On wikipedia's Battle of Cannae there is no mention of elephants, I've heard stories that the Romans fled in fear of Hannibal's elephants but I have no idea who made these claims, were the elephant bones or tusks ever on display anywhere in Rome? Was anything made of their ivory? Has anyone found an elephant corpse or fossilized elephant excrement anywhere on the Alps or Spain and France? If the Alpine ices can preserve Otzi from 3,300 BCE they must have preserved some part of the elephants if they were actually brought to Europe for his invasion
edit: Thank you all so much for these elegant and informed replies, confirming my view that this is by far the best and most professional subreddit. Learned a lot about the variety of elephants brought to and in use in the Mediterranean. Unfortunately it would seem there is no physical evidence of Hannibal's elephants that archaeologists have found, but we can hope for the best as responders have brought up writers like Polybius who discussed the elephants in question.
Hannibal did bring elephants to Italy, but they suffered greatly during the crossing of the Alps and afterwards. The elephants were not used to cold and snow, some of the paths were too narrow, and it was difficult to keep them fed.
The elephants which survived the crossing were used by Hannibal at the battle of the Trebia; Polybius reports they were placed in front of his wings to protect his flanks. (3.72.8) Shortly after the battle though all but one of the elephants perished due to their bad condition and the cold weather. Reportedly (again by Polybius, 3.79.11) this last elephant carried Hannibal through the marshes on the way to Lake Trasimene but died some time before the battle of Cannae, so there weren't any elephants present there.
I've never actually looked into evidence for the presence of elephants other than Roman reports. But I've never heard anyone doubt those reports either. We know the Carthaginians had access to elephants and used them in battle, so the reports that Hannibal brought elephants are very plausible.
I know very little of the archaeological work done in the Alps, besides the fact that it really doesn't show anything. However, the reason that elephants are not mentioned on the article on the Battle of Cannae is that by Cannae all of Hannibal's remaining elephants had died, as both Livy and Polybius point out. And there's no reason to suppose that Hannibal didn't cross with elephants, since its mentioned quite a lot in Roman literature and is one of the great motifs
Otzi was found pretty high up on a continuously ice-covered slope, not in a mountain pass. Mountain passes do not remain permanently frozen. At any rate. Indeed, we can't find much evidence that humans of ancient times were up there (no bones found right in the main passes that I know of).
Elephants are attested to as being in Rome on several occasions, and were imported into Rome by the Romans themselves, so even if we found ivory in Rome, we couldn't be sure the source was Hannibal.
So it's going to have to be historical research (documents) and the veracity of the writer upon which we depend for this answer. A researcher at the U of Chicago has put together an interesting page about elephants in Rome:
http://penelope.uchicago.edu/~grout/encyclopaedia_romana/gladiators/elephantus.html
There's absolutely nothing to indicate that the elephants didn't come with Hannibal, of course.