Modular Curve X_E⁻(p): Anti-Symplectic Twist

• Modular Curve X_E⁻(p) is the anti-symplectic twist of X(p) that parametrizes elliptic curves with Galois-equivariant, reversed Weil pairings.
• It provides a fine moduli space through anti-symplectic isomorphisms of p-torsion groups, highlighting key local invariants and Hasse principle failures.
• Advanced techniques including p-adic cohomology, explicit birational parametrizations, and Coleman theory link its arithmetic dynamics to equidistribution and rigidity phenomena.

The modular curve $X_E^-(p)$ is a twist of the classical modular curve $X(p)$, intricately connected to the arithmetic of elliptic curves and their Galois representations at the prime $p \geq 3$. Geometrically, its non-cuspidal $\mathbb{Q}$-points parametrize elliptic curves $E'/\mathbb{Q}$ equipped with a Galois isomorphism $\varphi: E[p] \to E'[p]$ that is anti-symplectic with respect to the Weil pairing. Unlike the "direct" ($+$) twist, which classifies symplectic isomorphisms (preserving the Weil pairing), the "reverse" ($-$) twist $X_E^-(p)$ classifies those that reverse the pairing up to a non-square factor. The absence of a natural $\mathbb{Q}$-point for the anti-symplectic twist—in contrast to the direct case—has deep implications for both local and global arithmetic, especially in the context of the Hasse principle and distribution of points.

1. Moduli Interpretation and Galois Structure

The modular curve $X_E^-(p)$ provides a fine moduli space for pairs $(E',\varphi)$ where $E'$ is an elliptic curve over $\mathbb{Q}$ and $\varphi$ is a $G_{\mathbb{Q}}$-equivariant, anti-symplectic isomorphism between the $p$-torsion groups $E[p]$ and $E'[p]$ (Freitas et al., 4 Sep 2025). The anti-symplectic condition means that $\varphi$ inverts the Weil pairing: for nonzero $x, y \in E[p]$, $e_p(\varphi(x),\varphi(y)) = e_p(x, y)^{-1}$. This difference is encoded by a twist of $X(p)$, yielding a cover that is nontrivial precisely when anti-symplectic isomorphisms fail to be realized over the base field. The moduli problem leads, for $p=6$, to the modular curve $X_E^-(6)$, parameterizing elliptic curves reverse $6$-congruent to $E$ (Chen, 2014).

2. Local Points: Classification and Criteria

Determining the existence of local points on $X_E^-(p)$ over various completions of $\mathbb{Q}$ ($\mathbb{Q}_\ell$ for $\ell\neq p$, and $\mathbb{Q}_p$) is governed by explicit invariants and reduction types of $E/\mathbb{Q}_\ell$ (Freitas et al., 4 Sep 2025). If $E/\mathbb{Q}_\ell$ admits a $\mathbb{Q}_\ell$-isogeny of prime degree $q \ne p$ with $(q/p) = -1$, there exists a $\mathbb{Q}_\ell$-point on $X_E^-(p)$. The criteria can be summarized:

Reduction Type	Existence of $\mathbb{Q}_\ell$-point	Invariant/Condition
Potentially multiplicative	Always	$q$-isogeny with $(q/p) = -1$
Good reduction ($e=1$)	Usually, up to special cases	$p \nmid \#E[p](\text{Frob}_\ell)$, $-p\Delta_\ell$ non-square, or such $q$ exists
Non-abelian inertia	Conditional	Explicit congruence and semistability checks

For primes $\ell > 4g^2$ ($g=$ genus of $X(p)$), the curve has points by Hensel’s lemma. In the special case of $p=3$ or $5$, $X_E^-(p)$ is genus zero, ensuring local solubility everywhere.

3. Explicit Models, Fiber Products, and Birational Parametrizations

Explicit equations for $X_E^-(p)$ (notably for $p=6$) are constructed as intersections of quadrics (for generic $n$-congruence twists), or via affine models described by equations $f(x,y,z)=0$, $g(x,y,z)=0$ with coefficients ultimately determined by the invariants of $E$ (Chen, 2014). The modular curve $X_E^-(n)$ is systematically built as a fiber product of lower-level modular curves ($X_E(2)$, $X_E(3)$ for $n=6$), via commutative diagrams relating their symplectic/anti-symplectic structure:

$$\begin{array}{ccc} X_E^-(6) & \longrightarrow & X_E^-(2) \ \downarrow & & \downarrow \ X_E^-(3) & \longrightarrow & X(1) \end{array}$$

The explicit birational models for $X_E^-(6)$ typically involve shifting from squares (direct twist) to cubes (reverse twist) in the parameterization of discriminants, reflecting the altered behavior under the Weil pairing.

4. Reduction Types, Local Invariants, and Symplectic Criteria

The detailed behavior of $X_E^-(p)$ over $\mathbb{Q}_\ell$ depends on the reduction type of $E$ at $\ell$ (potentially multiplicative, good, bad, non-abelian inertia) and further local invariants such as $c_4$, $c_6$, and $\Delta_m$ (factored as $\ell^{v_\ell(\Delta_m)}\tilde{\Delta}$) (Freitas et al., 4 Sep 2025). The local symplectic criterion is simplified:

• Once the $p$-torsion Galois representations are uniquely isomorphic as $G_{\mathbb{Q}_\ell}$-modules, determining symplectic/anti-symplectic nature reduces to a quadratic residue check, e.g.:

$$\left(\frac{\ell}{p}\right)^r \left(\frac{2}{p}\right)^t = 1$$

This refinement streamlines the explicit verification of local points and the determination of the moduli problem for $X_E^-(p)$.

5. p-adic Rigidity, CM Points, and Isolation Phenomena

The work on p-adic isolation phenomena (Habegger, 2012) establishes that for ordinary CM points, there is a uniform lower bound on the $p$-adic distance to subvarieties of modular curves such as $X_E^-(p)$: $$\operatorname{dist}_p(x, X) = \sup_{f \in I(X)} |f(x)|_p \geq \varepsilon > 0$$ for $x$ not lying on $X$, with $\varepsilon$ independent of $x$. The "ordinary reduction" condition is necessary—dropping it allows supersingular CM points to approximate any point arbitrarily closely, resulting in failure of isolation. Thus, in moduli spaces like $X_E^-(p)$, ordinary CM points exhibit rigidity akin to torsion points on semi-abelian varieties. This has deep implications for the philosophy of unlikely intersections and the diophantine structure of Hecke orbits.

6. p-adic Cohomology: Eichler–Shimura Maps and Coleman Theory

The $p$-adic Eichler–Shimura maps constructed in (Camargo, 2021) apply to all modular curves, including twists such as $X_E^-(p)$. The approach employs the Hodge–Tate period map to pull back $GL_2$-equivariant bundles from the flag variety $\mathbb{P}^1$, utilizing the dual BGG resolution and the Faltings extension to relate étale cohomology (via Tate modules) to overconvergent modular forms. Overconvergent ES maps: $$ES_A : H^1_{\text{proét}}(X_E^-(p), A^\delta_{\chi,\text{ét}}\widehat{\otimes}\widehat{\mathcal{O}_X}) \to H^0(X_E^-(p), \omega_E^{\chi+\alpha})$$ provide decompositions interpolating classical and overconvergent cohomology, compatible with Poincaré and Serre pairings—after restriction to the finite-slope part (controlled by higher Coleman theory). This "dictionary" extends to eigenvariety structures and serves as a unifying framework for rigid analytic geometry on $X_E^-(p)$.

7. Global Points, the Hasse Principle, and Density Results

$X_E^-(p)$ often fails the Hasse principle: i.e., possesses points over all local completions $\mathbb{Q}_\ell$ (including the archimedean place) but lacks global $\mathbb{Q}$-points (Freitas et al., 4 Sep 2025). For CM elliptic curves with certain discriminants and primes $p$ satisfying $p \equiv 5 \pmod{8}$ and $(D/p)=1$, the twist $X_E^-(p)$ is a counterexample for infinitely many $p$. Assuming the Frey–Mazur conjecture, the proportion of all rational elliptic curves generating such counterexamples is at least $60\%$, and for $50\%$ of primes. These statistical properties are established via explicit local computations and density theorems.

8. p-adic Dynamics, Equidistribution, and Modular Graphs

Recent results on $p$-adic equidistribution (Pérez-Piña, 25 May 2024) provide a dynamical perspective on $X_E^-(p)$, treating it as a space where packets of closed geodesics (Ihara–Shintani cycles) and Heegner points equidistribute in $p$-adic analytic coverings. The spaces investigated, such as $$T^{(p)}(Y_0(\mathbb{C})) = \Gamma^+\backslash (H \times \mathrm{PGL}_2(\mathbb{Q}_p))$$, model the "unit tangent bundle" for the modular curve in the $p$-adic setting. The natural identification of $p$-isogeny graphs (volcanoes) with reductions of modular curves like $X_E^-(p)$ establishes a bridge between the analytic, combinatorial, and arithmetic properties inherent to the moduli interpretation.

Summary Table: Core Attributes of $X_E^-(p)$

Aspect	Description
Moduli Points	Pairs $(E', \varphi)$, $\varphi$ anti-symplectic Galois isomorphism ($E[p] \cong E'[p]$)
Local Existence Criteria	Explicit via reduction invariants, congruence conditions, type of isogeny, quadratic residue checks
Rigidity (CM, p-adic)	Ordinary CM points are $p$-adically isolated from subvarieties unless contained; supersingular loss of bound
Failure of Hasse Principle	Positive density for $E$ and $p$; counterexamples constructed via local-global analysis
Explicit Equations	Given as intersections of quadrics or explicit affine models (fiber products for $n=2,3$ cases)
p-adic Cohomology	Overconvergent Eichler–Shimura maps, higher Coleman theory, compatibility with duality pairings
Dynamic/Equidistribution	$p$-adic equidistribution of cycles; modular graphs as reductions modeling analytic/geometric structure

The modular curve $X_E^-(p)$ thus synthesizes modern $p$-adic arithmetic geometry, Galois representation theory, explicit moduli problems, and dynamical phenomena, serving as a focal object for the paper of isogeny-based twists, local-global principles, and the distribution of special points on modular curves.