Half and half

0:00One curve, endless twists

Here is an elliptic curve, the first one in the tables, of conductor 11. Multiply one side by a whole number d, and you get a new curve: its twist by d. Every squarefree d, positive or negative, gives a different twist. Each twist has a rank, which measures how many independent rational points it has. Here are all 3,648 twists with d up to 3,000 in size. 44 percent have rank zero, 50 percent rank one, about 6 percent rank two, and just seven have rank three. In 1979, Dorian Goldfeld conjectured that the average rank tends to 1/2: in the limit, half the twists have rank zero and half have rank one, while the higher ranks, which keep appearing, shrink to a vanishing share. At this size, the share of rank two and up hasn’t visibly started to shrink. Result 006 in OpenAI’s math repository claims to prove it, for every elliptic curve over the rationals, with rank measured analytically.

1:00Rank, two ways

There are two ways to measure rank. The algebraic rank counts independent rational points. The analytic rank comes from the curve’s L-function: it is the order of the zero at s = 1. A twist of rank zero doesn’t vanish there. Rank one crosses zero. Rank two touches zero and turns back. The Birch and Swinnerton-Dyer conjecture says the two ranks always agree. It is one of the Millennium Prize Problems, and it is still open. Every twist also carries a sign, plus or minus one, from its functional equation. A plus sign forces an even analytic rank, and a minus sign forces an odd one. Across the twists, the signs split evenly. So Goldfeld’s conjecture says the rank is almost always as small as its sign allows.

1:49What was known

For decades, results came in pieces. For particular curves, a positive fraction of twists were shown to have rank zero, and a positive fraction rank one: a curve of conductor 19 in 1998, and every curve with a rational 3-isogeny in 2019. Assuming the generalized Riemann hypothesis, Heath-Brown bounded the average analytic rank by 3/2. Then Alexander Smith studied an algebraic shadow of the rank, the 2-power Selmer corank, which is defined by local conditions, checked one prime at a time. In a 2025 preprint, he proved Goldfeld’s split for it, for every curve: corank zero for half the twists, corank one for the other half. So if Birch and Swinnerton-Dyer is true, Goldfeld’s conjecture follows. What was missing was a converse theorem at the prime 2. As Smith noted, the few known there all required complex multiplication.

2:46The missing converse

Here is the gap. The Selmer corank is an upper bound for the algebraic rank, and it has the same parity as the analytic rank. In the late 1980s, Gross, Zagier and Kolyvagin proved one direction: if the analytic rank is zero or one, the algebraic rank matches it, and the Selmer corank does too. A converse goes the other way, from the Selmer corank to the analytic rank. Converses were known at many odd primes, under various conditions. At the prime 2, they were known only for curves with complex multiplication. Result 006 claims the full converse at 2. For every elliptic curve over the rationals, if the 2-power Selmer corank is zero or one, then the analytic rank and the algebraic rank both equal it. Combine that with Smith’s theorem, and Goldfeld’s split follows quickly: among twists by squarefree d, ordered by size, ranks zero and one each have density 1/2.

3:44A cube of twists

How do you prove the converse? The paper surrounds the curve with a cube of twists. Choose some auxiliary primes. Each corner of a many-dimensional cube is a twist by a product of some of them, and one corner, the origin, is the original curve. At every other corner, the paper arranges that a certain number, built from the L-value, or its derivative in the rank one case, is nonzero, with only a bounded number of factors of two. About half of the 130 pages go into building these corners, with modular forms, Hecke operators, and graphs of quadratic residues. Then a counting theorem from 1935 finishes the job. One algebraic object, built from Kato’s classes or from Heegner points, produces the number at every corner. If the cube has enough dimensions, the Chevalley–Warning theorem forces some other corner to agree with the origin to many binary digits. So the origin’s number has few factors of two as well. In particular, it isn’t zero.

4:46From density to average

Densities aren’t the whole story. A rare set of twists with huge ranks could still drag the average up. A companion paper rules that out: the share of twists with rank above k is at most a constant over k², and no rank in the range can exceed a constant times the logarithm of the size. Together, the average analytic rank of the twists tends to exactly 1/2, as Goldfeld predicted. The companion also shows that the average algebraic rank tends to 1/2.

5:17Why it matters

If correct, this settles Goldfeld’s conjecture, in the paper’s way of counting, for every elliptic curve over the rationals. And on almost every twist, the analytic and algebraic ranks agree. It also gives a finite test: when a single 2-descent finds a small Selmer group, it certifies the analytic rank. And it is load-bearing. Episode two’s claimed Birch and Swinnerton-Dyer formula uses this density theorem as an input, and episode four’s paper can use its converse in place of its own companion.

5:51The caveats

The caveats. Both papers are preprints, not peer reviewed, with no Lean proofs. The density step relies on Smith’s 2025 theorem, posted as a preprint. And both papers count twists by signed squarefree d, not by Goldfeld’s original discriminants. Readers have also flagged one step. In one lemma, a congruence modulo 8 fails numerically, and we reproduced the failures. A second check confirmed the numbers, but found that every case the report named falls under the proof’s other branch, where this step isn’t used. Neither report claims a counterexample to the theorem. It remains an open question about the write-up. Half rank zero, half rank one. If the proof holds.