Application: finding rank and rational points.
Fact
If P ∈ F(Q(T)), we can compute the trace P efficiently (Thanks to
Nicolas Mascot).
Let
E : Y 2
= C(X) = X3
+ a2X2
+ a1X + a0
an elliptic curve over Q.
Search A, B ∈ Q[X] st F : Y 2
= A(X)T2
+ B(X)T + C(X) has
rank r ≥ 1 and compute a point P.
Take T = 0. Generalization of brute force search (take A = 0 and
B = b1X + b0).
Variant: if we know (x0, y0) ∈ E(Q), search A, B ∈ Q[X] on the form
A = (a1X + a0)2
B = (X − x0)(b1X + b0) + 2(a1x0 + a0)y0
st F : Y 2
= A(X)T2
+ B(X)T + C(X) has rank ≥ 2.