Goldfeld’s conjecture: densities and mean analytic rank

The explanation

A whole number n is called congruent if it is the area of a right triangle with rational sides. The 3, 4, 5 triangle has area 6, so 6 is congruent. The triangle with sides 3/2, 20/3 and 41/6 has area 5. The triangle with sides 24/5, 35/12 and 337/60 has area 7. Fermat showed that 1 is not congruent. A classical fact links this to curves. The number n is congruent exactly when the curve y² = x³ − n²x has infinitely many rational points.

Those curves form one family. Start from the elliptic curve y² = x³ − x. Twisting it by n gives ny² = x³ − x, which is the same curve as y² = x³ − n²x after rescaling. In general, a quadratic twist of y² = x³ + ax + b by d is dy² = x³ + ax + b. The papers use every squarefree d, meaning no square larger than 1 divides it. Both positive and negative d count, ordered by size.

Each twist has a rank, the number of independent rational points that generate infinitely many others. It also has an analytic rank, the number of times its L-function vanishes at the centre. The L-function is built from point counts modulo every prime. The Birch–Swinnerton-Dyer conjecture predicts the two ranks agree. A sign in the L-function's symmetry decides whether the analytic rank is even or odd.

In 1979, Goldfeld conjectured that the average analytic rank in such a family is 1/2. The density version says ranks 0 and 1 each occur half the time. For the congruent family, 1, 2 and 3 have rank 0, while 5, 6 and 7 have rank 1. Ranks of 2 or more should be rare enough to vanish in the average.

Earlier results gave only partial answers. In 1998, Ono and Skinner showed that many twists have nonzero central value, but not a positive share. Positive shares of both ranks were known for special curves, such as those with a rational 3-isogeny. Smith's 2025 preprint gives that Selmer coranks 0 and 1 each occur half the time, for every curve. A Selmer group is a computable group that bounds the rank. Smith's result for analytic ranks assumed the Birch–Swinnerton-Dyer conjecture.

The first paper fills that gap with a 2-converse. It claims that if the 2-power Selmer corank is 0 or 1, the analytic rank and the rank both equal it. It also claims the Tate–Shafarevich group, a group of hidden solutions, is finite. Combined with Smith's statistics, it claims ranks 0 and 1 each have density 1/2Source 1,Source 2. The rank and analytic rank then agree for almost every twistSource 3.

The second paper handles the average. A rare set of twists could still have huge ranks and pull the average up. The paper claims a bound on how much these rare high-rank twists can add. Together with the densities, the average analytic rank tends to 1/2Source 4. The proof uses neither the Birch–Swinnerton-Dyer conjecture nor the generalized Riemann hypothesisSource 5.

The hard step is going from the Selmer group to the L-function. The first paper builds finite families of twists indexed by strings of 0s and 1s. The curve under study sits at the all-zero string. Every other member is arranged to have the right analytic rank, with controlled 2-adic size. An interpolation argument then recovers the missing value at the all-zero stringSource 6.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. The density claim depends on Smith's preprint, and no specialist review of these papers has been published.

What it doesn't show. This does not decide the rank of every single twist, cover other orderings of twists, or give higher moments of analytic rank.