Low rank specialisations of elliptic surfaces

Let E/ℚ(T) be a nonisotrivial elliptic curve of rank r. A theorem due to Silverman [ Heights and the specialization map for families of abelian varieties , J. reine angew. Math. 342 (1983), 197 211] implies that the rank rt of the specialisation Et/ℚ is at least r for all but finitely many t ∈ ℚ. Moreover, it is conjectured that rt ≤ r + 2, except for a set of density 0. When E/ℚ(T) has a torsion point of order 2, under an assumption on the discriminant of a Weierstrass equation for E/ℚ(T), we produce an upper bound for rt that is valid for infinitely many t.We also present two examples of nonisotrivial elliptic curves E/ℚ(T) such that rt ≤ r + 1 for infinitely many t.