Modularity of Elliptic Curves over Totally Real Fields

Modularity of Elliptic Curves over Totally Real Fields

Definitions and mathematical statement

A number field is totally real if every embedding into ${\mathbb{C}}$ has image in ${\mathbb{R}}$. A Hilbert cuspidal newform of parallel weight two is a cuspidal Hecke eigenform whose weight is two at every real embedding, normalized and new at its level. The conjecture asks, for every elliptic curve $E/K$, for such an $f$ with \[ L(E,s)=L(f,s). \] The equality uses compatible local-factor normalizations at good and bad places, not just equality of a few computed coefficients. The field $K$ and the curve are arbitrary within the totally real scope.

Formulation and concept references: [S1].

Short English statement

Does every elliptic curve over a totally real number field have a matching weight-two Hilbert modular form?

Research attempt

Current frontier.

Freitas--Le Hung--Siksek prove modularity over real quadratic fields and finiteness of the possible nonmodular geometric isomorphism classes over each fixed totally real field [E1], Theorems 1 and 5. Yoshikawa's formulation source treats specified composites of real quadratic fields [S1]; his cyclotomic-extension result retains explicit algebraic $L$-value hypotheses [E2], Theorem 1.1. None is replaced by a universal theorem over arbitrary totally real fields. The attempt combines finite exceptional sets with residual-image criteria and checks whether twisting or isogenies can remove the exceptional behavior.

Attack route.

Potential modularity over an extension needs a descent theorem to recover the original field. A residual method needs a usable representation and a modularity theorem with the right hypotheses. A finite-exception argument needs to eliminate every exceptional geometric class, not merely almost every curve. We investigate the last route and test a natural repair: replace a difficult curve by a twist or an isogenous curve for which a small-prime modularity criterion applies.

Attempt: a finite elimination criterion.

Fix $K$. The exceptional modular curves in the finite-exception proof give a finite set $J_K$ of possible $j$-invariants; no effective list or height bound is presumed [E1], Section 8. If $j\ne0,1728$, two $K$-curves with the same $j$ are quadratic twists. Indeed, an isomorphism over $\bar K$ has Galois discrepancy in the automorphism group $\{\pm1\}$, hence defines a quadratic character. Modularity is preserved by quadratic twisting of the associated automorphic representation, with compatible local twists at bad as well as good places. CM curves, including $j=0,1728$, are covered by the Hecke-character construction; their larger twist groups are not declared quadratic [E1], discussion before Lemma 11.1.

Consequently, proving modularity of one $K$-curve representing each non-CM $j\in J_K$ suffices for every elliptic curve over this fixed $K$. This is a finite conditional elimination criterion, not an algorithm computing $J_K$ or a theorem removing its remaining members. The target still requires this for every totally real $K$.

First escape attempt: quadratic twists.

For a quadratic character $\chi_d$, the residual representations of $E$ and its twist satisfy \begin{equation}\label{wsref-0-72-36-37-3a-74-77-69-73-74} \bar\rho_{E^{(d)},p}(g)=\chi_d(g)\bar\rho_{E,p}(g). \end{equation} Thus the projective representations are the same. For any subgroup $H\le G_K$, their invariant subspaces over $\overline{\mathbb{F}}_p$ coincide: scalar multiplication by a nonzero scalar does not change whether a subspace is preserved. In particular, absolute irreducibility on $G_{K(\zeta_p)}$ is unchanged by twisting. Twists may change local reduction and conductors, so this does not say that twisting is useless; it says that this specific residual-image failure cannot be repaired in that way.

Second escape attempt: isogenies of arbitrary degree.

Prime-to-$p$ isogenies identify the $p$-torsion representations directly. A $p$-power isogeny does not, but it still preserves semisimplification. Here is the needed integral-lattice argument.

\begin{notebooklemma} For two $H$-stable ${\mathbb{Z}}_p$-lattices $\Lambda,\Lambda'$ in one finite-dimensional ${\mathbb{Q}}_p$-representation, the reductions modulo $p$ have isomorphic semisimplifications. This remains true after scalar extension to $\overline{\mathbb{F}}_p$. \end{notebooklemma}

A $K$-isogeny identifies rational Tate modules. Apply the lemma to their integral Tate lattices and restrict to any subgroup of $G_K$ before reducing. Absolute irreducibility on $G_{K(\zeta_p)}$ is therefore invariant under $K$-isogeny, even of degree divisible by $p$. This proves invariance of composition factors and irreducibility, not equality of full projective images or extension classes for arbitrary $p$-isogenous curves.

The resulting obstruction to a small-prime repair.

The criteria in [E1], Theorems 3--4, give modularity when the indicated restriction is absolutely irreducible for a suitable $p\in\{3,5,7\}$. If a curve fails all three tests, every curve reached by any sequence of $K$-isogenies and quadratic twists still fails all three. Thus the proposed procedure "change the curve until a small-prime test succeeds" cannot remove that exceptional absolute-reducibility condition.

Failure of a sufficient criterion is not nonmodularity. For example, \[ E:\quad y^2+xy+y=x^3 \] has discriminant $-26$. The tangent at $(0,0)$ is $y=0$, meeting the cubic there with multiplicity three. The chord-and-tangent law gives $3(0,0)=O$, and $(0,0)\ne O$, so the point has order three. Hence $E[3]$ has a rational invariant line, while the curve is modular over ${\mathbb{Q}}$. The same invariant-line obstruction at three persists under the operations above, without making any curve in this example nonmodular. The script checks the discriminant and the tangent substitution exactly.

Stress test.

Finiteness supplies neither an effective enumeration of $J_K$ nor proofs for its remaining points. Enlarging $K$ changes the desired conclusion and introduces descent. Absolute irreducibility at a large prime by itself does not prove residual modularity over an arbitrary totally real field. Special $j$-values have larger automorphism groups and were handled separately as CM, not folded into the quadratic-twist argument. The lattice lemma preserves semisimplified residual representations but not a preferred integral lattice.

No finite set of matching Euler coefficients is used as a certificate of full modularity. A successful final elimination would need an actual comparison theorem for compatible Galois or automorphic representations and the bad-place local factors. The notebook supplies no such identification for an arbitrary residual exceptional point.

Outcome.

Outcome: Conditional reduction.

The finite-$j$ criterion specifies what would suffice for each field, and the twisting and lattice arguments show why one natural residual-image repair cannot deliver it. The outstanding work is a different modularity argument for every exceptional geometric class, or an exhaustive elimination on the modular curves with valid stopping information. No new general modularity theorem or counterexample is claimed; the finite-exception theorem and the general lattice principle are established mathematics, explicitly applied and proved here where needed.

Sources

• S1 Sho Yoshikawa, "On the modularity of elliptic curves over a composite field of some real quadratic fields," Research in Number Theory 2 $2016$, article 31. https://arxiv.org/pdf/1607.05549. Catalogue formulation source.
• S2 Modularity of elliptic curves over cyclotomic Z_p-extensions of real quadratic fields. https://arxiv.org/html/2206.12860v1. Catalogue research/status reference; not a proof endorsement.
• S3 Reliability of elliptic curve data over number fields (reviewed). https://www.lmfdb.org/knowledge/show/dq.ecnf.reliability. Catalogue research/status reference; not a proof endorsement.

• E1 Nuno Freitas, Bao V. Le Hung, and Samir Siksek, Elliptic curves over real quadratic fields are modular, Inventiones Mathematicae 201 $2015$, 159--206; arXiv:1310.7088, Theorems 1 and 3--5, Sections 8 and 11. https://arxiv.org/html/1310.7088. Real quadratic modularity, small-prime criteria, finite geometric exceptions, and CM scope. Consulted 27 September 2026.
• E2 Sho Yoshikawa, Modularity of elliptic curves over cyclotomic ${\mathbb{Z}}_p$-extensions of real quadratic fields, arXiv:2206.12860 $2022$, Theorem 1.1. https://arxiv.org/abs/2206.12860. The specified algebraic $L$-value hypothesis is retained; not general totally real modularity. Consulted 27 September 2026.