sentence construction - What about you?

• Amy: What day is it today?


• Tom: It's Monday. Hmm... I like Monday.


• Amy: What do you have today?


• Tom: We have math, science and art.What about you?



• Amy: We have English and P.E. What do you have tomorrow?


• Tom: We have art and P.E.tomorrow.


• Amy: Oh, great.

I can understand that "What about you" means "What do you have today?" But I wonder whether "What about you" is an elliptical sentence. If it is, what is the full sentence? If it isn't, how to parse this structure?