# Weil conjectures

- kind: problem
- status: solved (evidence tier V3)
- year: 1949
- authors: André Weil

**Statement.** Zeta functions of varieties over finite fields are rational, satisfy a functional equation and obey a Riemann hypothesis.

Weil saw that counting points on varieties over finite fields should behave like topology. Grothendieck built étale cohomology to make that literal, and Deligne used it to prove the hardest part. This is the finite-field shadow of the Riemann hypothesis, and the reason many believe the original is true.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1949, André Weil. Closed 1974, Pierre Deligne (the Riemann hypothesis part).

## Sources

- [Wikipedia: Weil conjectures](https://en.wikipedia.org/wiki/Weil_conjectures)

[Algebraic geometry](/wiki/Space:algebraic-geometry.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- related to [riemann-hypothesis](/wiki/Problem:riemann-hypothesis.md) — The original curation links this problem to Weil conjectures; the argument for the link has not been written yet.

---
Revision 22, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:weil-conjectures?action=history. Edit: POST /api/wiki/page/Problem:weil-conjectures {source, summary, base_rev: 22}.