On inertial types of elliptic curves

Technical Summary: On Inertial Types of Elliptic Curves

Problem Statement
The paper addresses the classification of inertial Weil-Deligne types arising from elliptic curves defined over finite extensions $F/\mathbb{Q}_p$. While the third author, along with Dembélé and Voight [12], provided a complete explicit description of these types for the base field $F = \mathbb{Q}_p$ (for $p \neq \ell$), this work extends that classification to arbitrary finite extensions $F/\mathbb{Q}_p$. The study is particularly motivated by the need to understand inertial types in the context of modularity, the classification of Galois representations, and Diophantine equations. A specific challenge arises in the 2-adic and 3-adic cases, where new phenomena emerge, such as exceptional types with images isomorphic to the quaternion group $Q_8$ or $SL_2(\mathbb{F}_3)$, and "triply imprimitive" types that cannot be described by induction from a single quadratic extension.

Methodology
The authors employ a unified approach based on the classification of local Galois representations into principal series, Steinberg, and supercuspidal representations. Unlike previous works that relied on specific generators of unit groups for $\mathbb{Q}_p$, this paper generalizes results to any finite extension $F$.

Key methodological steps include:

1. Characterization via Inertia Fields: The authors establish that an inertial type $\tau$ arising from an elliptic curve is completely determined by its kernel (Theorem 3.4). This implies that the extension of $F$ cut out by $\tau$ uniquely identifies the type, provided the image is compatible with the possible Galois groups of elliptic curve torsion fields (Theorem 2.16).
2. Necessary and Sufficient Conditions: The paper derives necessary conditions for a type to arise from an elliptic curve (Section 5) and proves these conditions are sufficient (Section 6). This sufficiency proof involves constructing continuous representations $\rho$ with specific finite images (subgroups of $GL_2(\mathbb{F}_\ell)$ for $\ell=3$ or $5$) and applying a theorem by Khare and Wintenberger [20] to guarantee the existence of an associated elliptic curve.
3. Handling Exceptional Cases: For $p=2$, the authors handle exceptional types (projective image $A_4$ or $S_4$) by base-changing to a suitable cubic extension $L/F$ where the type becomes non-exceptional (triply imprimitive), applying the developed theory, and then descending the results back to $F$.
4. Algorithmic Implementation: The theoretical classification is translated into an algorithm (Algorithm 8.1) implemented in Magma. This algorithm computes all inertial types by searching for inertia characters $\chi$ satisfying specific order and conductor conditions derived from the main theorems.

Key Results
The paper provides a complete classification of inertial types for elliptic curves with potentially good reduction over $F/\mathbb{Q}_p$:

• For $p \geq 5$: The types are uniform and well-known, limited to principal series and supercuspidal types with images isomorphic to cyclic groups $C_2, C_3, C_4, C_6$.
• For $p = 3$ (Theorem 1.1): An inertial type $\tau$ arises from an elliptic curve if and only if it has trivial determinant and falls into one of three categories:
  • Principal series with image $C_2, C_3, C_4, C_6$.
  • Unramified supercuspidal with image $C_3, C_4, C_6$.
  • Ramified supercuspidal with image $C_3 \rtimes C_4$.
• For $p = 2$ (Theorem 1.3): The classification is more complex due to exceptional types. A type arises from an elliptic curve if it has trivial determinant and is:
  • Principal series or unramified supercuspidal with images $C_2, C_3, C_4, C_6$.
  • Ramified supercuspidal with image $Q_8$, subject to specific imprimitivity conditions depending on whether $\mu_3 \subset F$.
  • Exceptional: Arising from a representation $\rho$ over a cubic extension $L/F$ (satisfying property H1) with image $Q_8$, which is triply imprimitive and Galois-invariant.

A significant finding is that any type satisfying these theorems automatically satisfies the known conductor bound for elliptic curves ($v_F(N_E) \leq 2 + 3v_F(3) + 5v_F(2)$), despite the theorems not imposing a conductor bound explicitly.

Computational Contributions
The authors have implemented a Magma package [6] that computes all inertial types for a given field $F$. They have precomputed and tabulated these types for all quadratic and cubic extensions of $\mathbb{Q}_2$ and $\mathbb{Q}_3$. The algorithm efficiently handles the search for inertia characters and determines the corresponding inertial fields.

Significance and Applications
The paper claims significance in several areas:

1. Completeness and Explicitness: It provides a fully explicit description of inertial types over arbitrary finite extensions, a generalization of the results in [12].
2. Algorithmic Efficiency: By proving that the classification conditions are sufficient, the authors avoid the computationally heavy step of finding an explicit elliptic curve for every type (a bottleneck in [12]). This allows for the rapid generation of a database of all inertial types for fields of degree up to 3.
3. Diophantine Applications: The results are directly applicable to Diophantine equations. The paper notes that partial information about an inertial type (e.g., being supercuspidal induced from a specific quadratic extension) can be used to narrow search spaces in Diophantine problems.
4. Beyond Elliptic Curves: The classification is independent of the source of the representation. The authors highlight an application in the work of Pacetti–Torcomian [19], where their classification was used to determine inertial types of Jacobians of hyperelliptic curves in the study of the Fermat equation $x^5 + y^p = z^3$.

The work establishes that the inertial field determines the type, providing a robust framework for analyzing Galois representations associated with elliptic curves and other arithmetic objects over local fields.