Let E be an elliptic curve over Q(t), let E(A^1) denote the elliptic curve over the function field Q(t), and let E(t) denote its specialization at t∈Q. Order rational numbers t by height. Silverman's conjecture. For almost all t∈Q, one has operatornamerank E(A^1)leqslant operatornamerank E(t)leqslant 1+operatornamerank E(A^1). This is presented as a natural analogue of Goldfeld's conjecture for quadratic twists. The supplied text does not state a resolution, so the conjecture is recorded as open.

Silverman's rank conjecture for elliptic curves in one-parameter families

About 12 years old · traced to 2014

Let $E$ be an elliptic curve over $\mathbb{Q}(t)$ , let $E(\mathbb{A}^1)$ denote the elliptic curve over the function field $\mathbb{Q}(t)$ , and let $E(t)$ denote its specialization at $t\in\mathbb{Q}$ . Order rational numbers $t$ by height. Silverman's conjecture. For almost all $t\in\mathbb{Q}$ , one has

\operatorname{rank} E(\mathbb{A}^1)\leqslant \operatorname{rank} E(t)\leqslant 1+\operatorname{rank} E(\mathbb{A}^1).

This is presented as a natural analogue of Goldfeld's conjecture for quadratic twists. The supplied text does not state a resolution, so the conjecture is recorded as open.

References

Primary source

Peter Koymans, Carlo Pagano and Efthymios Sofos, “Elliptic fibrations and 3 2^k”, arXiv:2409.02080 (2024).

Additional references

3 papers in this index state this conjecture (2014–2024). The statement above is taken from the most recent of them; the others are arXiv:1710.00294, arXiv:1401.6677.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A February 2025 paper reports the first unconditional example of infinitely many specializations whose rank does not increase, but the general conjecture remains open.

Silverman's conjecture predicts that, in almost every rational specialization of a one-parameter elliptic-curve family, the rank is either the generic rank or exactly one higher. The supplied literature records the conjecture but no general resolution.

Known results

Silverman's specialization theorem gives $\operatorname{rank} E_{\eta}(K(T))\leq \operatorname{rank} E_t(K)$ for all but finitely many $t$ (2004 source).
A 2022 paper states the expected density formulation $\operatorname{rank} X_b(\mathbb{Q})\in\{R,R+1\}$ outside a density-zero set, with only conditional examples then known.
The Neumann–Setzer and Legendre families provide conditional examples, respectively assuming Bunyakovsky's conjecture and infinitely many Mersenne primes.

February 2025 unconditional example

A paper reports an unconditional family with generic rank $0$ and infinitely many rational specializations of rank $0$ . This proves only the weaker existence of infinitely many no-rank-jump fibres, not Silverman's almost-all upper bound for arbitrary families.

Current status (as of August 2026): The specialization lower bound is known and one family has infinitely many no-rank-jump fibres, but Silverman's conjectured upper bound for almost all specializations remains open.

Sources

No solutions have been posted yet.