---
title: 'Modular Curve X_E⁻(p): Anti-Symplectic Twist'
url: https://www.emergentmind.com/topics/modular-curve-x_e-p
type: topic
---

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

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 $E'[p]$ [2509.04294]. 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 $E$ [1403.7557].

## 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 $E/\mathbb{Q}_\ell$ [2509.04294]. 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 $E$ [1403.7557]. 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 $\ell^{v_\ell(\Delta_m)}\tilde{\Delta}$) [2509.04294]. 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 [1210.3299] 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 [2102.13099] 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 [2509.04294]. 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 [2405.16032] 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 study of isogeny-based twists, local-global principles, and the distribution of special points on modular curves.

Source: https://www.emergentmind.com/topics/modular-curve-x_e-p