Hadrian's Statue on The Temple Mount was The Abomination of Desolation

The one referred to by Jesus which was a repeat of Antiochus Epiphanes actions in Daniel 11.

First I want to state that Luke 21 (which isn't an Olivet Discourse, it was given in in The Temple Complex) verses 20-24 is about 70 AD, but the "Times of The Gentiles" we are still in.

Likewise in Matthew 24:21 the "Great Tribulation" is a period that began in Ancient times and which we are still in now, which is the same as the Tribulation referred to in Matthew 24:29 and Mark 13:24.   In Revelation 7 ALL of the Martyrs are said to have come out of Great Tribulation, not just those of a specific brief time period. 

70 AD Preterists obsess over an argument that a Biblical Generation is 40 years because the wandering in the wilderness was to kill off a generation.  But not all of them actually died, that statement was hyperbole, it was mostly just about the 10 spies who gave the bad report.  Numbers 14:33-24 clarifies it was 40 years because the spy mission was 40 days.  Genesis 6 and the lifespan of Moses support making a Biblical generation up to 120 years.

 Quadratus of Athens in his apology to Hadrian written for Hadrian's visit to Athens in 124 or 125 AD says that some of those healed and risen from the dead by Jesus were still alive at that time.  Today it is verified as being possible to live to 122.  Pliny using documents related to a Roman Census of 74 AD says in one region of Italy there were many people who were over 100, 4 were 130 and some up to 140.  So I have no doubt that in Judea some people born BC lived through the Bar Kochba Revolt and that some people who were healed by Jesus and then witnessed Him Risen made it even into the reign of Antonius Pius.

With Luke 21 it's unique characteristics are what makes it most applicable to 70 AD.  Only Luke 21 actually uses the name of Jerusalem at all, when foretelling it's desolation which is language borrowed from Jeremiah about the fall to Nebuchadnezzar indicating what happened to Jerusalem then will happen again.

But Luke 21 does NOT contain a statement that this time of trouble is will never be surpassed.

The Bark Kochba revolt did not add anything to the destruction of Jerusalem since this time the Rebels never even had Jerusalem to begin with.  But for Judea as a whole that war was far more catastrophic and destructive then the 66-73 AD war and over a shorter period of time.   Many historians consider this the real beginning of the Diaspora.  It is only the fact that it doesn't have it's own Josephus that makes it less analyzed by historians and scholars and less romanticized by artists and poets.

Luke 21 is about things that happened before the "beginnings of sorrows", Matthew and Mark about things that happen during or after.  Meanwhile the second time Matthew and Mark's discourses bring up the issue of False Christs has no parallel in Luke at all, likewise no False Prophets in Luke 21.

This is significant because contrary to popular opinion the era leading up to and during the 66-73 AD war was NOT filled with would be Messiahs.  Josephus only ever uses the word Christ when describing what Jesus was called. There were would be prophets, and secular revolutionaries, but no claimed Messiahs.  Jewish prophetic expectations of the time were generally that the Messiah can't come till after Rome has already fallen.

Bar Kochba was the first to ever claim to be the Messiah as a rebel leader, that was his innovation.  And he really was the second person after Jesus to ever truly claim that title at all.  Meanwhile since Preterists don't take literally the stuff involving the Sun, Moon and Stars, maybe Stars falling from heaven is also wordplay on the name of Bar Kochba?  Kukbe is the word used in the Peshita?

The Abomination of Desolation is a very specific phrase, that has connotations more specific then just the etymological meanings of the words used to construct it.  Of the two places where the phrase appears in Daniel the one in chapter 12 is probably what Jesus is revealing to still have at least one more yet future fulfillment.  But it's the context in Daniel 11 that defines it.

There are three or four different Hebrew words that get translated "Abomination" in the KJV, the one used in Daniel is not even related to the one used in Leviticus 18-20 and Ezekiel 40-48.  But more importantly to the topic at hand, the precise word used in Daniel is everywhere it appears a synonym for an Idol or False god, from Deuteronomy 29:17 to 1 Kings 11 to Jeremiah 32:34.

But what makes the Abomination of Desolation special is it's being placed inside The Temple (not near it) by a Pagan ruler who had outlawed their faith.  The history of the Hasmonean revolt was to first century Jews not just the reason behind Hanukkah, it was to them as the Revolutionary War or French Revolution is for modern America and France.  When Jesus used this phrase he knew exactly what imagery he was evoking and so did His audience.

Now I'm open to a more "creative" interpretation of what a Historicist (or Futurist) fulfillment of this may look like, but that's about redefining what this would mean for the New Testament Church with the help of II Thessalonians 2 just as we redefine a number of Hebrew Bible concepts under the doctrine that now we are The Temple.  If you're going to insist this is about the Judea of that time, then you have to be specific to what that idea meant to those Judeans.

70 AD Preterists bend over backwards coming up with every excuse they can to apply that phrase to something that happened in 70 AD.  They take a passage from the Talmud claiming Titus had sex with a whore on a Torah scroll and sliced open the veil with his sword.  Leaving aside how I doubt Titus would have had the means, motive or opportunity to do that from what the actual eyewitness Historian tells us, even this Talmud passage doesn't call that an Abomination of Desolation or compare it to Antiochus Epiphanes in any way.

The timing is also wrong, by the time Titus was able to do anything anywhere near The Temple it was already too late to run.  Jesus spoke of the Abomination of Desolation as an event that begins the time of trouble not occurring at the middle or end of it. That fit Hadrian who's said to have set up the initial Idol in 31 AD sparking the Rebellion even though the full Temple to Jupiter is built after.

An alternate 70 AD Preterist take on the Abomination of Desolation makes it about things Josephus described Zealot leaders doing in 68 or 69, but again the specific word for Abomination in this phrase is about Idolatry, not mere ceremonial desecration or uncleanness.

Preterists aren't the only ones refusing to distinguish between the Olivet Discourses, there are also Futurists who want to use Luke 21 to say Jerusalem will be surrounded by armies again.

Yes the three discourses are "parallel" in a lot of ways, but the differences are there for a reason and ignoring them because you don't want to think Jesus was foretelling more then one thing is simply not respecting the text.  In the case of Luke it has to do with how this isn't even the only place that Gospel records Jesus talking about the fall of Jerusalem, that is a theme of the entire Gospel in a way it's not in the others.

So plenty of people want to argue that Luke 21:20 is about the same thing as The Abomination of Desolation because Jesus then advises basically the same reaction.  As if there can't be more then one good reason to get out of Dodge.

Remember the OG Abomination of Desolation preceded that Jewish revolt, but since they won that war the city was never surrounded by armies.

One of the oldest examples of Patristic support for viewing the Abomination of Desolation as already re-fulfilled is Jerome applying the term to the Statue of Hadrian set up where The Temple formally stood which was seemingly still standing when he wrote his commentary on Matthew.  Jerome may have been off on saying it was specifically over the Holy of Holies, in the Bordeaux Pilgrim the two Statues he saw were separate from the "stone" the Jews anointed which I think may have been where the Ark once rested.  Epiphanes' statue was on the Brazen Altar according to 1 Maccabees 1:54-59.

We even have a secular pagan gentile source on this happening, Cassius Dio.

[69.12.1] At Jerusalem, Hadrian founded a city in place of the one which had been razed to the ground, naming it Aelia Capitolina, and on the site of the temple of the [Jewish] god, he raised a new temple to Jupiter. This brought on a war of no slight importance nor of brief duration, 

[69.12.2] for the Jews deemed it intolerable that foreign races should be settled in their city and foreign religious rites planted there. So long, indeed, as Hadrian was close by in Egypt and again in Syria, they remained quiet, save in so far as they purposedly made of poor quality such weapons as they were called upon to furnish, in order that the Romans might reject them and they themselves might thus have the use of them. But when Hadrian went farther away, they openly revolted.

Meanwhile somewhat less reliable sources like the Historia Augusta say Hadrian also banned Circumcision and sacrificed Pigs to this Idol making it echo Antiochus Epiphanes even more.  And like then this caused the war rather then being caused by it.  It seem Pigs were depicted on Coins minted in Aelia Capitolina.

And like in 70 AD the Jewish Christians of Jerusalem did as Jesus advised and fled, becoming the Nazarenes of later generations, some may have went to Mesopotamia and also became among the ancestors of the "Nestorians" or other Syraic Rite sects.