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Silverman’s minimalist conjecture for elliptic fibrations

Prove Silverman’s conjecture that for an elliptic fibration E over the projective line P^1 defined over a number field, 100% of fibers have Mordell–Weil rank equal to either the generic rank rk(E) or rk(E) + 1, with the rank rk(E) + 1 occurring when the specialized root number forces opposite parity.

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Background

The authors situate their work within the minimalist philosophy, which predicts that ranks in natural families tend to be as small as parity allows. Silverman’s conjecture formalizes this by asserting that almost all fibers of an elliptic fibration realize the minimal rank compatible with the root number.

Their results address existence questions within twist families but the broader distributional assertion in Silverman’s conjecture remains a core open problem in arithmetic statistics.

References

A precise instance of the minimalist philosophy can be found in Silverman's conjecture asserting that $100\%$ of the fibers in an elliptic fibration $\mathcal{E}$ on $\mathbb{P}1$ should have rank equal to either the generic rank $\mathrm{rk} \, \mathcal{E}$ or $\mathrm{rk} \, \mathcal{E} + 1$.

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