Modularity of elliptic curves over imaginary quadratic fields

Every elliptic curve over every imaginary quadratic field is claimed to match an automorphic form, place by place.

1 paper October 4, 2026 Explained by AI

An elliptic curve is a cubic equation whose solutions can be counted modulo each prime. Modularity says these counts are the coefficients of a highly symmetric function, an automorphic form. Over the rational numbers this was proved by 2001, completing work that gave Fermat's Last Theorem. The paper claims to prove it over every imaginary quadratic field, such as the numbers a + bi with a, b rational. It also claims the match holds at every prime, including the bad ones.

First step

1995

A. Wiles and R. Taylor–A. Wiles, who proved modularity for semistable elliptic curves over the rationals

Significance

Landmark

Modularity over imaginary quadratic fields was known only for some curves or some fields, if this proof holds.

Trust

Claimed, not checked

Field

Number theory

1 paper

The explanation

Work with numbers of the form a + bi, where i² = −1 and a, b are fractions. Take the curve y² = x³ + x + i. To count its points modulo 5, note that 5 = (2 + i)(2 − i). Working modulo 2 + i turns i into 3, since 3² = 9 leaves −1 after dividing by 5. The curve becomes y² = x³ + x + 3 with remainders mod 5. It has 3 solutions with ordinary x and y, plus one point at infinity, so 4 points. The number 5 + 1 − 4 = 2 is the curve’s count at this prime.

An elliptic curve is an equation like this one, with no repeated roots on the right. Do the same count at every prime and you get an endless list of numbers. Modularity says one symmetric object produces that whole list. That object is an automorphic representation (a highly symmetric function built from the field, here acting by 2 × 2 matrices). The conjecture asks for even more. At the bad primes, where the curve breaks, the full local data must match tooSource 1.

The field of numbers a + bi is one imaginary quadratic field. Others use √−2, √−3, and so on. Over the ordinary fractions, Wiles and Taylor–Wiles proved modularity for semistable curves in 1995. Breuil, Conrad, Diamond and Taylor finished every curve in 2001Source 2. A key method lifts a match from a simpler “residual” version of the curve’s data, its pattern modulo one primeSource 3.

Imaginary quadratic fields resisted this method. The relevant data occur in more than one cohomology degree, a technical layer the method usesSource 4. Potential modularity was proved: every curve becomes modular after enlarging the fieldSource 5. But proving it over the original field is a further problemSource 6. Caraiani and Newton proved modularity for some curves and fields. Their results include every curve over a field where a certain curve, X₀(15), has rank zeroSource 7. The paper claims to remove the remaining restrictions on the curve and the fieldSource 8. It still relies on Caraiani and Newton’s lifting theorem as an essential inputSource 8.

The paper claims the full statementSource 9. For every imaginary quadratic field and every elliptic curve over it, a matching automorphic representation exists. The L-function of the curve (a function that packages all its point counts) equals the automorphic one, shifted by 1/2Source 10. In the example, the count 2 at the prime 2 + i is one coefficient of a single automorphic object.

The obstacle is reaching a residual pattern that the known theorem accepts. The paper builds a chain of linked objects from one family of curves of higher genusSource 11. Each link agrees with the next modulo a prime: first modulo p, then modulo 3, then modulo p again. Matches pass along the chain, from a curve at one end that is shown modular first, to the given curve. This echoes Wiles’s switch between the primes 3 and 5Source 12.

The claim stays inside its setting. It is about imaginary quadratic fields only. It does not cover real quadratic fields or larger fields.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. The proof combines a new construction with deep published theorems. No specialist review has been published.

What it doesn't show. This does not cover other fields, such as real quadratic fields or fields of higher degree.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark only if the proof holds. No Lean formalization is listed and no specialist review is published. It covers imaginary quadratic fields only.

Words used

Elliptic curve: The solutions of a cubic equation like y² = x³ + ax + b with no repeated roots, plus one point at infinity.
Imaginary quadratic field: The numbers a + b√−d with a, b fractions, for a fixed positive whole number d, such as a + bi.
Modular: Matched by an automorphic form, so the curve’s point counts are that form’s coefficients.
Automorphic representation: A highly symmetric object built from a number field whose numbers can match the counts of a curve.
L-function: A function that packages all the point counts of a curve, or all the numbers of an automorphic object.
Residual representation: The pattern of a curve’s symmetry data reduced modulo one prime.
Potential modularity: Modularity that holds only after the field is enlarged.

The original paper

Modularity of elliptic curves over imaginary quadratic fields (PDF, opens in a new tab)
October 4, 2026

All quoted lines (12)

At a place of good reduction this requires the Hecke polynomial to recover the point count of the reduced curve; at a bad place it requires the full local parameter, including its inertia and monodromy.
Modularity of elliptic curves over imaginary quadratic fields Back to text
Over $\Q$ , Wiles and Taylor–Wiles proved modularity of semistable elliptic curves, and Breuil–Conrad–Diamond–Taylor proved modularity of every elliptic curve.
Modularity of elliptic curves over imaginary quadratic fields Back to text
Over $\Q$ , Wiles and Taylor–Wiles proved modularity of semistable elliptic curves, and Breuil–Conrad–Diamond–Taylor proved modularity of every elliptic curve. A central method is to lift automorphy from a residual Galois representation.
Modularity of elliptic curves over imaginary quadratic fields Back to text
For imaginary quadratic fields, the relevant cohomology occurs in more than one degree.
Modularity of elliptic curves over imaginary quadratic fields Back to text
The potential automorphy theorem of Allen–Calegari–Caraiani–Gee–Helm–Le Hung–Newton–Scholze–Taylor–Thorne implies potential modularity of every elliptic curve over a CM field.
Modularity of elliptic curves over imaginary quadratic fields Back to text
Potential modularity permits an extension of the ground field; automorphy over the original field is a further problem.
Modularity of elliptic curves over imaginary quadratic fields Back to text
Their results give modularity under suitable residual-image hypotheses in characteristic $3$ or $5$ , modularity for $100\%$ of Weierstrass equations, ordered by height, over each Galois CM field not containing $\zeta_5$ , and modularity of every elliptic curve over an imaginary quadratic field $K$ for which $X_0(15)(K)$ has rank zero.
Modularity of elliptic curves over imaginary quadratic fields Back to text
Theorem … removes the remaining restrictions on the curve and the imaginary quadratic field. Caraiani–Newton's lifting theorem and residual-image criterion for modularity of elliptic curves remain essential inputs.
Modularity of elliptic curves over imaginary quadratic fields Back to text
We prove this conjecture for every $K$ and $E$ .
Modularity of elliptic curves over imaginary quadratic fields Back to text
In particular every local factor agrees, and $L(E/K,s)=L(\pi_E,s-1/2).$
Modularity of elliptic curves over imaginary quadratic fields Back to text
We construct the intermediate representations as character pieces of a single family of Jacobians.
Modularity of elliptic curves over imaginary quadratic fields Back to text
Wiles's $3$ – $5$ switch also shows how an auxiliary elliptic curve, chosen with prescribed torsion and local behavior, can replace a residual representation that does not satisfy the initial lifting hypotheses.
Modularity of elliptic curves over imaginary quadratic fields Back to text

Other sources

016Zilber–Pink in abelian varieties and the Siegel threefoldNumber theory
006Goldfeld’s conjecture: densities and mean analytic rankNumber theory
002The full BSD formula from low Selmer corankNumber theory
001Milne’s rationality conjecture and algebraic specializationNumber theory
032The rational Hodge conjecture for CM abelian varietiesAlgebraic and complex geometry

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