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Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images

Matthew Bisatt, Lorenzo Furio, Davide Lombardo

Abstract

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the local Galois structure of the $p$-power torsion of $E$. We pay special attention to the case where $E$ has potentially supersingular reduction, where we give an algorithm to determine the corresponding filtered $(\varphi,\operatorname{Gal}(K/\mathbb{Q}_p))$-module from a Weierstrass model (which appears to be novel), and introduce alternative division polynomials that may be of independent interest. We deduce global consequences for elliptic curves $E/\mathbb{Q}$: when the mod $p$ representation of $E$ has non-split Cartan image and $E$ doesn't have CM, the $p$-adic image must be the full preimage of the normaliser of a mod $p^n$ non-split Cartan for some $n \geq 1$. As an application, we sharpen existing bounds on the adelic image in terms of the Weil height of the $j$-invariant.

Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images

Abstract

Let be an elliptic curve whose mod Galois image is contained in the normaliser of a non-split Cartan. We classify the possible -adic images of using tools from -adic Hodge theory via a careful analysis of the local Galois structure of the -power torsion of . We pay special attention to the case where has potentially supersingular reduction, where we give an algorithm to determine the corresponding filtered -module from a Weierstrass model (which appears to be novel), and introduce alternative division polynomials that may be of independent interest. We deduce global consequences for elliptic curves : when the mod representation of has non-split Cartan image and doesn't have CM, the -adic image must be the full preimage of the normaliser of a mod non-split Cartan for some . As an application, we sharpen existing bounds on the adelic image in terms of the Weil height of the -invariant.
Paper Structure (38 sections, 67 theorems, 128 equations)

This paper contains 38 sections, 67 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Explicit $p$-adic Hodge theory approach
  3. Relation to existing literature
  4. Layout
  5. Preliminaries
  6. Cartan subgroups and their normalisers
  7. Supersingularity and canonical subgroups
  8. N-Cartan lifts
  9. Description of the Galois image in special cases
  10. Review of Volkov's results
  11. The ring $R$
  12. The ring of Witt bivectors $BW(R)$
  13. The filtered $(\varphi, G_{K/\mathbb{Q}_p})$-modules $D_\alpha$
  14. Description of $E[p^k]$ in terms of the roots of a polynomial
  15. Lattices in $V_\alpha$ and a description of $E[p^k]$
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $E/\mathbb{Q}$ be an elliptic curve without CM and let $p > 7$ be a prime. Suppose that $\operatorname{Im} \rho_{E, p} \subseteq C_{ns}^+(p)$. Then there exists $n \geq 1$ such that where $\pi_n : \operatorname{GL}_2(\mathbb{Z}_p) \to \operatorname{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ is the canonical projection.

Theorems & Definitions (171)

  • Theorem 1.1: =\ref{['thm: no sharp groups over Q']}
  • Theorem 1.2
  • Theorem 1.3: =\ref{['lemma: p-adic image in the twisted crystalline case']} + \ref{['cor: explicit p-adic image']}
  • Remark 1.4
  • Theorem 1.5: =\ref{['thm: p2 torsion points correspond to roots of g']} + \ref{['rmk: twisted Galois action 2']}
  • Theorem 1.6: =\ref{['prop: beta algorithm', 'prop: valuation beta', 'prop: beta parameter', 'prop: epsilon via discriminant']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 161 more