A genus-2 curve with 648 rational points

GPT-6 Astra found a genus-2 curve with 648 rational points, beating the record of 642 set by Elkies and Stoll in 2008; the deeper uniformity question remains open.

Problem
Is there a genus-2 curve over Q\mathbb{Q} with more than 642 rational points?
Credit
GPT-6 Astra (OpenAI) · Found during Epoch AI's pre-release evaluation
Checked
Passed Epoch AI's verifier
Timeline
Elkies–Stoll record 642 (2008) → 648 (2026; exact date not given)
Sources
Epoch AI problem page
PARTIALNew record · verified by Epoch AI

The problem

By Faltings' theorem, a curve of genus 2 over Q\mathbb{Q} has only finitely many rational points, but how many it can have is not known. Since 2008 the record has been a curve with 642 rational points found by Elkies and Stoll. Epoch AI's FrontierMath Open Problems asked for a curve with more.

What AI did

During Epoch AI's pre-release evaluation of OpenAI's GPT-6 Astra, the model found a genus-2 curve with 648 rational points. Epoch AI's problem page gives the curve's equation. Epoch does not give the date, so we list only the year.

How it was checked

The curve passed Epoch AI's verifier. Anyone can check a claimed list of rational points on an explicit curve directly.

Limitations

This is a new record, not a resolution of the underlying question: whether the number of rational points on a genus-2 curve is bounded uniformly (the question behind the work of Caporaso, Harris and Mazur) remains open. Problem description paraphrased from Epoch AI (CC BY 4.0).

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