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Inertial Types of Elliptic Curves Over $\mathbb{Q}_{p^{2}}$

Jose Castro-Moreno, Nuno Freitas

TL;DR

This work delivers a complete, explicit classification of inertial Weil–Deligne types arising from elliptic curves over the unramified quadratic extension $\mathbb Q_{p^2}$ for all primes $p$. It separates the analysis into the tame case $p\ge5$, the $p=3$ case, and the $p=2$ case (including exceptional and triply imprimitive representations), with detailed treatment of quadratic twists and their effect on inertial data. The authors provide exhaustive realizations by constructing elliptic curves over $\mathbb Q_{p^2}$ corresponding to every inertial type, together with explicit base-change towers, character data, and Magma resources. The results have applications to modularity questions, representation theory of $G_{F}$, and Diophantine problems, and extend prior work on inertial types over $\mathbb Q_p$ to the quadratic extension setting. The paper also supplies extensive computational infrastructure (MAGMAFiles) to realize and identify inertial types from given elliptic curves and to study inertia fields explicitly.

Abstract

We provide a complete, explicit description of the inertial Weil-Deligne types arising from elliptic curves over $\mathbb{Q}_{p^{2}}$ for p prime

Inertial Types of Elliptic Curves Over $\mathbb{Q}_{p^{2}}$

TL;DR

This work delivers a complete, explicit classification of inertial Weil–Deligne types arising from elliptic curves over the unramified quadratic extension for all primes . It separates the analysis into the tame case , the case, and the case (including exceptional and triply imprimitive representations), with detailed treatment of quadratic twists and their effect on inertial data. The authors provide exhaustive realizations by constructing elliptic curves over corresponding to every inertial type, together with explicit base-change towers, character data, and Magma resources. The results have applications to modularity questions, representation theory of , and Diophantine problems, and extend prior work on inertial types over to the quadratic extension setting. The paper also supplies extensive computational infrastructure (MAGMAFiles) to realize and identify inertial types from given elliptic curves and to study inertia fields explicitly.

Abstract

We provide a complete, explicit description of the inertial Weil-Deligne types arising from elliptic curves over for p prime

Paper Structure

This paper contains 31 sections, 38 theorems, 25 equations, 20 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Computational software
  4. Acknowledgements
  5. Background material
  6. Notation
  7. Weil--Deligne representations
  8. Nonexceptional supercuspidal representations
  9. Exceptional (supercuspidal) representations
  10. Inertial types
  11. Elliptic curves
  12. Quadratic extensions
  13. The effect of quadratic twists
  14. Inertial types for
  15. Inertial Types over
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $E/\mathbb Q_{p^2}$ be an elliptic curve with additive potentially good reduction and inertial type $\tau_E$. (A) If $p=3$, then $\tau_E$ is classified according to Table Table Pot Good p=3; (B) If $p=2$, then $\tau_E$ is exceptional if and only if the 3-torsion field of $E$ has degree 24 over $

Theorems & Definitions (86)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 76 more