Goldfeld's conjecture on analytic ranks of quadratic twists

Lineage and corrections

Revised edition (October 6, 2026): The mean analytic rank of quadratic twists of elliptic curves evidence for this event

Goldfeld 1979, Conjectures on elliptic curves over quadratic fields evidence for this event

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Same family: The mean analytic rank of quadratic twists of elliptic curves evidence for this event

Goldfeld's conjecture on analytic ranks of quadratic twists parent of this event

The Selmer p-converse entry, whose p = 2 case is proved here evidence for this event

Input (family 006): Goldfeld's analytic density conjecture and the 2-converse evidence for this event

Density paper: for every $E/\mathbb{Q}$ and $j\in\{0,1\}$, the proportion of squarefree $0<|d|\le X$ with $a(E^{(d)})=j$ tends to $1/2$; on a density-one set analytic and Mordell-Weil ranks agree and Sha is finite. The main new input is the 2-converse: if the $2^\infty$-Selmer corank is 0 or 1 it equals the analytic and Mordell-Weil ranks and Sha is finite. Mean paper: $\frac{1}{\#\mathcal D(Y)}\sum_{d}a(E^{(d)})\to1/2$, via a tail bound on high-rank twists, plus algebraic-rank moments. Both use signed squarefree twists ordered by $|d|$, not Goldfeld's discriminant ordering; no higher analytic-rank moments are claimed. parent of this event

The BSD formula entry that depends on this result evidence for this event

VibeMathed record: Goldfeld's conjecture on analytic ranks of quadratic twists evidence for this event

Revised edition (October 7, 2026): Goldfeld's analytic density conjecture and the 2-converse for elliptic curves evidence for this event