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18 Introduction to Rational Points on Plane Curves

(6.4) Remark. The Mordell conjecture and Faltings’ proof of it are really for curves
of genus strictly greater than 1. The curve must be nonsingular but not necessarily a
plane curve.

Finally, a closely related subject is the study of integral points on rational curves.
In terms of an equation f (x, y) = 0 we can ask if there are finitely many (x, y)
with f (x, y) = 0and x, y rational integers. With the solution of the Mordell conjec-
ture this problem has less interest, but historically it was Siegel who established the
finiteness result.

(6.5) Theorem of Siegel. The number of integral points on a nonsingular curve of
genus strictly greater than 0 and defined over the rational numbers is finite.
In particular this applies to nonsingular cubic curves, but not to the singular cubic
3
2
3
2
y = x which has infinitely many integral points of the form (n , n ), where n is
any integer.
(6.6) Remark. For certain explicit elliptic curves there are bounds on the size of the
2
3
integral points. For exmaple, for y = x − k one has:
4 9 3
max(|x|, |y|) ≤ exp 2 7·2 k 10 ·2 .
3
2
(6.7) Example. The only integral solutions of y + k = x for k = 2occurwhen
y =±5 and for k = 4 occur when y =±2, ±11. This question goes back to
Diophantus and was taken up by Bachet in 1621. For the case k = 2 we will give an
√
argument based on properties of the ring Z[ −2]. We factor
√ √ 3
y + −2 y − −2 = x .

For the equation to hold mod 4, x and y must both be odd. If a prime p divides x, then
3 √ √
p divides (y + −2)(y − −2).If p divides both factors, then it would divide the
3
2
sum 2y, and this is impossible since y + 2 = x . Since this holds for each p, both
√ √ √ √ 3
factors y+ −2and y− −2 must be perfect cubes. Thus y+ −2 = (a+b −2) ,
from which we deduce that

3 2 2 2 2 2
y = a − 6ab = a a − 6b , 1 = b 3a − 2b .
The last equation gives b =+1and a =±1, and hence y =±(−5) as was asserted.
√
We have used that p has to be a prime of the unique factorization domain Z[ −2].
Finally we return to Remark (1.6) concerning the definition of a rational curve. A
rational curve in the sense of geometry is a curve of genus 0. This definition makes
sense over any field and has nothing to do with the rational numbers. The concept of
genus is also extended to singular curves where it is called the arithmetic genus. The
3
2
singular cubic y = x is a curve of arithmetic genus = 0.