In his article on the Nativity in Luke, Richard Carrier argues that "It is beyond reasonable dispute that Luke dates the birth of Jesus to 6 A.D" and claims that he has surveyed "all the evidence both to this effect and against all known attempts to reconcile these authors." Of course, most historians do dispute such a late dating and believe that Luke places Jesus' birth prior to 3 BCE.
Although Carrier does cover a lot of ground, I do not believe he has given adequate attention to the announcement narrative itself or Mary's encounter with Elizabeth the mother of John the Baptist. From those two accounts, it seems clear that Luke places Mary's conception very shortly after the announcement that she will conceive (which took place while Elizabeth was pregnant with John).
The appearance of Gabriel to Mary announcing her pregnancy, as well as Mary's visit to her cousin (pregnant with John the Baptist) and finally the birth of John the Baptist, are all clearly placed "In the days of Herod, King of Judaea." It seems that Luke places Jesus' birth in this same time. It is very unlikely that Luke intends to give the impression that Jesus was not conceived at this time, or even close to this time. The theory Luke places Jesus' birth in 6 CE means that the announcements of Mary's pregnancy -- as well as interaction with the pregnancy and birth of John the Baptist -- occurred at least 10 years before Jesus' conception. In fact, this would require that John the Baptist was 12 when Jesus was born and that poor Joseph had to wait 10-12 years before "consummating" his marriage with Mary.
There are other features about Luke's first chapter that indicate that the birth followed shortly after the announcement. First, Luke uses Mary's response to the announcement to indicate that conception will take place shortly:
And behold, you will conceive in your womb and bear a son, and you shall name Him Jesus. "He will be great and will be called the Son of the Most High; and the Lord God will give Him the throne of His father David; and He will reign over the house of Jacob forever, and His kingdom will have no end." Mary said to the angel, "How can this be, since I am a virgin?" The angel answered and said to her, "The Holy Spirit will come upon you, and the power of the Most High will overshadow you; and for that reason the holy Child shall be called the Son of God.
Luke 1:31-35 (emphasis added). The bold portion shows that Mary certainly understands the angel to mean that the conception to occur very soon. If it was to take place at some indeterminate point in the future, as much as 10 years later, she would have assumed that this would occur after her marriage to Joseph. Because she was already betrothed to Joseph (Luke 1:27), this places the conception very soon after the announcement. See Darrell L. Bock, Jesus According to Scripture, page 59 ("The note about betrothal means that the marriage probably would be consummated within a year."). As Bock comments:
Mary obviously takes the announcement not in terms of what the future marriage will bring but in terms of something that will happen in the near future, for she notes that currently she has no husband.
Bock, op. cit., page 60.
Next, Elizabeth's response to Mary after John leapt in her womb indicates that Mary was already pregnant.
When Elizabeth heard Mary's greeting, the baby leaped in her womb; and Elizabeth was filled with the Holy Spirit. And she cried out with a loud voice and said, "Blessed are you among women, and blessed is the fruit of your womb! "And how has it happened to me, that the mother of my Lord would come to me?
Luke 1:41-43. As the bold portion shows, Elizabeth already refers to Mary as the "mother of my Lord." This makes much more sense after conception than before. Note also that Elizabeth says of Mary, "And blessed is she who believed that there would be a fulfillment of what had been spoken to her by the Lord." Luke 1:45. Note the use of the past tense. Elizabeth does not say that Mary expects fulfillment or has faith that there will be a fulfillment, the affair is placed in the past tense.
Carrier raises other points which deserve attention, such as the dating of John's and Jesus' ministry, but the above discussion gives reason to doubt his understanding of Luke.