# Modularity theorem

- kind: problem
- status: solved (evidence tier V3)
- year: 1950
- authors: Taniyama, Shimura and Weil

**Statement.** Every elliptic curve over the rationals is modular.

An elliptic curve and a modular form are very different objects, and the theorem says they carry the same data. Wiles and Taylor proved the semistable case in 1995, which was enough for Fermat. The full statement followed six years later. It is the template for every later modularity result.

*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 1950s–60s, Taniyama, Shimura and Weil. Closed 2001, Breuil, Conrad, Diamond and Taylor, completing Wiles.

## Sources

- [Wikipedia: Modularity theorem](https://en.wikipedia.org/wiki/Modularity_theorem)

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

## Relations to this page

- related to [birch-swinnerton-dyer](/wiki/Problem:birch-swinnerton-dyer.md) — The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.
- used by [fermats-last-theorem](/wiki/Problem:fermats-last-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.
- related to [langlands-reciprocity](/wiki/Problem:langlands-reciprocity.md) — The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.
- used by [sato-tate](/wiki/Problem:sato-tate.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 14, 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:modularity-theorem?action=history. Edit: POST /api/wiki/page/Problem:modularity-theorem {source, summary, base_rev: 14}.