Motivations and Notations
• Family of elliptic curves: an elliptic curve over Q(T).
F : Y 2
= X3
+ α2(T)X2
+ α4(T)x + α6(T), with αi(T) ∈ Z[T],
non-isotrivial.
r ∈ N the rank Q(T).
• Specialization: for all but finitely many t ∈ Z, F(t) is an e. c. over Q:
r(t) the rank of F(t) over Q.
Goal: study F when deg αi(T) ≤ 2.