills tha he was capable of reviving the dead, and as a consequence of this infringement on godly powers, he was slain by Zeus.
    Apollo was enraged, being the father of Aesclepius, and though he could not bring scathe to the god himself, exacted vengeance by killing off the Cyclops (or at least some of them) who were the manufacturers of Zeus' thunderbolts. This did not go unpunished and Apollo was sentenced to labor under Admetos for some duration, as mentioned by the quote above.

The Katasterismoi epitome goes on to identify Apollo's arrow with that of Abaris:

    According to most ancient sources, Abaris traveled about Greece carrying Apollo's arrow as a badge. Heraklides^*3, however, said that Abaris traveled about by means of Apollo's arrow, using it, apprently, like a witch's broom-stick. Heraklides is cited by the Katasterismoi only in support of the great size of the arrow. (p.152)

And as to the arrow bearing "fruitful Demeter" (or "fruits of the season"), this is explained in terms of the Hyperborean custom of wrapping their offerings in sheafs of wheat straw and sending them on to Delos by having it passed on to neighboring Scythia and so on (Herodotus 4.36)

§ Hyginus Poeticon astronomicon

Hyginus's Poeticon Astronomicon ^*1 also presents a quote out of Eratosthenes, yet the details differ somewhat: Apollo buried the arrow in a Hyperborean mountain (whereas the epitome is unclear, although the incidental mention of the "feather temple" made of beeswax and feathers may mean that was where it was hid). After Zeus forgave him, the arrow came of its own accord bearing a fruits of the time(season).]

§ Abaris and the Palladium of Troy

There is an story that Abaris sold to the Trojans the sacred Palladium, an idol of Pallas Athena. He purported that the idol was made from the bones of Pelops^*1.
Pelops was a man once cut up and boiled by his father Tantalus to be served up to the gods. The gods restored him to whole however.

The Trojan Palladium rendered the city impregnable while it was in its possession, and so Odysseus and Diomedes carried it away (Aeneid ii, 164 etc.).
In another account, Aeneas brought the Trojan Palladim when he escaped to Italy.

The story that Abaris fashioned this Palladium is told by Firmicus Maternus ^*2:

"Abaris sold this god-image to the Trojans, making deceitful promises to the the gullible men. A God was sold in order be of use to the buyer, and the buyer prayed submissively to something he saw at an auction a little earlier. Moreover, [he alleged that] the stuff of this god-image was made from the bones of Pelops.."

—Firmicus Maternus, De errore profanarum religionum, XV

There is also a lore reported by Pausanias that in Lacedaemonia, the aforementioned land Abaris once rid of pestilence, one finds opposite the temple to Olympian Aphrodite, and opposite this is a temple dedicated to the "Savior Maid" κόρη Σωτείρας and that according to some Abaris was responsible for the erection of a temple.^*3.