Summary: Points on Curves
There is surprising depth, but also surprisingly accessible questions, when investigating integer and rational points on simple nonlinear curves.
- We start with rational points on conics. Fact 15.1.2 gives a famous parametrization of the points on the unit circle, though we also see in Fact 15.1.8 that some conics don’t have any rational points at all.
- In Section 15.2 we explore a few more fun, though less crucial, cubic questions.
- The next section begins our exploration of integer points, including facts such as Fact 15.3.5 about some curves with none or one.
- Then in Section 15.4 the conic (quadratic) cases begin.
- We use hyperbolas to bring in the wonderful geometric Algorithm 15.5.1 for using existing points to get us more and more of them.
- Can this strategy be made algebraic? The final section does so, culminating in the most general proposition Fact 15.6.4 of this type we present.
The Exercises focus a lot on filling in proof details, as well as the excitement of exploring for actual integer points.