I thought I heard once that it was tied to English archers. That when they were captured by the enemy their middle fingers were removed to make them unable to use a bow. Is this true?
This one has classical origins, actually. It comes from ancient Greece, and was a very literal gesture used to convey offence; the idea was that the extended middle finger was to represent the shaft of the penis, and the rest of the hand, the testicles. Translated into current language, it was more or less a non-verbal version of "Fuck you!". We have several mentions of it in Greek writing; in The Clouds, a comedy play of the period, none other than Socrates is the target of the finger.[1]
The gesture carried on through Rome as well, and was still regarded as a rather obscene gesture; Suetonius tells us that emperor Augustus Caesar expelled an actor from not only Rome, but the whole of Italy, for the offence of giving the finger to an audience member that booed him.[2] [Edit: Worthy of note is that this was an unusually harsh punishment for this offence; Suetonius' language when discussing Augustus' handling of the event conveys his own surprise at such a firm exile, and this occurence is mentioned in a series of examples that outline the emperor's unusually strict moral standards for behavior in theatrical productions and other performances of the time. I don't want to mislead people into thinking that getting punted out of Italy was "normal" for giving the finger, it certainly was not.]
The gesture was not quite the same as we currently treat it, however. Today, most people point the finger upward, with their fist balled up beneath it; in antiquity, the gesture seems to have been more vague. In the above Suetonius example, the Latin is demonstrasset digito conspicuumque fecisset; an inelegant, modern translation of this would be something like "he had pointed out [the audience member] with the finger" (digito conspicuum certainly refers to the middle finger, and is analogous to digitus impudicus; the impudent finger, if you like). This seems to imply that in some contexts, at least, the gesture consisted of pointing the middle finger directly at the target of offense, rather than raising it vertically.
At any rate, the gesture is certainly much, much older than the battle of Agincourt; there is a running folklore that as you say, captured archers in the 1415 battle had their fingers removed to prevent archery. The story varies; sometimes people say that it was the middle removed, and sometimes both middle and index, but the invariable claim following is that uncaptured archers then gestured back at the French with either the middle finger or the middle and index finger in an upturned V shape (the latter being an offensive gesture today in the UK), to taunt the French with the fact that they can still shoot. This story seems to have been a purely apocryphal invention, probably born of British patriotism more than anything else. Nevertheless, you'll still hear it a lot -- I myself was taught this in secondary school!
Finally, there is the question of how the middle finger came to be an internationally recognized gesture. This is getting into modern history, which I don't know very much about, but Desmond Morris, an anthropologist who writes about the history of gestures (including the finger and the above examples), commented in a BBC interview that the gesture probably migrated to the United States through Italian immigrants, where it presumably blossomed to international fame via mass media exposure.[3] For more reading on that part of the origin story, I would suggest his book, which goes into much greater detail on the topic.[4]
Edit: Reference list formatting.