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Supersingular isogeny graphs and Hecke modules with level structure

Leonardo Colò, David Kohel

Abstract

We study supersingular isogeny graphs with level structure and their associated Galois representations.

Supersingular isogeny graphs and Hecke modules with level structure

Abstract

We study supersingular isogeny graphs with level structure and their associated Galois representations.

Paper Structure

This paper contains 4 sections, 5 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Graphs with level structure
  3. Explicit Isogeny graphs
  4. Hecke modules on supersingular points and monodromy

Key Result

Proposition 1

Let $G$ be an open subgroup of $\mathrm{GL}_2(\widehat{\mathbb{Z}})$ of level $N$. For $M \in \mathbb{N}$, let $\pi_M: \mathrm{GL}_2(\widehat{\mathbb{Z}}) \rightarrow \mathrm{GL}_2(\mathbb{Z}/M\mathbb{Z})$ be the reduction map, and set $G_M = \pi_M^{-1}(\pi_M(G))$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Supersingular $2$-isogeny twisted Cartan graphs covers for $X_{ns}^t(2) \to X(1)/\mathbb{F}_{11^2}$
  • Figure 2: Supersingular $3$-isogeny graph of level $5$ over $\mathbb{F}_{2^2}$

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • Proposition 6