MathDBThe semistable Taniyama–Shimura conjecture
The semistable Taniyama–Shimura conjecture
Let denote the field of rational numbers, and call an elliptic curve over semistable if it is semistable at every prime, meaning that at each prime it has either nonsingular reduction or a singular point with two distinct tangent directions. Call an elliptic curve over modular if it admits a holomorphic map from some modular curve onto it. Semistable Taniyama–Shimura conjecture. Every semistable elliptic curve over is modular. This is the semistable portion of the Taniyama–Shimura conjecture used in the argument for Fermat's Last Theorem; the source explains that Wiles's work proves the relevant result, but the supplied status is unknown.
Progress summary
The conjecture was proved in 1995, so the semistable case used in the proof of Fermat’s Last Theorem is settled.
The conjecture asserts that every semistable elliptic curve over is modular, meaning that it is a quotient of some modular curve . It was the case needed in Wiles’s strategy for Fermat’s Last Theorem.
Known results
- Wiles, with Richard Taylor, proved modularity for all semistable elliptic curves over in 1995.
- Breuil, Conrad, Diamond, and Taylor completed the modularity theorem for all elliptic curves over by 1999.
1995 proof
Wiles’s 1993 announcement initially relied on an incomplete Euler-system construction and a result not yet fully checked. The repaired work with Taylor established the semistable theorem; no later counterexample, gap, retraction, or competing claim was found.
Current status (as of August 2026): The semistable Taniyama–Shimura conjecture is settled by Wiles’s 1995 work with Taylor; no unresolved part of this semistable case remains.
Sources & referencesView supporting material
Primary source
Karl Rubin and Alice Silverberg, “A report on Wiles' Cambridge lectures”, arXiv:math/9407220 (1994).
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