I can't help you with French language books, unfortunately, but I have a few suggestions for English books:

Ewart Oakeshott's The Archaeology of Weapons is a good introduction to medieval European weapons, especially the sword. Oakeshott is an expert on swords, and so his book focuses primarily on the development of that weapon, with only moderate attention paid to other weapons. His Sword in the Age of Chivalry is probably his most impressive work, although an actual sword expert may be able to provide more insight into that.

Matthew Strickland and Robert Hardy's The Great Warbow is the best work on the English longbow. It's a long book and can be difficult to find a copy of. Jim Bradbury's The Medieval Archer is a much shorter and easier to find substitute.

Kelly DeVries' Medieval Military Technology (which recently got a second edition) provides a really broad survey of pretty much every medieval weapon you could want. Individual weapons often only get a couple of pages, but it covers pretty much everything.

The crossbow is a tricky thing. If you speak German I can recommend some books, but in English most of the good books are out of print. Ralph Payne-Gallwey's The Crossbow is a good book, but it was written at the turn of the twentieth century so to say it's dated would be a radical understatement. Still, copies are easy to find and it works as an alright general history.

I'm not sure what the best book to learn about the axe is. My gut would guess that a viking book would probably have a good discussion of the weaopn (since it's famously associated with viking warriors) but I'm no expert so I couldn't say for certain.