Rank one elliptic curves and rank stability

TL;DR

The paper proves that for any number field $K$ and any quadratic extension $L/K$, there exist infinitely many elliptic curves $E$ over $K$ with $\mathrm{rank}\,E(K)=\mathrm{rank}\,E(L)=1$, by specializing a nonisotrivial rank-one family and performing explicit $2$-descent to fix ranks. The core method combines a carefully constructed family of curves, precise local conditions at a finite set of primes, and Kai’s number-field generalization of Green–Tao–Ziegler to produce suitable parameters $a,b$; this yields exact rank control and a proof that rank stability under quadratic base change occurs abundantly. Consequently, the results generalize prior theorems of Koymans–Pagano and Alpöge–Bhargava–Ho–Shnidman and connect with Hilbert’s tenth problem via families that witness rank 1 for both the base field and quadratic extension. The work also supplies an explicit, infinitely-parameterized mechanism to generate rank-stable curves, with consequences for the arithmetic of elliptic curves over number fields and the distribution of ranks under base change.

Abstract

For any quadratic extension $L/K$ of number fields, we prove that there are infinitely many elliptic curves $E$ over $K$ so that the abelian groups $E(K)$ and $E(L)$ both have rank $1$. In particular, there are infinitely many elliptic curves of rank $1$ over any number field. This result generalizes theorems of Koymans-Pagano and Alpöge-Bhargava-Ho-Shnidman which were used to independently show that Hilbert's tenth problem over the ring of integers of any number field has a negative answer. Our approach differs since we are obtaining our elliptic curves by specializing a nonisotrivial rank $1$ family of elliptic curves and we compute all the ranks involved.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Koymans--Pagano KP*Theorem 2.4
• Theorem 1.4: Alpöge–Bhargava–Ho–Shnidman ABHS*Theorem 1.1
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3