On p. 123 of Algebraic Geometry: A Problem Solving Approach, it says that
Many areas of mathematics are concerned with the computation of the order of various points on a cubic curve.
What are some examples of this?
I am not familiar with any examples from my previous studies. The text refers obliquely to applications to number theory at times, which might explain this discrepancy -- I know next to nothing of number theory. Still, it seems from the above that there should also be applications outside of number theory, whereas I am not aware of any yet.