On the infinitude of elliptic curves over a number field with prescribed small rank

TL;DR

The paper proves that over any number field $K$ and for each $0\le r\le 4$, there exist infinitely many elliptic curves over $K$, up to $\overline{K}$-isomorphism, of rank $r$. The strategy uses a carefully chosen nonisotrivial elliptic curve $E$ over the function field $K(T)$ with rank $r$ and specializes it at infinitely many $t\in K$ to obtain $E_t/K$ with rank exactly $r$, validated by a $2$-descent analysis of the associated isogeny pair $(\phi,E,E')$ and controlled by Kai's number-field generalization of Green–Tao–Ziegler to fix the bad primes. The argument combines geometric Mordell–Weil computations, a sieve-driven selection of primes, explicit choices of $E/K(T)$ realizing ranks $0$ through $4$, and Selmer-group computations to pin down the rank in each specialization. Consequently, the paper delivers the first uniform infinitude results for small ranks over arbitrary number fields and furnishes explicit families achieving each rank up to $4$, with the $j$-invariants varying to produce infinitely many nonisomorphic curves. The methods have potential implications for understanding rank distribution and the construction of high-rank families via function-field techniques and local-global control.

Abstract

For any number field $K$ and integer $0\leq r \leq 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the function field $K(T)$. We use a result of Kai, which generalizes work of Green, Tao and Ziegler to number fields, to choose our specializations so that we have control over the bad primes and can perform a $2$-descent to compute ranks.

Theorems & Definitions (38)

• Theorem 1.1
• Corollary 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 4.1
• Lemma 5.1