Most disagreements between religions cannot be settled by evidence, because they concern things no evidence could reach. This one is different, and that is what makes it worth spending time on.
Whether a particular man was executed by a particular method in a particular decade is a historical question. It is the kind of question historians answer routinely about far less well-documented figures.
What the Quran says
The relevant text is a single verse in Surah an-Nisa, which states that they did not kill him and did not crucify him, but that it was made to appear so to them — and that those who differ about it are in doubt, following conjecture.
Classical commentators generally elaborated this into a substitution account: someone else was made to resemble Jesus and was crucified in his place, while Jesus himself was taken up to God. The identity of the substitute varies between commentators, which is itself worth noticing.
What the historical record shows
Here the picture is unusually uniform, and it is important to be clear about who is saying it.
The crucifixion of Jesus under Pontius Pilate is accepted by effectively the entire field of critical historical scholarship, including scholars who are not Christians and do not believe in the resurrection, miracles, or anything supernatural whatsoever. It is not a Christian claim defended by Christians. It is a historical consensus held across the ideological range, and it is stronger than the consensus on most events of the period.
The reasons are worth setting out.
Non-Christian sources report it. The Roman historian Tacitus, writing in the Annals around 116 CE and with no sympathy for Christianity whatsoever, records that Christus was executed under Pilate during the reign of Tiberius. The Jewish historian Josephus refers to it as well; his passage on Jesus shows signs of later Christian interference, but the scholarly consensus is that a core notice underlies it, and it includes the execution.
The Christian sources are early. The creed Paul quotes in his first letter to the Corinthians — that Christ died, was buried, and was raised — is material he says he received rather than composed, and is dated by scholars across the spectrum to within a very few years of the events. That is an extraordinarily short interval by ancient standards.
Nobody would invent it. This is the argument historians tend to find most compelling. Crucifixion was a punishment of maximum humiliation, reserved for slaves and insurgents. A movement inventing a messiah would not invent one who had been executed in the most degrading manner the Roman state possessed. The claim was an obstacle to be explained, not an asset, and the earliest Christian writings show them labouring to explain it.
The difficulty for the substitution account
The substitution reading has to carry several loads at once.
It must account for the unanimous testimony of those present, both followers and enemies. It must account for the Roman execution detail, since the sources describe a procedure with soldiers, a formal charge, and a body. It must explain why a movement built on the claim spread from Jerusalem itself, where the events were recent and public and contradiction would have been easy.
And it appears roughly six centuries after the events, in a different language, from a different region, with no earlier attestation.
Historians do sometimes prefer a later source over earlier ones — but only with a specific reason to think the later source has access to better information. Here the argument would have to be theological rather than historical, which is a legitimate move but a different kind of move, and it should be named as such.
An honest note on the Muslim side of the argument
It would be unfair to present this as though Muslim scholarship had nothing to say.
Some modern Muslim scholars have argued that the verse does not deny the physical event at all — that its force is to deny that Jesus' enemies truly killed him, in the sense of defeating him or ending him, since God raised him. On this reading the verse is about who was really in control, not about what happened on the hill. It is a minority position and it is contested within Islam, but it exists, and it is held by people who are neither ignorant nor insincere.
There is also a straightforward theological argument underneath the mainstream reading, and it deserves respect: that God would not permit a prophet of his to suffer such a defeat and such a humiliation. That is a real intuition about what God is like, and it is worth engaging with rather than dismissing — not least because Christianity's answer is that this is precisely the point, that the humiliation was chosen rather than suffered helplessly.
Why this question decides so much
Because almost everything else follows from it.
If Jesus did not die, there is no resurrection to investigate, and Christianity is straightforwardly false. If he did die, the question of what happened next becomes unavoidable, and it is the question we take up in the evidence a skeptic can weigh.
It is also, by his own account, one of the two questions that Nabeel Qureshi could not get around — see his story. And it connects directly to the tahrif objection, since a reader who finds the historical case strong will next be told the documents are corrupted: that objection is answered here.
Common questions
Could the disciples have been mistaken about who was on the cross?
The sources describe a public execution over several hours in a known location, with followers, enemies, and Roman soldiers present. Mistaken identity is difficult to sustain under those conditions.
Is the swoon theory a serious option?
It is not held by any significant body of scholarship. Roman execution squads were professionally competent, and a man surviving scourging and crucifixion could not plausibly have presented himself days later as risen and victorious.
What did the earliest Christians think it meant?
That it was deliberate rather than accidental, and substitutionary — that he died in the place of others. That claim, not merely the fact of the death, is what the New Testament is centrally about.