many rational points on an algebraic curve

Question

Given polynomial $f(x,y)$ with integer coefficients, may be reducible, but without linear factors. For positive integer $n$ denote by $a_n$ the number of points $(x,y)\in \frac1n \mathbb{Z}^2$ on a curve $f(x,y)=0$. May it appear that $a_n$ tends to infinity (when $n$ increases taking all positive integer values), but is always finite? Similar question: if we consider only bounded part of our curve, say $\{ (x,y):f(x,y)=0,x^2+y^2 < R^2 \}$, and define $b_n$ as the cardinality of the intersection of this set with lattice $\frac1n \mathbb{Z}^2$, may $b_n$ tend to infinity?

Answer 1

I'll assume the curve is irreducible, otherwise look the components.

If your $a_n$'s are unbounded then your curve has infinitely many rational points, so has genus zero or one. In the case of genus one (or genus zero with more than two points at infinity), the $a_n$ will always be finite and, by finiteness of $S$-integral points, the $a_n$ are bounded if e.g. $n$ ranges through integers with a fixed set of prime factors. So $a_n$ does not go to infinity with $n$. But, even in this case, the sequence $a_n$ can be unbounded. E.g. take an elliptic curve with rank one, such as $y^2=x^3+3$.

Maybe you should look at the case of the unit circle for both of your problems. That's not covered by what I wrote above and probably you can figure out precisely what is happening in this case.