So, who made Jesus arbiter and judge?

There are no doubt many ways of reading this fascinating little exchange in Luke 12 which leads into the Parable of the Rich Fool. What follows is no doubt a guide to Jesus'  thinking here. But consider the question and answer and at least 2 ways of reading it, which could be seen as antithetical or harmonizable.

³ Someone in the crowd said to him, “Teacher, tell my brother to divide the inheritance with me.”

¹⁴ Jesus replied, “Man, who appointed me a judge or an arbiter between you?” ¹⁵ Then he said to them, “Watch out! Be on your guard against all kinds of greed; life does not consist in an abundance of possessions.”

"Who appointed me as a judge or an arbiter between you?", Jesus asks.

Answer 1 is rightly or wrongly, "nobody". 

Jesus might be saying this belongs to a secular or civil realm.

Or I have no legal authority here.

Or this is not my concern. Or not my chief concern.

Or why can't you sort this out between you?

Or go to the courts or priests or whatever.

The section which follows certainly implies that Jesus cares more about the heart and eternal destiny of the questioner than about the precise financial and legal settlement required.

But Jesus often asks questions or makes statements which are ambiguous and permit a fuller or deeper meaning, or even which are meant to provoke a more profound response.

Answer 2 might be "God the Father". You are the Messiah, the New and Better Moses, the Son of God come in the flesh. The judge of all the world is before me, so, maybe, though I understand you have bigger fish to fry, you could please just quickly help us with this one which is a big deal to us and our families. You have all authority in heaven and on earth, Jesus, so you are Lord of family disputes and inheritance rights.

But as far as we know the man didn't say that.