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LOW RANK SPECIALISATIONS OF ELLIPTIC SURFACES
Published online by Cambridge University Press: 13 January 2025
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.
MSC classification
Primary:
14J27: Elliptic surfaces
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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