The BSD Rank Conjecture Implies that is Computable
Proposition 1.3
Let be an elliptic curve over .
If Conjecture 1.1 is true, then
there is an algorithm to compute the rank of .
Next we show that given the rank , the
full group is computable. The issue is that
what we did above might have only computed a subgroup
of finite index. The argument below follows
[Cre97, §3.5] closely.
The naive height of a point
is
The Néron-Tate
canonical height of is
Note that if has finite order then .
Also, a standard result is
that the height pairing
defines a nondegenerate real-valued quadratic form on
with discrete image.
Proposition 1.5
Let be an elliptic curve over .
If Conjecture 1.1 is true, then
there is an algorithm to compute .
William
2007-05-25