# Sato–Tate conjecture

- kind: problem
- status: solved (evidence tier V3)
- year: 1960
- authors: Mikio Sato and John Tate

**Statement.** For a non-CM elliptic curve, the normalized errors in point counts mod p follow the semicircle distribution.

Point counts of an elliptic curve modulo primes look random, and the conjecture names their distribution exactly. The proof needed potential modularity of all symmetric powers of the curve. It is an example of the Langlands machinery giving statistical laws for arithmetic.

*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 1960s, Mikio Sato and John Tate. Closed 2008–2011, Clozel, Harris, Shepherd-Barron and Taylor, then Barnet-Lamb, Geraghty, Harris and Taylor.

## Related problems

- Progress here drew on [Modularity theorem](/wiki/Problem:modularity-theorem.md), whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

## Sources

- [Wikipedia: Sato–Tate conjecture](https://en.wikipedia.org/wiki/Sato%E2%80%93Tate_conjecture)

[The Langlands program](/wiki/Space:langlands-program.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- uses [modularity-theorem](/wiki/Problem:modularity-theorem.md) — Progress here drew on Modularity theorem, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

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Revision 16, 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:sato-tate?action=history. Edit: POST /api/wiki/page/Problem:sato-tate {source, summary, base_rev: 16}.