Let $E$ be a nonisotrivial elliptic curve over $\mathbb{ Q}(T)$ and denote the rank of the abelian group $E(\mathbb{ Q}(T))$ by $r$. For all but finitely many $t\in \mathbb{ Q}$, specialization will give an elliptic curve $E_t$ over $\mathbb{ Q}$ for which the abelian group $E_t(\mathbb{ Q})$ has rank at least $r$. Conjecturally, the set of $t\in\mathbb{ Q}$ for which $E_t(\mathbb{ Q})$ has rank exactly $r$ has positive density. We produce the first known example for which $E_t(\mathbb{ Q})$ has rank $r$ for infinitely many $t\in\mathbb{ Q}$. For our particular $E/\mathbb{ Q}(T)$ which has rank $0$, we will make use of a theorem of Green on $3$-term arithmetic progressions in the primes to produce $t\in\mathbb{ Q}$ for which $E_t$ has only a few bad primes that we understand well enough to perform a $2$-descent.
Categories: math.NT