LOW RANK SPECIALISATIONS OF ELLIPTIC SURFACES

Abstract

Listen

Translate article

Abstract Let $E/\mathbb {Q}(T)$ 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 $r_t$ of the specialisation $E_t/\mathbb {Q}$ is at least r for all but finitely many $t \in \mathbb {Q}$ . Moreover, it is conjectured that $r_t \leq r+2$ , except for a set of density $0$ . When $E/\mathbb {Q}(T)$ has a torsion point of order $2$ , under an assumption on the discriminant of a Weierstrass equation for $E/\mathbb {Q}(T)$ , we produce an upper bound for $r_t$ that is valid for infinitely many t. We also present two examples of nonisotrivial elliptic curves $E/\mathbb {Q}(T)$ such that $r_t \leq r+1$ for infinitely many t.