1 · The essentials
Start with an elliptic curve defined over the rational numbers and consider a family of related curves. Each curve in this family has a number called its “analytic rank.” The first manuscript claims that, in the limit, half of these curves have analytic rank zero and half have analytic rank one. The second claims that the average of these ranks tends to one half. These statements concern every starting elliptic curve over the rationals, using a specific rule for counting the related curves. Proportions and averages are distinct conclusions: rare but large values can affect an average. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine cards bearing scores, examined in a fixed order. Saying that scores zero and one each occupy half the cards in the limit does not by itself determine the average: increasingly rare cards bearing large scores could still affect it.
Why it matters
The two manuscripts claim the density and mean versions of Goldfeld’s conjecture under the stated counting convention. Together, they describe both the distribution of small ranks and the behavior of the average. The first also announces a connection between several measures of a curve’s rank, under a specific hypothesis.
2 · Going further
Fix an elliptic curve over . The family consists of its quadratic twists , with ranging over positive and negative squarefree integers, ordered by . The first manuscript claims that analytic ranks zero and one each have density in this family. It identifies this statement as Goldfeld’s analytic density conjecture.
The companion manuscript claims that, for every curve over , the average analytic rank of the with tends to as tends to infinity. It presents this as a resolution of Goldfeld’s mean analytic-rank conjecture under this counting convention. This complements the density statement: the limiting frequencies of ranks zero and one do not, by themselves, control the contribution of rare higher ranks to the average.
The first manuscript also announces a low-corank “2-converse”: if the -Selmer corank of is zero or one, it equals both the analytic rank and the Mordell–Weil rank, and the Tate–Shafarevich group is finite. The supplied abstracts do not describe the argument or explain how this result enters the density argument. They therefore support an account of the claims and how they complement one another, but not a reconstruction of the strategy.
Useful notions
- Elliptic curves over the rationals
- Quadratic twists
- Analytic rank
- Limiting densities and averages
- Selmer groups and Mordell–Weil rank
What was known before / what is new
Before
The source identifies the density and mean statements as conjectures of Goldfeld. It supplies no historical account, earlier partial results, or description of what had already been established.
New
The first manuscript claims densities of for analytic ranks zero and one for every elliptic curve over . It also announces the 2-converse under the hypothesis of corank zero or one. The second claims convergence of the mean analytic rank to under the same counting convention.
Limits and assumptions
The claims concern quadratic twists of elliptic curves over , counted using squarefree parameters of both signs ordered by absolute value. The densities and mean are limiting statements, not claimed equalities for every finite set of parameters. The 2-converse requires Selmer corank zero or one. The main result is not formalized in Lean and has not been formally verified; the abstracts do not supply the arguments needed to assess it or specify which questions remain open.
Mini glossary
- Quadratic twists —
- Curves related to a starting elliptic curve, forming here the family indexed by the parameters .
- Squarefree integer —
- An integer not divisible by the square of any prime number.
- Density —
- A limiting proportion, obtained here by counting parameters up to an increasing bound on their absolute value.
- Mean analytic rank —
- The sum of the analytic ranks of the curves counted, divided by the number of those curves.
Keywords : Goldfeld’s conjecture · Quadratic twists · Analytic rank
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Goldfeld's analytic density conjecture and the 2-converse for elliptic curves
07/10/2026We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the -Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.
The mean analytic rank of quadratic twists of elliptic curves
06/10/2026For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.