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

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.

💡 Research Summary

The paper proves that for any number field K and any integer r with 0 ≤ r ≤ 4, there exist infinitely many elliptic curves over K whose Mordell–Weil rank is exactly r. The construction proceeds by choosing a non‑isotrivial elliptic curve E defined over the rational function field K(T) that already has rank r. The curve is taken in a Weierstrass form
 y² = x³ + a(T)x² + b(T)x,
with (0,0) a rational 2‑torsion point. This point yields a degree‑2 isogeny φ:E→E′ and its dual φ̂:E′→E, where E′ has coefficients a′ = −2a and b′ = a² − 4b.

Specializing the parameter T to a value t∈K (avoiding the finite set B of roots of the discriminant Δ(T)) gives an elliptic curve E_t/K. By Silverman’s specialization theorem, for all but finitely many t the specialization map E(K(T))→E_t(K) is injective, so rank E_t(K) ≥ r. To force equality, the authors perform a 2‑descent via the isogeny φ. They define homomorphisms
 δ_E : E(K)→K×/(K×)², δ_{E′} : E′(K)→K×/(K×)²,
whose kernels are the images of the dual isogenies. Lemma 3.1 shows that
 rank E(K) = dim_F₂ δ_E(E(K)) + dim_F₂ δ_{E′}(E′(K)) − 2.
Thus the rank can be read off from the sizes of the φ‑Selmer group Sel^φ(E/K) and the dual Selmer group Sel^{φ̂}(E′/K).

Controlling these Selmer groups requires precise knowledge of the primes of bad reduction of E_t. The paper uses a recent theorem of Kai (2025), a number‑field analogue of the Green–Tao–Ziegler theorem on linear equations in primes. Kai’s result guarantees the existence of elements m, n ∈ 𝒪_{K,S} such that the ratios t = m/n have the property that each prime divisor of m − e n (with e∈B) is a distinct prime ideal of 𝒪_{K,S}. By carefully choosing local open sets U_v at each place v∈S∪S_∞, the authors can prescribe the reduction type of E_t at every prime in a finite set S, ensuring that the local conditions defining the Selmer groups are satisfied.

The goal is to make the natural maps
 E(K(T)) → E_t(K) → Sel^{φ̂}(E′_t/K) (2.1)
 E′(K(T)) → E′_t(K) → Sel^φ(E_t/K) (2.2)
surjective. When both are surjective, the dimensions of the Selmer groups equal the known rank r, and Lemma 3.1 forces rank E_t(K)=r. If one map fails to be surjective, the authors show that the 2‑torsion part of the Tate–Shafarevich group of either E_t or E′_t vanishes, which still yields the desired rank.

In Section 6 the authors exhibit five explicit families E/K(T), one for each r∈{0,1,2,3,4}. For each family the discriminant Δ(T) splits completely into linear factors, so B consists of rational points. The conductor N of the associated elliptic surface satisfies deg N − 4 = r, giving the “maximal rank” condition that simplifies the analysis. The most intricate case is r=4, where
 y² = x³ − 70(T² − 25²)x² + 2⁴·7²(T² − 11²)(T² − 25²)x,
has discriminant
 Δ = 2¹⁴·3²·7⁶ · (T ± 11)² · (T ± 25)³ · (T ± 39).
Using Kai’s theorem the authors produce infinitely many t∈K for which the local conditions force both (2.1) and (2.2) to be surjective, and a direct 2‑descent calculation shows that the Selmer groups have dimensions 4, yielding rank 4 for infinitely many specializations. The other families are treated analogously, establishing the infinitude of curves of rank 0, 1, 2, and 3 as well.

Thus the paper extends previously known results over ℚ to arbitrary number fields, showing that the combination of a carefully chosen high‑rank non‑isotrivial family, Kai’s linear‑forms‑in‑primes theorem, and explicit 2‑descent provides a robust method for producing infinitely many elliptic curves of any prescribed small rank r≤4 over any number field. This represents a significant advance in the understanding of rank distribution in families of elliptic curves over global fields.