PPVW boundedness conjecture for ranks over a fixed number field

Establish that the Mordell–Weil ranks of elliptic curves over any fixed number field K are uniformly bounded across all elliptic curves defined over K, as predicted by the Park–Poonen–Voight–Wood conjectures.

Background

The authors reference the conjectural framework of Park–Poonen–Voight–Wood, which predicts uniform boundedness of ranks over any fixed number field. This sweeping statement is foundational to current heuristics on the distribution of ranks.

While the present paper proves the existence of infinitely many rank-one curves over any number field, confirming the PPVW boundedness conjecture would give a structural ceiling on ranks, complementing existence with global control.

References

This belief was made rigorous in the striking conjectures of Park--Poonen--Voight--Wood . In particular, their work predicts that elliptic curves over a given number field should have uniformly bounded ranks.

— Elliptic curves of rank one over number fields (2505.16910 - Koymans et al., 22 May 2025) in Section 1.1 (Introduction: Conjectural boundedness of ranks)