来る is most often read as
くる. However, it can also be read as
きたる mostly an archaic reading? Does it carry a different, or any additional meanings?? When are "appropriate" (socially, grammatically, etc.) times to use it?
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Basically, it is literary (and archaic), but there are some situations when it is still appropriate to use it in a modern context.
Outside of these usages, it has a literary flavor, sort of like "is come" or "cometh" (rather than "has come") in modern English.
Incidentally, I learned that although it looks like 来る + past/perfect aux. たる, it is actually a separate word derived from 来る + 至る; I just checked the 広辞苑 and 日本国語大辞典, and they agree with this (although obviously, sometimes the word きたる will in fact be 来る + たる). Special bonus trivia: the 来る + 至る /kitaru/ got its big break in the world of kanbun 漢文, as a pinch hitter for 来(る). (And it is true that the character 来 is generally rendered /kitaru/ in kanbun.) I can't find a reliable source for why kanbun kundoku doesn't just use /ku(ru)/, but some (e.g. fontomanie at Chiebukuro) suggest that it was because /ku(ru)/ didn't have enough syllables for convenient okurigana affixation (cf /nasu/ instead of /su(ru)/ for 為).
So, the specific literary flavor is that of kanbun or heavily Chinese-influenced Japanese, not "pure" classical Japanese.