Episode 79 · The Machine's Theorems
Elliptic Curves and Hodge Classes: Near the Millennium Line
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(004), the rational Hodge conjecture for CM abelian varieties (032) and Milne's rationality conjecture (001), plus the withdrawn K3 chain and the Kuznetsov counterexamples (054). Every claim is labelled: verified, Lean-checked, contested, unrefereed or withdrawn. Expert voices come from MathOverflow: Will Sawin, Olivier and Timothy Chow.
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Inside this episode
- Cold open: a long-distance call about doughnuts
- Doughnuts with an addition law
- Counting points modulo primes: the L-function
- The conjecture, said in words
- Sha and Selmer: the ghost and the net
- Selmer's cubic: a ghost you can write down
- A balance you can check by hand
- What was known before
- What family two claims
- How a referee would read it
- The experts react: Sawin, Olivier and the house of cards
- Why no machine has checked this
- Goldfeld's conjecture: half and half
- Hilbert's tenth problem over the rationals
- Halfway: what we know so far
- The Hodge conjecture: shadows that should be cast by real objects
- Abelian varieties and complex multiplication
- What family thirty-two claims
- Milne's rationality conjecture
- The withdrawn chain: a sign error in the eightfolds
- Cutting the other way: Kuznetsov's conjecture
- Why it matters to an engineer
- Closing: Augustin's fences
Production credits & AI disclosure
Scripts generated with Claude Opus. Synthetic narration generated with Edge TTS. Reviewed, listened to, and approved by Sankalp Thakur.