Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 237

element of , then it is usually di cult to show that C a,b,c does not have
rational points.

Suppose we have an elliptic curve on which we want to find rational points.
If we do a 2-descent, then we encounter curves C a,b,c . Ifwesearchforpoints
on a curve C a,b,c and also try congruence conditions, both with no success,
then perhaps ( a, b, c ) represents a nontrivial element of 2 .Orwemight
need to search longer for points. It is dicult to decide which is the case.
Fortunately for Fermat, the curves on which he did 2-descents had trivial
2 .
The possible nontriviality of the group 2 meansthatwedonothavea
general procedure for finding the rank of the group E ( Q ). The group S 2 can
be computed exactly and allows us to obtain an upper bound for the rank.
But we do not know how much of S 2 is the image of φ and how much consists
of triples ( a, b, c ) representing elements of a possibly nontrivial 2 .Since
the generators of E ( Q ) can sometimes have very large height, it is sometimes
quite di cult to find points representing elements of the image of φ . Without
this information, we don't know that the triple is actually in the image.

The Shafarevich-Tate group is often called the Tate-Shafarevich group
in English and the Shafarevich-Tate group in Russian. Since comes after
T in the Cyrillic alphabet, these names for the group, in each language, are
the reverse of the standard practice in mathematics, which is to put names
in alphabetical order. The symbol was given to the group by Cassels (see

[23, p. 109]).

REMARK 8.27

The Hasse-Minkowski theorem (see [104]) states that a

quadratic form

n n

Q ( x 1 ,...,x n )= a ij x i x j

i =1 j =1

with a ij ∈ Q represents 0 nontrivially over Q (that is, Q ( x 1 ,...,x n )=0for
some (0 ,..., 0) =( x 1 ,...,x n ) ∈ Q n ) if and only if it represents 0 nontrivially
in Q p for all p ≤∞ . This is an example of a local-global principle .
For a general algebraic variety over Q (for example, an algebraic curve), we
can ask whether the local-global principle holds. Namely, if the variety has a
p -adic point for all p , does it have a rational point? Since it is fairly easy

to determine when a variety has p -adic points, and most varieties fail to have
p -adic points for at most a finite set of p , this would make it easy to decide
when a variety has rational points. However, the local-global principle fails in
many cases. In Section 8.8, we'll give an example of a curve, one that arises in
a descent on an elliptic curve, for which the local-global principle fails.

≤∞