An elliptic surface with infinitely many fibers for which the rank does not jump

TL;DR

This paper constructs an elliptic surface with infinitely many fibers where the rank remains constant, providing the first example of such a phenomenon and exploring its implications for rank distribution in families of elliptic curves.

Contribution

It presents the first known example of an elliptic surface with infinitely many fibers maintaining the same rank, advancing understanding of rank behavior in elliptic curve families.

Findings

Infinite fibers with constant rank demonstrated

Application of Green's theorem on prime progressions

Explicit examples with controlled bad primes

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.