The density of ℚ-curves over a fixed quadratic field

Abstract

For a fixed quadratic field, we conjecture that non-CM $\mathbb Q$-curves have density zero among all non-CM elliptic-curve isogeny classes ordered by conductor norm. The assertion includes base changes from $\mathbb Q$ as well as strict $\mathbb Q$-curves. Numerical data over the deepest imaginary quadratic fields show a consistent decline in both components, while the main theoretical difficulty is obtaining conductor-aspect counts uniform over all modular degrees and twist families.

Statement

Fix a quadratic field $K$. Let $\mathcal E_K(X)$ be the set of non-CM $K$-isogeny classes $\mathcal C$ of elliptic curves satisfying

$$\mathrm N_{K/\mathbb Q}\mathfrak N_{\mathcal C}\le X.$$

Let $\mathcal Q_K(X)\subseteq\mathcal E_K(X)$ be the subset of $\mathbb Q$-curve classes, namely those for which an elliptic curve in the class is isogenous over $\overline{\mathbb Q}$ to each of its Galois conjugates. Base changes from $\mathbb Q$ are included.

Conjecture 1. For every quadratic field $K$,

$$\lim_{X\to\infty} \frac{\#\mathcal Q_K(X)}{\#\mathcal E_K(X)}=0.$$

Both the conductor and the $\mathbb Q$-curve property are invariant under $K$-isogeny, so the formulation is intrinsic to isogeny classes.

Numerical evidence

The complete scan contains $297834$ non-CM quadratic-field isogeny classes, of which $21253$ are $\mathbb Q$-curves: $11712$ base-change classes and $9541$ strict $\mathbb Q$-curves. In every one of the thirteen largest fixed-field datasets, the cumulative $\mathbb Q$-curve proportion decreases as the conductor cutoff grows. Table 1 gives the five deepest fields.

$D_K$	first conductor decile	full range	terminal shell
$-3$	$6.633\%$	$3.220\%$	$1.970\%$
$-4$	$6.607\%$	$2.920\%$	$2.344\%$
$-7$	$6.742\%$	$2.968\%$	$1.893\%$
$-8$	$7.772\%$	$3.471\%$	$2.076\%$
$-11$	$9.452\%$	$3.949\%$	$2.063\%$

Decline of the $\mathbb Q$-curve proportion in the deepest fixed-field datasets.

The decline persists after separating base changes and strict $\mathbb Q$-curves. In the five fields of Table 1, base-change shares fall from roughly $3.5\%$–$5.6\%$ in the first shell to $0.9\%$–$1.6\%$ in the last, while strict-$\mathbb Q$-curve shares fall to approximately $0.7\%$–$1.0\%$.

Geometric and automorphic motivation

Being a $\mathbb Q$-curve is a strong descent condition. A non-CM $\mathbb Q$-curve is associated with a modular object carrying an inner-twist structure, and central $\mathbb Q$-curves of a fixed squarefree degree are parametrized by suitable Atkin–Lehner quotients or twists of modular curves. The union over all degrees is countable but highly structured; it should be thin inside the two-parameter moduli of elliptic curves over $K$.

The conductor ordering prevents a direct transfer of geometric height-density statements. A fixed geometric $\mathbb Q$-curve class has infinitely many quadratic twists, all of which remain $\mathbb Q$-curves. At a new good prime ramified in the twisting character, the conductor exponent is typically two. Thus a single seed already generates a family on an $X^{1/2}$ scale. Conjecture 1 predicts that the full family of elliptic curves over $K$ grows faster.

Relation with known results

Ribet characterizes non-CM $\mathbb Q$-curves as geometric factors of abelian varieties of $\mathrm{GL}_2$-type over $\mathbb Q$. Modular-curve and Chabauty methods determine $\mathbb Q$-curve loci for specified degrees, and modern computations determine quadratic points on many individual curves $X_0(N)$. These are level-by-level results. They do not give a count uniform in the degree of the isogeny to the conjugate.

There is no general asymptotic count of all elliptic curves over a quadratic field by conductor norm. Even over $\mathbb Q$, conductor-aspect asymptotics are known only in restricted families. Consequently, neither the numerator nor the denominator in Conjecture 1 is presently accessible at the required precision.

Main obstacles

A proof needs a numerator estimate that beats the expected growth of the ambient family. One possible target is

$$\#\mathcal Q_K(X)=o\bigl(X^{5/6}(\log X)^{-C_K}\bigr),$$

paired with a lower bound of the indicated size for non-$\mathbb Q$ curves. Establishing such an estimate requires uniform control over

• all squarefree central $\mathbb Q$-curve degrees;

• rational points on the associated Atkin–Lehner quotients and twists;

• all quadratic twists of every geometric class;

• the passage from geometric or modular height to conductor norm.

A uniform bound on possible $\mathbb Q$-curve degrees would not by itself solve the problem, because each surviving geometric class has infinitely many twists. Conversely, a sufficiently strong conductor-to-height inequality would permit geometric thinness results to enter, but such inequalities are tied to unresolved Szpiro/abc-type phenomena.

References

• K. A. Ribet, Abelian varieties over $\mathbb Q$ and modular forms, in Algebra and Topology 1992, Korea Advanced Institute of Science and Technology, 1992, 53–79.

• J. E. Cremona and F. Najman, $\mathbb Q$-curves over odd degree number fields, Res. Number Theory 7 (2021), Paper No. 62.

• N. Adžaga, T. Keller, P. Michaud-Jacobs, F. Najman, E. Özman and B. Vukorepa, Computing quadratic points on modular curves $X_0(N)$, Math. Comp. 94 (2025), 1501–1535.

• A. N. Shankar, A. Shankar and X. Wang, Families of elliptic curves ordered by conductor, Compos. Math. 157 (2021), 1538–1583.

• B. S. Banwait, F. Najman and O. Padurariu, Cyclic isogenies of elliptic curves over fixed quadratic fields, arXiv:2206.08891.