2.3. Integral points 23
is an invertible rational map, defined over Q, that sends (0,1) to
O, and ψ(0, −1) = (0,0). See Exercise 2.12.4. More generally, any
quartic
C : V
2
= aU
4
+ bU
3
+ cU
2
+ dU +
q2
for some a, b, c, d, q Z is isomorphic over Q to a curve of the form
E :
y2
+ a1xy + a3y =
x3
+
a2x2
+ a4x + a6, also defined over Q. The
isomorphism is given in [Was08], Theorem 2.17, p. 37.
Let E be an elliptic curve over Q given by a Weierstrass equation
E :
y2
+ a1xy + a3y =
x3
+
a2x2
+ a4x + a6, ai Q.
With a change of variables (x, y)
(u−2x, u−3y),
we can find the
equation of an elliptic curve isomorphic to E given by
y2
+ (a1u)xy +
(a3u3)y
=
x3
+
(a2u2)x2
+
(a4u4)x
+
(a6u6)
with coefficients aiui Z for i = 1,2,3,4,6. By the way, this is one
of the reasons for the peculiar numbering of the coefficients ai.
Example 2.2.6. Let E be given by
y2
=
x3
+
x
2
+
5
3
. We may
change variables by x =
X
62
and y =
Y
63
to obtain a new equation
Y
2
=
X3
+ 648X + 77760 with integral coefficients.
2.3. Integral points
In 1929, Siegel proved the following result about integral points E(Z),
i.e., about those points on E with integer coordinates:
Theorem 2.3.1 (Siegel’s theorem; [Sil86], Ch. IX, Thm. 3.1). Let
E/Q be an elliptic curve given by
y2
=
x3
+ Ax + B, with A, B Z.
Then E has only a finite number of integral points.
Siegel’s theorem is a consequence of a well-known theorem of Roth
on Diophantine approximation. Unfortunately, Siegel’s theorem is not
effective and provides neither a method to find the integral points on
E nor a bound on the number of integral points. However, in [Bak90],
Alan Baker found an alternative proof that provides an explicit upper
bound on the size of the coefficients of an integral solution. More
concretely, if x, y Z satisfy
y2
=
x3
+ Ax + B, then
max(|x|, |y|)
exp((106
· max(|A|,
|B|))106
).
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