Low rank specializations of elliptic surfaces

Let $E\mathrm{/}\mathbb{Q}(T)$ be a non-isotrivial elliptic curve of rank $r$. A theorem due to Silverman implies that the rank $r_t$ of the specialization $E_t\mathrm{/}\mathbb{Q}$ is at least $r$ for all but finitely many $t\in \mathbb{Q}$. Moreover, it is conjectured that $r_t\le r+2$, except for a set of density $0$. In this article, when $E\mathrm{/}\mathbb{Q}(T)$ has a torsion point of order $2$, under an assumption on the discriminant of a Weierstrass equation for $E\mathrm{/}\mathbb{Q}(T)$, we produce an upper bound for $r_t$ that is valid for infinitely many $t$. We also present two examples of non-isotrivial elliptic curves $E\mathrm{/}\mathbb{Q}(T)$ such that $r_t\le r+1$ for infinitely many $t$.