Number theorist digests OpenAI's modularity breakthrough; Hodge Conjecture papers pulled over sign error
ChrSzegedy · x · 2026-10-11
Blogger Persiflage reviews OpenAI's new algebraic number theory results. OpenAI cracked the remaining cases of modularity of elliptic curves over imaginary quadratic fields (reducible mod 3 and mod 5 representations) with a clever trick that Bao Le Hung has already reproduced and extended to all totally real fields and CM fields not containing ζ₅ — a nice trick, per the author, but not a fundamental break. The quasi-RH proof crucially uses Q(√-3) throughout, vindicating the view that more algebraic ideas were needed. Also: OpenAI reportedly withdrew some partial Hodge Conjecture papers after a sign error was found.
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