A genus-2 curve with a rational torsion point of prime order at least 31

The problem

For a genus-2 curve over $\mathbb{Q}$, the rational points of finite order on its Jacobian form a finite group. Which prime orders can occur is not known; the largest previously known was 29, due to Leprévost. Epoch AI's FrontierMath Open Problems asked for a curve with a rational torsion point of prime order at least 31.

What AI did

On 30 July 2026, OpenAI's GPT-5.6 Sol submitted a curve that passed Epoch AI's verifier. The curve, however, was already in a public database: Edgar Costa's ModularAbelianSurfaces repository, which collects work by Costa, Elkies, Hashimoto, Jha, Martin, Poonen and Voight. It also appears in a repository maintained by Timo Keller.

How it was checked

The verifier confirmed the curve and the torsion point; such a claim can be checked by direct computation.

Limitations

Epoch AI itself says the solution "was latent in the existing literature" and has raised the target to order 37. The model found a known answer rather than a new one, so we grade this "prior". Problem description paraphrased from Epoch AI (CC BY 4.0).

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