Minimalist conjecture on rank distribution of elliptic curves over number fields

Determine whether, when elliptic curves over a fixed number field K are ordered by height, the density of curves with Mordell–Weil rank 0 equals 1/2 and the density of curves with rank 1 equals 1/2, as predicted by the minimalist conjecture consistent with the parity conjecture.

Background

The paper proves that for any number field K there exist infinitely many elliptic curves of rank 1, and more strongly, it establishes rank stability across quadratic extensions L/K for infinitely many curves. These results address existence but not the global distribution of ranks among elliptic curves over K when ordered by height.

The minimalist conjecture is a widely discussed prediction in the arithmetic of elliptic curves suggesting that, subject to the parity conjecture, rank 0 and rank 1 should each occur with density 1/2. The authors reference this conjecture to contextualize their existence results within broader expectations about rank distribution, highlighting that their theorems do not resolve this conjectural density statement.