Locally imprimitive points on elliptic curves

1. Problem Statement and Context

The paper investigates the Lang-Trotter conjecture (Case III) regarding elliptic curves, specifically focusing on the existence of locally imprimitive points.

• Background:

  • Case I (Multiplicative): Under the Generalized Riemann Hypothesis (GRH), a globally primitive element $x \in K^*$ (not a perfect power) is a primitive root modulo a set of primes of positive density (Hooley, 1967).
  • Case II (Cyclic Reduction): For an elliptic curve $E/K$, the set of primes where the reduction $E(\mathbb{F}_p)$ is cyclic has a conjectural positive density (Serre).
  • Case III (Elliptic Primitive Root): For a point $P \in E(K)$ of infinite order, does $P$ generate the group $E(\mathbb{F}_p)$ for a set of primes of positive density?

• The Core Question: Can a point $P$ be globally primitive (not contained in any proper cyclic subgroup of $E(K)$ generated by roots of $P$) yet locally imprimitive (fail to generate $E(\mathbb{F}_p)$ for almost all primes $p$)?

• The Phenomenon: While obstructions to local primitivity in Cases I and II are well-understood (often "trivial" obstructions like the field containing roots of unity or full torsion), the authors identify a non-trivial obstruction unique to Case III. This obstruction arises from the specific structure of the Galois representation associated with the point $P$.

2. Methodology

The authors employ a combination of Galois representation theory, algebraic geometry, and explicit construction of isogenies.

• Galois Representations:

  • They define the $m$-division field of a point $P$ as $K_m = K(m^{-1}P)$, the field generated by points $Q$ such that $mQ \in \langle P \rangle$.
  • This yields a Galois representation $\rho_{E/K,P}: G_K \to \text{GL}_3(\hat{\mathbb{Z}})$.
  • The condition that $P$ generates $E(\mathbb{F}_p)$ is translated into a condition on the Frobenius element $\text{Frob}_p$: specifically, $\text{rank}(\text{Frob}_p - I) \geq 2$ in the 3-dimensional vector space $V_\ell = \langle \ell^{-1}P \rangle / \langle P \rangle$.
  • If the image of the Galois group consists entirely of matrices with rank $(\sigma - I) \leq 1$, the density of such primes is zero.

• Characterization of Obstructions (Theorem 1.2):
  The authors prove that a non-torsion point $P$ is locally $\ell$-imprimitive if and only if one of three conditions holds:

  • Condition A (Trivial): $E(K)$ has an $\ell$-torsion point and $P \in \ell E(K)$.
  • Condition B (Trivial): $E$ has full rational $\ell$-torsion ($E[\ell] \subset E(K)$).
  • Condition C (Non-trivial): There exists an $\ell$-isogeny $\phi: E' \to E$ defined over $K$ with kernel generated by an $\ell$-torsion point, such that $P \in \phi(E'(K))$.
  • Key Insight: Condition C is unique to elliptic curves. It implies that while $P$ is not an $\ell$-multiple in $E(K)$, it is the image of a point under an isogeny, causing its reduction to always lie in a specific subgroup of index divisible by $\ell$.

• Explicit Constructions:

  • The authors use Weierstrass models and isogeny formulas (e.g., Vélu's formulas) to construct families of curves satisfying Condition C.
  • They utilize Hilbert's Irreducibility Theorem to show that for infinitely many specializations of parameters, the resulting points are globally primitive but locally imprimitive.

3. Key Contributions and Results

A. Theoretical Characterization (Theorem 1.2)

The paper provides a complete classification of when a point $P$ fails to be a local generator. Crucially, it identifies Condition C as a "non-trivial" obstruction. Unlike Cases I and II, where vanishing density usually implies trivial arithmetic reasons (like $K$ containing $\zeta_\ell$), Condition C allows for local imprimitivity even when $E(K)$ has no $\ell$-torsion and $P$ is not an $\ell$-multiple.

B. Construction of Non-Trivial Examples

The authors construct infinite families of such points for various primes $\ell$:

• Case $\ell = 2$ (Theorem 1.3):

  • For any elliptic curve $E/K$ with exactly one rational point of order 2, there are infinitely many quadratic twists $E_D$ possessing a non-trivial locally 2-imprimitive point.
  • The construction relies on the explicit description of 2-isogenies and the fact that the $x$-coordinate of the image point must be a square.

• Case $\ell = 3$ (Theorem 6.2):

  • Using Deuring normal forms, the authors construct a 1-parameter family of curves $E_b$ over $K(X)$ with a point $P_b$ that is globally 3-primitive but locally 3-imprimitive.
  • They prove that for infinitely many specializations $t \in K$, the specialized point remains non-torsion and retains the local imprimitivity property.

• Cases $\ell = 5, 7$ (Section 7):

  • Using Tate's normal forms, they provide explicit (though computationally heavy) formulas for curves with locally 5-imprimitive and 7-imprimitive points.
  • They provide concrete numerical examples over $\mathbb{Q}$ (e.g., curve 5835.c2 for $\ell=5$ and 20622.j1 for $\ell=7$) where the generator of the Mordell-Weil group is locally imprimitive.

C. Composite Level Obstructions (Section 8)

The paper discusses obstructions arising from the entanglement of division fields at composite levels (e.g., level 6).

• They construct an example over $\mathbb{Q}$ (curve 12100.j1) where a point $P$ is never a primitive root because the Frobenius elements are forced to fix subspaces due to the interaction between the 2-division and 3-division fields.
• This demonstrates that obstructions can exist even when no single prime level $\ell$ provides an obstruction, a phenomenon not seen in the multiplicative case over $\mathbb{Q}$.

4. Significance

1. Refutation of a Naive Analogue: The results show that the direct analogue of Artin's primitive root conjecture (that globally primitive elements are locally primitive at a positive density of primes) fails for elliptic curves. There exist globally primitive points that are locally imprimitive at all primes (density 0).
2. New Arithmetic Phenomenon: The identification of Condition C reveals a structural property unique to elliptic curves involving isogenies. This highlights a fundamental difference between the multiplicative group $K^*$ and the group of points $E(K)$.
3. Density Theory: The paper clarifies the conditions under which the conjectural density $\delta_{E/K,P}$ vanishes. It moves beyond "trivial" vanishing (due to torsion) to "structural" vanishing (due to isogenies and field entanglement).
4. Explicit Examples: By providing explicit equations and specific curves from the LMFDB database, the authors move the theory from abstract existence to concrete verification, allowing for further computational study.

Conclusion

Jones, Pappalardi, and Stevenhagen demonstrate that the Lang-Trotter problem is significantly more subtle than its multiplicative or cyclic reduction counterparts. They prove that locally imprimitive points are not just rare anomalies but can be systematically constructed via isogenies. This work fundamentally alters the understanding of the distribution of primitive points on elliptic curves, showing that global primitivity does not guarantee local primitivity, even under GRH.