An elliptic surface with infinitely many fibers for which the rank does not jump
David Zywina
TL;DR
The paper addresses whether a nonisotrivial elliptic curve $E/\mathbb{Q}(T)$ can have infinitely many specializations $E_t/\mathbb{Q}$ with the same minimal Mordell–Weil rank $r$ as $E/\mathbb{Q}(T)$. It constructs an explicit rank-$0$ example and uses a detailed $2$-descent, together with local invariants and Tamagawa numbers, to bound the rank of specializations. A Green–Tao type result on 3-term arithmetic progressions of primes is leveraged to ensure infinitely many admissible specializations with only a controlled set of bad primes, for which the descent can be carried out. Consequently, the authors produce the first unconditional infinite family of $t$ with $E_t(\mathbb{Q})$ of rank $r$, supporting the conjecture that $\mathcal{N}(E)$ is infinite for such $E$. The work also indicates the potential for multiple infinite families with higher $r$ in follow-up studies.
Abstract
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.