Inertial Weil–Deligne Types in Elliptic Curves

• Inertial Weil–Deligne types are invariants derived from restricting 2-dimensional local Galois representations, capturing both the Weil group action and a nilpotent monodromy operator.
• They are explicitly classified into principal series, Steinberg, and supercuspidal types, each corresponding to distinct elliptic curve reduction types and conductor exponents.
• This classification underpins local-global compatibility in the Langlands program and supports precise computations in elliptic curve arithmetic and deformation theory.

An inertial Weil–Deligne type is a character-theoretic invariant associated to the restriction of a local Galois representation, arising from the $\ell$-adic cohomology of an elliptic curve over a local field, to the inertia subgroup. For an elliptic curve $E$ over $F = \mathbb{Q}_\ell$ or $F = \mathbb{Q}_{p^2}$, the attached 2-dimensional Weil–Deligne representation $(r, N)$ encodes both the action of the Weil group $W_F$ and the monodromy operator $N$, with the inertial type being the isomorphism class of the pair $(r|_{I_F}, N)$. The classification of all inertial Weil–Deligne types arising in this context is both explicit and exhaustive; it is central in the study of local-global compatibility in the Langlands program, the analysis of deformation rings, and the moduli of elliptic curves.

1. Definition and Structure of Inertial Weil–Deligne Types

Given a finite extension $F$ of $\mathbb{Q}_p$, the absolute Galois group $G_F = \operatorname{Gal}(\overline{F}/F)$ has the Weil group $W_F$ as a dense subgroup, with inertia $I_F \subset W_F$ as the kernel of the natural projection to the Galois group of the residue field. Let $E$ be an elliptic curve over $F$, and fix $\ell \ne p$ a prime. The $\ell$-adic Tate module $T_\ell E$ gives rise to a continuous 2-dimensional representation $r : W_F \to \operatorname{GL}_2(\mathbb{C})$ with open kernel; it comes equipped with a nilpotent monodromy operator $N$ of order at most $2$, satisfying the relation $r(g) N r(g)^{-1} = |g| N$ for the norm character $|\cdot|$.

The pair $(r, N)$ is a Weil–Deligne representation; its inertial type is $(r|_{I_F}, N)$ up to equivalence. When $N = 0$, the type is called unipotent. The conductor exponent of $E$ is determined by this inertial type and, in the context of elliptic curves, is always less than or equal to $2$ except in certain wild cases for $\ell = 2$ or $3$.

2. Classification of 2-Dimensional Inertial Types

Every such inertial Weil–Deligne type arising from $E/F$ falls into one of the following categories:

1. Principal Series: $(r, N)$ is a direct sum $\chi_1 \oplus \chi_2$, $N=0$, with $\chi_1, \chi_2$ quasicharacters of $W_F$, $\chi_1/\chi_2 \neq |\cdot|^{\pm 1}$. Restricting to inertia, $\tau = \chi_1|_{I_F} \oplus \chi_2|_{I_F}$.
2. Special (Steinberg): $r = \chi \otimes \operatorname{St}_2$, $N \neq 0$ of rank $1$. On inertia, $\tau = (\chi|_{I_F}) \otimes (\mathbbm{1} \oplus \mathbbm{1})$, $N$ is the standard $2\times2$ nilpotent Jordan block.
3. Supercuspidal: $N=0$, $r$ irreducible:
   • Non-exceptional: $r = \operatorname{Ind}_{W_K}^{W_F}\psi$ for $K/F$ quadratic, and $\psi$ a quasicharacter of $W_K$ not factoring through the norm.
   • Exceptional (for $p=2$): Projective image $A_4$ or $S_4$, constructed from specific cubic-quadratic extensions; these types only occur for certain wild ramification invariants.

This partition is exhaustive; each inertial type uniquely determines a reduction type for $E$ via the Kodaira–Néron classification (Dembélé et al., 2022, Castro-Moreno et al., 4 Dec 2025).

3. Explicit Correspondence with Kodaira–Néron Reduction Types

The reduction type of $E$ dictates the inertial type:

• Good Reduction: Inertia acts trivially, $N=0$—the trivial type.
• Split or Non-split Multiplicative Reduction: $\tau$ is Steinberg type, $N \neq 0$; if over a ramified quadratic extension, a quadratic twist appears.
• Additive but Potentially Multiplicative: Special type with quadratic twist by the discriminant of the relevant quadratic extension.
• Additive but Potentially Good Reduction: $N=0$, $r|_{I_F}$ finite image of order $e_E = [L : F^{\operatorname{ur}}]$, with the precise family determined by divisibility relations: $e_E\mid(p-1)$ indicates principal series, while $e_E\mid(p+1)$ gives non-exceptional supercuspidal, and $e_E=24$ (for $p=2$) gives exceptional supercuspidal (Dembélé et al., 2022, Castro-Moreno et al., 4 Dec 2025).

4. Concrete Parameterizations and the Influence of Residue Characteristic

The explicit parameterization depends on $p$ and $F$:

• For $p \geq 5$: Additive potentially good reduction only allows orders $e \in \{3,4,6\}$ in the inertia quotient; principal series appear if $e\mid(p-1)$, supercuspidals if $e\mid(p+1)$.
• For $p=3$: Possible inertia orders are $3,4,6,12$; both principal series and (ramified or unramified) supercuspidal types occur, as well as their twists. Each is explicitly constructed via characters on inertia of prescribed order and conductor, and listed in (Dembélé et al., 2022, Castro-Moreno et al., 4 Dec 2025).
• For $p=2$ or $F = \mathbb{Q}_4$: Wild ramification allows inertia orders up to $24$. Both principal series and non-exceptional supercuspidal types are realized by induction from quadratic extensions or their explicit characters. Exceptional (primitive) types appear only for $e=8$ and $e=24$; these have projective image $Q_8$ or $\operatorname{SL}_2(\mathbb{F}_3)$. Parameterizations use explicit data on unramified and ramified quadratic (and, for exceptional types, cubic) extensions.

Types, orders, conductors, and parameterizations are tabulated in full in (Dembélé et al., 2022, Castro-Moreno et al., 4 Dec 2025).

5. The Case of Unramified Quadratic Base Field

For $F = \mathbb{Q}_{p^2}$, the classification mirrors the $\mathbb{Q}_p$ case with refinements due to the larger unramified base:

• Principal Series: Types $\operatorname{PS}(1,1,e)$, parameterized by characters of $I_F$ of precise order dividing $q\pm1$, where $q=p^2$.
• Steinberg: As above, for potentially multiplicative reduction.
• Unramified and Ramified Supercuspidal: Induced from quadratic extensions (unramified or ramified); parameterization involves explicit choice of $\theta$ with prescribed order and restriction to inertia.
• Exceptional Types ($p=2$): Triply-imprimitive supercuspidals, built from a minimal cubic $K/F$ and a character on a quadratic $M/K$ of prescribed conductor.

The semistability defect $e$ remains the key invariant; possible values and the resulting type are tabulated (see (Castro-Moreno et al., 4 Dec 2025), Tables 1–10).

6. Summary Table of Inertial Types by Reduction Type

Reduction Type	Inertial Type	Conductor Exponent
Good reduction	Trivial	0
Multiplicative (split/non-split)	$\operatorname{St}(1)$ or $\operatorname{St}(\varepsilon_K)$	1 or 2
Additive, potentially multiplicative	$\operatorname{St}(1)\otimes\varepsilon_d$	2, 4, or 6 (wild)
Additive, potentially good	Principal Series or (Supercuspidal, Exceptional)	2, possibly higher

All such types are realized by explicit curves of matching Kodaira–Néron reduction, and are fully classified in (Dembélé et al., 2022, Castro-Moreno et al., 4 Dec 2025).

7. Applications and Explicit Computations

The explicit description and classification of inertial Weil–Deligne types underlies multiple arithmetic applications:

• The determination of local factors, conductors, and root numbers in $L$-functions attached to elliptic curves (Dembélé et al., 2022).
• Local-global compatibility for modular and automorphic Galois representations.
• Exhaustiveness checks and explicit curves for each inertial type and conductor exponent, including explicit modular forms and tables for characteristic $2$ and $3$ wild ramification (Castro-Moreno et al., 4 Dec 2025).
• Algorithmic and computational verification using explicit local field extensions, as tabulated in supplementary computational materials.

The methodology combines local class field theory, explicit determination of the image of Galois on torsion points, and verification against Tate's algorithm (Dembélé et al., 2022, Castro-Moreno et al., 4 Dec 2025).