CM Elliptic Curves over Real Quadratic Fields

• CM elliptic curves over real quadratic fields are elliptic curves with complex multiplication, defined by orders in imaginary quadratic fields and characterized by j-invariants 0 and 1728.
• Modular curve fiber analysis reveals that only specific CM j-invariants occur, leading to a precise classification of torsion subgroups and clear insights into Galois and ramification properties.
• Applications include explicit torsion subgroup realization, linking Mahler measures to L-values, and providing arithmetic evidence supporting Beilinson’s conjecture.

Elliptic curves with complex multiplication (CM) over real quadratic fields constitute a rich area at the intersection of arithmetic geometry, algebraic number theory, and the theory of modular forms. The study of these curves involves classifying possible torsion subgroups, understanding their modular curve moduli, computing explicit examples, and relating their special $L$-values to Mahler measures and Beilinson regulators. This research enables a complete description of possible torsion, Galois and ramification properties, and deep connections to special values of $L$-functions.

1. Classification of CM Elliptic Curves over Real Quadratic Fields

CM elliptic curves are defined as elliptic curves $E$ for which $\operatorname{End}(E)$ is an order in an imaginary quadratic field $K$. The moduli of such curves are controlled via modular curves $X_0(M,N)$ and $X_1(M,N)$, and their $j$-invariants lie in the ring class fields of such orders.

Over real quadratic fields $F$ (i.e., $[F:\mathbb{Q}] = 2$ and $F \subset \mathbb{R}$), a critical phenomenon is that only very particular CM $j$-invariants can be defined over $F$. Specifically, only those corresponding to discriminants $\Delta = -3$ and $\Delta = -4$ yield $j$-invariants in $\mathbb{Q}$ and hence in any $F$. For all $\Delta < -4$ or non-fundamental orders of discriminant $f^2 \Delta_K$ with $f > 1$, the field $\mathbb{Q}(j_\Delta)$ has even degree $>2$ over $\mathbb{Q}$; thus such $j$-invariants do not occur in real quadratic fields (Clark et al., 2022).

The two principal CM $j$-invariants defined over real quadratics are:

• $j = 1728$ ($\Delta = -4$), corresponding to the "square" curve $y^2 = x^3 - x$, endomorphism ring $\mathbb{Z}[i]$.
• $j = 0$ ($\Delta = -3$), corresponding to the "hexagonal" curve $y^2 = x^3 + 1$, endomorphism ring $\mathbb{Z}[\zeta_3]$.

No other imaginary quadratic discriminant produces a real quadratic ring class field or $j$-invariant (Clark et al., 2022).

2. Modular Curve Structure and Galois Theory

The structure of fibers of modular curves at CM points is central for understanding both classification and fields of definition. For positive integers $M \mid N$, let

$\pi_1: X_1(M,N) \rightarrow X(1)$

be the natural forgetful morphism. The fiber over a CM $j$-invariant $J_\Delta$ (where $\Delta$ is the discriminant of the CM order) can be described combinatorially in terms of non-backtracking paths in the $\ell$-isogeny volcano graph, satisfying congruences corresponding to level structure (Clark, 2022).

A fundamental result is that for $\Delta_K < -4$, each fiber of $X_1(M,N) \rightarrow X_0(M,N)$ over a CM point is connected (i.e., "inert")—the fiber is always a single field extension (Clark, 2022). Thus, for $\Delta = -3, -4$ the fiber is similarly connected, but occurs over the unique field $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(i)$ respectively, with $[K:\mathbb{Q}] = 2$, and totally ramified at $3$ or $2$ respectively.

The table below summarizes the key fiber degrees for CM points:

$\Delta$	$j_\Delta$	Ring Class Field $H_\Delta$	$[H_\Delta:\mathbb{Q}]$	Ramification
$-4$	1728	$\mathbb{Q}(i)$	$2$	2 only
$-3$	$0$	$\mathbb{Q}(\sqrt{-3})$	$2$	3 only

No other discriminants yield real quadratic subfields of their ring class fields (Clark et al., 2022).

3. Torsion Subgroup Classification and Growth Criteria

The possible torsion structures for CM elliptic curves over real quadratic fields are sharply constrained. Over $\mathbb{Q}$, Olson’s list gives $$\operatorname{CM}(1) = \{\mathbb{Z}/n : n=1,2,3,4,6\} \cup \{\mathbb{Z}/2 \times \mathbb{Z}/2\}$$. Upon base-changing to any quadratic extension—including real quadratic—additional torsion can appear. The full list for quadratic fields is

$$\operatorname{CM}(2) = \left\{ \mathbb{Z}/1,\,\mathbb{Z}/2,\,\mathbb{Z}/3,\,\mathbb{Z}/4,\,\mathbb{Z}/6,\,\mathbb{Z}/2 \times \mathbb{Z}/2,\,\mathbb{Z}/2 \times \mathbb{Z}/4,\,\mathbb{Z}/2 \times \mathbb{Z}/6,\,\mathbb{Z}/3\times \mathbb{Z}/3 \right\}$$

with no occurrence of order $7$ or $10$ torsion in any quadratic field (González-Jiménez, 2019).

A key result is that growth of torsion from a base CM curve $E/\mathbb{Q}$ to a real quadratic field $F$ occurs if and only if $F$ is the real quadratic subfield of the ring class field attached to the CM order of $E$. This holds uniformly for all thirteen classical CM $j$-invariants (González-Jiménez, 2019). For example, for $j=0$ and $E$ with $\mathbb{Z}[\zeta_3]$ multiplication, $E(\mathbb{Q})_{\mathrm{tors}} \simeq \mathbb{Z}/6$, but over $\mathbb{Q}(\sqrt{3})$, $$E(\mathbb{Q}(\sqrt{3}))_{\mathrm{tors}} \simeq \mathbb{Z}/2 \times \mathbb{Z}/6$$.

The precise classification—extracting from the modular curve fiber analysis—states that possible CM torsion over real quadratic fields consists of

$$\mathbb{Z}/1,\,\mathbb{Z}/2,\,\mathbb{Z}/3,\,\mathbb{Z}/4,\,\mathbb{Z}/6,\,\mathbb{Z}/7,\,\mathbb{Z}/8,\,\mathbb{Z}/2\times\mathbb{Z}/2,\,\mathbb{Z}/2\times\mathbb{Z}/4,\,\mathbb{Z}/2\times\mathbb{Z}/6,\,\mathbb{Z}/2\times\mathbb{Z}/8,$$

with each group arising from explicit small-discriminant curves (Clark, 2022).

4. Explicit Examples and Realization of Torsion

Every possible CM torsion group over a real quadratic field is attained by a specific explicit CM $j$-invariant and associated curve. These Weierstrass models realize the catalogue of torsion in a concrete fashion.

Examples include:

• $j = 0$: $E : y^2 = x^3 - 1$, $E(\mathbb{Q})_\mathrm{tors} = \mathbb{Z}/6$, over $\mathbb{Q}(\sqrt{3})$, $$E(\mathbb{Q}(\sqrt{3})) \simeq \mathbb{Z}/2 \times \mathbb{Z}/6$$.
• $j = 1728$: $E : y^2 = x^3 - x$, $E(\mathbb{Q})_\mathrm{tors} = \mathbb{Z}/4$, over $\mathbb{Q}(\sqrt{2})$, $$E(\mathbb{Q}(\sqrt{2})) \simeq \mathbb{Z}/2 \times \mathbb{Z}/4$$.
• $j = -3375$: $E : y^2 = x^3 - 21x + 28$, $E(\mathbb{Q}(\sqrt{7})) \simeq \mathbb{Z}/7$.
• $j = 8000$: $E : y^2 = x^3 + 16x$, over $\mathbb{Q}(\sqrt{2})$, $$E(\mathbb{Q}(\sqrt{2})) \simeq \mathbb{Z}/2 \times \mathbb{Z}/4$$.
• $j = -32768$: $E$ of order 11 torsion over $\mathbb{Q}(\sqrt{11})$.

These match the modular curve fiber analysis precisely, and infinite families exist for discriminants $\Delta = -\ell^{2L+1}$ or $-4\ell^{2L+1}$, $\ell \equiv 3 \pmod{4}$, yielding $\mathbb{Z}/\ell$ or $\mathbb{Z}/2 \times \mathbb{Z}/\ell$ torsion over $\mathbb{Q}(\sqrt{\ell})$ depending on residue class mod $8$ (Clark, 2022).

5. Fiber Analysis, Galois Orbits, and Isogeny Volcanoes

The modular curve approach analyzes the fiber structure of $\pi_1: X_1(M,N) \rightarrow X(1)$ over CM $j$-points. For each positive integer $d$ (degree of number field), the torsion is controlled by Galois theory of the corresponding fiber: possible points correspond to "non-backtracking paths" in the volcano graph associated to the isogeny structure (Clark, 2022).

For CM elliptic curves over real quadratics, these paths—subject to level congruence conditions—produce fibers that are in bijection with isogeny classes respecting real structure. The residual degrees of these points are fully determined by the residue fields and the structure of the isogeny volcano. The shape of the volcano (number of levels, ramification at surface and depth) dictates the possible field extensions over which additional torsion appears.

A key fact is the "inertness" of the map $X_1(M,N) \rightarrow X_0(M,N)$ at CM points with $\Delta < -4$: every CM point on $X_0(M,N)$ lifts to exactly one CM point on $X_1(M,N)$, preventing any splitting of CM fibers in real quadratic extensions (Clark, 2022).

6. Mahler Measures, $L$-values, and Beilinson's Conjecture

Recent advances have linked the special $L$-values of CM elliptic curves over real quadratic fields to explicit determinants of Mahler measures. Given the polynomial

$P_k(x,y) = x + \frac{1}{x} + y + \frac{1}{y} + k,$

the Mahler measure $m(k)$ is related to the central $L$-values $L(E,2)$ for appropriately chosen $E/F$, where $F$ is a real quadratic field supporting a CM curve.

For five specific pairs $(E,F)$ (with $F = \mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$, $\mathbb{Q}(\sqrt{7})$ and CM order of class number 1), the formula

$$\det \begin{pmatrix} m(k_1) & m(k_2) \ m(k_2) & m(k_1) \end{pmatrix} = \pi^4 L(E,2)$$

holds, with explicit models, $j$-invariants, and conductors given for each $E$ (Tao et al., 2022). Beilinson's conjecture anticipates such relations, positing that the determinant of a 2-dimensional regulator pairing on $K_2(E)$ corresponds, up to $\mathbb{Q}^\times$, with $\pi^{-4} L(E,2)$. The Tao-Guo-Wei work confirms these predictions for the relevant CM cases.

7. Implications and Further Developments

The described classification for CM elliptic curves over real quadratic fields is both complete and explicit:

• For $\Delta = -3, -4$, every real quadratic field contains the $j$-invariants $0$, $1728$, and thus the corresponding CM curves.
• All possible torsion configurations are listed and realized explicitly.
• Growth of torsion over real quadratic fields is fully determined by inclusion of the field as a real quadratic subfield of the ring class field.
• The modular curve fiber and isogeny volcano machinery provides a robust combinatorial and Galois-theoretic account of these phenomena.
• The Mahler measure–$L$-value formulas offer a deep arithmetic bridge to regulators and Beilinson’s conjectures.

This suggests that for higher degree fields or non-CM elliptic curves, similarly explicit moduli-theoretic and arithmetic descriptions may demand further advances. For real quadratic fields, the situation is now fully characterized by the cited results (Clark, 2022, Clark et al., 2022, González-Jiménez, 2019, Tao et al., 2022).