Shimura–Taniyama conjecture for elliptic curves over totally real fields

Let $K$ be a totally real number field, and let $E$ be an elliptic curve over $K$ . Say that $E$ is modular if there exists a Hilbert modular form $f$ over $K$ of parallel weight $2$ such that

L(E,s)=L(f,s).

Shimura–Taniyama conjecture. Any elliptic curve over $K$ is modular.

This is the natural totally real generalization of the original Shimura–Taniyama conjecture over $\mathbb{Q}$ , which was proved by Wiles, Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor. The statement for arbitrary totally real fields is presented here as a conjectural generalization.

Sources & referencesView supporting material

Primary source

Sho Yoshikawa, “On the modularity of elliptic curves over a composite field of some real quadratic fields”, arXiv:1607.05549 (2016).

Additional references

2 papers in this index state this conjecture (2016). The statement above is taken from the most recent of them; the others are arXiv:1606.06597.

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Progress summary

Refreshed 22 hours ago

Partially solved

The conjecture is proved for several important families of totally real fields, but no general proof is known and the remaining cases begin in degree four.

The conjecture says that every elliptic curve over a totally real number field comes from a suitable Hilbert modular form. The general statement remains unproved, although substantial low-degree and restricted-family cases are settled.

Known results

Freitas, Le Hung, and Siksek proved modularity over real quadratic fields in 2013.
Modularity is proved for totally real fields of degree at most $3$ .
Box proved the degree- $4$ case when the field does not contain $\sqrt{5}$ .
Modularity is proved for totally real abelian fields unramified above $3$ , $5$ , and $7$ .

October 2022 degree-five result

A paper dated October 9, 2022 proves modularity for all but finitely many pairs consisting of a totally real degree- $5$ field and a possible $j$ -invariant. It leaves the finitely many exceptional pairs, and the conjecture in general degree at least $4$ , unresolved.

Current status (as of August 2026): Modularity is settled in degrees at most $3$ and in several restricted degree- $4$ and degree- $5$ families, but the conjecture for arbitrary totally real fields remains open.

Sources

No solutions have been posted yet.