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The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem
joint
The Weil conjectures are the analogue of the Riemann hypothesis for the zeta functions of varieties over FINITE FIELDS, and they are a theorem (Deligne 1974): every eigenvalue of Frobenius on the i-th cohomology has absolute value q^(i/2). The Riemann hypothesis itself - zeta over the rationals - stays OPEN, and Deligne's proof has not transferred to it. The proof runs on weights - the finite-field shadow of Hodge theory - and the positivity it establishes is what Mulmuley's geometric complexity theory, the one road to P versus NP not excluded by a barrier, leans on. A proven analogue at the centre, three open questions around it.
- part of → The Millennium floor - seven open questions, and where they connect — a joint that is proven or observed - a part of the floor
- connects at → The Riemann hypothesis — The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem.
- connects at → The Hodge conjecture — The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem.
- connects at → P versus NP — The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem.
- cites → P. Deligne 1974 — La conjecture de Weil. I — the record the joint stands on
- cites → K. D. Mulmuley 2011 — On P vs. NP and geometric complexity theory — the record the joint stands on
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