We do have an account of Him physically writing! It is what He was writing that is in question! I could easily believe He was writing the four beautiful sentences that he spoke. I have heard it preached that more than likely the same men accusing her, had been in sin with her. Maybe Jesus was writing the Ten Commandments to remind them of the law

...
Or maybe....Jesus
---loves--
The World
Since Jesus and the Father are one, in essence He did write. lol There you go making me think! My preacher would envy you! Maybe no book was authored by Jesus own hand, (understand I believe it is all inspired by Him,) because He was/is the Word living among us.
This they said, tempting him, that they might have to accuse him. But Jesus stooped down, and with his finger wrote on the ground, as thought he heard them not.
So when they continued asking him, he lifted up himself, and said unto them, He that is without sin among you, let him first cast a stone at her.
And again he stooped down, wrote on the ground.
And they which heard it, being convicted by their own conscience, went out one by one, beginning at the eldest, even unto the last: and Jesus was left alone, and the woman standing in the midst.
When Jesus had lifted up himself, and saw none but the woman, he said unto her, Woman, where are those thine accusers? hath no man condemned thee?
She said, No man, Lord. And Jesus said unto her, Neither do I condemn thee: go, and sin no more. John 8: 6-11