Elliptic Curves and Hodge Classes: Near the Millennium Line

Deep End Era. Engineering with Sankalp · Oct 12, 2026 · 56m

Part 11 of "The Machine's Theorems". Farah and Augustin take on elliptic curves, "doughnuts with an addition law", and the Birch and Swinnerton-Dyer formula, then the Hodge conjecture, "shadows that should be cast by real objects". We read OpenAI's manuscripts for the full BSD formula in Selmer corank zero and one (family 002), Goldfeld's conjecture (006), Hilbert's tenth problem over the rationals (004), the rational Hodge conjecture for CM abelian varieties (032) and Milne's rationality conjecture (001), plus the withdrawn K3 chain and the Kuznetsov counterexamples (054). Every claim is…