Non-CM Elliptic Curves in Number Fields

• Non-CM elliptic curves are curves whose endomorphism ring equals the integers, distinguishing them in arithmetic geometry.
• The study uses Galois representations and modular curve analysis to bound torsion growth, classify isogeny degrees, and examine division field properties.
• Recent research provides criteria for isogeny degrees, detects non-monogeneity in division fields, and explores Frobenius trace density for these curves.

Non-CM elliptic curves are curves over number fields whose endomorphism ring equals the integers and which therefore lack complex multiplication (CM). The paper of such curves over number fields intersects deep topics in arithmetic geometry, Galois representations, torsion growth, isogenies, division fields, and density conjectures. This article focuses on the arithmetic phenomena and invariants governing non-CM elliptic curves, especially those defined over number fields not admitting rationally defined CM ("no RCM"), and outlines the key theorems, methodologies, and arithmetic consequences known for their torsion, isogeny degrees, division field structure, and Galois-theoretic behaviors.

1. Rationally Defined CM and Non-CM Over Number Fields

An elliptic curve $E/K$ is CM if $\operatorname{End}_{\overline{K}}(E)$ is an order strictly containing $\mathbb{Z}$ (necessarily in an imaginary quadratic field). A quadratic field $K$ is said to admit rationally defined CM (RCM) if some CM elliptic curve is defined over $K$; otherwise, $K$ is "no RCM." As established by Im–Kim (Im et al., 2023), for $K$ quadratic with no RCM and any extension $L/K$ of degree $d$ whose minimal prime divisor exceeds a field-dependent threshold $p_K$, no non-CM $E/K$ can acquire new torsion over $L$. This recovers the case $K=\mathbb{Q}$ and generalizes prior conditional results by removing the necessity of assuming the GRH. The threshold $p_K$ is explicitly constructed as

$$p_K = \max \bigl\{ M(K),\ N_{K},\ \max R(K) \bigr\}$$

where $M(K)$ is Merel's uniform bound for torsion, $N_K$ is a large-prime cutoff (Galois-theoretic), and $R(K)$ is a finite set derived from Momose's finiteness theorem for isogenies of specific types over $K$.

2. Growth of Torsion Subgroups in Extensions

The arithmetic of torsion subgroups of non-CM elliptic curves over number fields is governed by global and local Galois representations, as well as modular curve rational points. By Merel’s theorem, for any $K$, there exists a bound $M(K)$ such that the torsion in $E(K)$ is contained in $E(K)[M(K)]$, i.e., all torsion primes satisfy $\ell \leq M(K)$. Im–Kim’s main result (Im et al., 2023) quantifies torsion growth in extensions as follows:

• For $L/K$ of degree $d$, with all prime divisors of $d$ exceeding $p_K$, one has $E(L)_{\mathrm{tors}} = E(K)_{\mathrm{tors}}$ for any $E/K$. The proof splits by $\ell$ relative to $p_K$: small $\ell$ are controlled by uniform boundedness (Merel), and large $\ell$ can only provide new torsion where the mod-$\ell$ Galois representation falls into an exceptional image (Borel with unipotents), which requires $[L:K]$ to be divisible by primes in $R(K)$—but by construction, this does not occur. The key inputs are Momose’s finiteness for isogenies of type 2 and algebraic control of subgroup structure in $\mathrm{GL}_2(\mathbb{F}_\ell)$.

3. Classification of Isogeny Degrees in Non-CM Curves

The possible degrees of cyclic isogenies for non-CM elliptic curves with rational $j$-invariant over number fields of bounded degree have now been classified in full generality for prime degree extensions (Novak, 5 Nov 2024), with previous results for $d\leq 7$ (Najman, 2015). Let

$$\Psi_{\mathbb{Q}}(d) = \{n \geq 1 : \exists\, E/\mathbb{Q},\ j(E)\in\mathbb{Q},\ \text{cyclic }n\text{-isogeny defined over }[K:\mathbb{Q}]=d \}$$

For $d=1$ (Mazur–Kenku),

$$\Psi_{\mathbb{Q}}(1) = \{1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,21,25,37\}$$

Theorems for prime $p$ yield:

• For $p=3$: $$\Psi_{\mathbb{Q}}(3) = \Psi_{\mathbb{Q}}(1) \cup \{2k: k \text{ odd}\} \cup \{3k: 3|k\} \cup \{28\}$$
• For $p>3$: $$\Psi_{\mathbb{Q}}(p) = \Psi_{\mathbb{Q}}(1) \cup \{p\,k: p|k,\,k \in \Psi_{\mathbb{Q}}(1)\}$$ Galois-theoretic arguments show odd-prime-degree extensions cannot create new odd-prime isogenies except in isolated cases and that new degrees must be multiples of those already present in the base field. For $d \leq 7$, (Najman, 2015) proves unconditionally that no new primes occur beyond Mazur’s list. The case $p=2$ adds degrees $\{20,24,32,36\}$.

4. Division Fields and Monogeneity

Given $E/\mathbb{Q}$ non-CM, the $n$-division field $K_n = \mathbb{Q}(E[n])$ is rarely monogenic. The main theorem of (Smith, 2020) shows infinite families of $E/\mathbb{Q}$ with $K_n$ non-monogenic for specified $n$, and, moreover, for any $E/\mathbb{Q}$ without CM, there exist infinitely many $n$ (specifically $n = p+1$ for supersingular primes $p$) such that $K_n$ is not monogenic. The obstruction is detected via Dedekind’s splitting criterion and the explicit description of Frobenius elements in $\operatorname{Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ per Duke–Tóth. If the splitting type of a prime $p$ in $\mathcal{O}_{K_n}$ cannot match that of any monic irreducible polynomial of degree $[K_n:\mathbb{Q}]$ modulo $p$, then $p$ is an essential discriminant divisor and $K_n$ fails to be monogenic. Explicit algorithms and examples illustrate infinite obstructed families, such as $E_t: y^2 + xy = x^3 + t$ (for $n=11,23$) and semistable families for $n=13,41$. The phenomenon is generic for non-CM curves by the density of supersingular primes.

5. Galois Representations and Genus 0 Adelic Images

The adelic Galois representation

$$\rho_E: \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\widehat{\mathbb{Z}})$$

encodes the action of absolute Galois on all torsion points simultaneously. For non-CM $E/\mathbb{Q}$, Serre’s open-image theorem ensures the representation is open in $\mathrm{GL}_2(\widehat{\mathbb{Z}})$. Rakvi (Rakvi, 2023) classifies all open subgroups of $\mathrm{GL}_2(\widehat{\mathbb{Z}})$ arising as $\pm \rho_E(G_\mathbb{Q})$ (modulo $-I$) with genus 0, listing ninety-eight explicit cases with rational parametrizations of the associated modular curves. The modular curve $X_G$ for $G$ of genus 0 parametrizes elliptic curves whose adelic image lands in $G$, with rational points characterized by explicit $j$-maps and field intersection constraints—a precise link between Galois image and potential rational level structure.

6. Frobenius Trace Density and Lang–Trotter for Products

For non-CM $E/\mathbb{Q}$, the density of primes $p$ such that $a_p(E) = t$ is conjecturally controlled by the Lang–Trotter asymptotic

$\pi_{E,t}(x) \sim C(E,t)\,\frac{\sqrt{x}}{\log x}$

with $C(E,t)$ a product of Sato–Tate, exceptional, and universal (Euler product) factors. The generalization to products $E_1 \times E_2$ (non-CM, non-isogenous) gives joint trace and sum-of-trace statistics:

• For joint traces, $$\pi_{E_1,E_2}(x;t_1,t_2) \sim C_{E_1,E_2}(t_1,t_2)\,x/(\log x)^2$$
• For sum-of-traces, $$\pi_{A,T}(x)\sim C(E_1\times E_2,T)\,\sqrt{x}/\log x$$ Explicit formulas for constants are available. Computational evidence (e.g., (Chen et al., 2020)) demonstrates the predicted distribution and inspects entanglement phenomena—when division fields intersect, certain traces $T$ never occur.

7. Arithmetic Implications and Future Directions

For non-CM elliptic curves, the arithmetic landscape is shaped by uniform boundedness results, Galois representation image structure (full image, Cartan-normalizer exceptions, commutator index), and the distribution of torsion, isogeny degrees, and division field arithmetic. The current classification provides:

• Explicit bounds and lists for torsion growth and cyclic isogeny degrees in prime and small degree extensions.
• Algorithms for detecting non-monogeneity of division fields, with infinite families of obstructions.
• Systematic enumeration of genus 0 modular curve parametrizations of possible adelic images.
• Density conjectures for Frobenius traces in higher-dimensional abelian varieties. Continued research focuses on composite degree extensions, higher genus modular curves, improvements in effective Galois representation uniformity, and the detailed structure of entanglements and exceptional behaviors in division fields and trace densities.