Unbounded rank (or maximal rank) of elliptic curves over Q

Ascertain whether the supremum of Mordell–Weil ranks of elliptic curves over the rational field Q is infinite; if it is finite, determine the exact largest possible Mordell–Weil rank of an elliptic curve over Q.

Background

The paper opens by highlighting fundamental uncertainties about the size of Mordell–Weil ranks even over Q. Despite record constructions of high-rank curves, there is no general proof of unboundedness or a determination of a maximum rank. The authors reference recent progress on explicit large-rank examples but emphasize that the main qualitative question remains unsettled.

This problem is central to the arithmetic of elliptic curves and interacts with deep conjectures about rank distributions and the behavior of L-functions. It provides the context in which the authors’ result (guaranteeing infinitely many rank-one curves over any number field) sits as complementary to the global landscape of rank questions.

References

Even for $K := Q$, it is unknown whether one can find curves with arbitrarily large rank or, alternatively, what the largest possible rank should be.

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

Unfortunately, this is a hard problem in general (there are plenty of open problems about this topic; for example, we neither know if the set of possible algebraic ranks is bounded above or not).

— Introduction to the Birch and Swinnerton-Dyer Conjecture  (2608.25975 - Ruffino, 26 Aug 2026) in Section 1, Introduction