Goldfeld's conjecture on analytic ranks of quadratic twists

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.

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.