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Deep hole lattices and isogenies of elliptic curves

Lenny Fukshansky, Pavel Guerzhoy, Tanis Nielsen

TL;DR

For the period lattice corresponding to an isomorphism class of elliptic curves, this work produces a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of $\operatorname{SL}_2(\mathbb{Z})$ on the upper halfplane.

Abstract

Given a lattice $L$ in the plane, we define the affiliated deep hole lattice $H(L)$ to be spanned by a shortest vector of $L$ and a deep hole of $L$ contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices. In particular we investigate conditions on $L$ under which $H(L)$ is well-rounded and prove that $H(L)$ is defined over the same field as $L$. For the period lattice corresponding to an isomorphism class of elliptic curves, we produce a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of $\operatorname{SL}_2(\mathbb{Z})$ on the upper halfplane. In the case of CM elliptic curves, we prove that all elliptic curves generated by this sequence are isogenous to each other and produce bounds on the degree of isogeny. Finally, we produce a counting estimate for the planar lattices with a prescribed deep hole lattice.

Deep hole lattices and isogenies of elliptic curves

TL;DR

For the period lattice corresponding to an isomorphism class of elliptic curves, this work produces a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of on the upper halfplane.

Abstract

Given a lattice in the plane, we define the affiliated deep hole lattice to be spanned by a shortest vector of and a deep hole of contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices. In particular we investigate conditions on under which is well-rounded and prove that is defined over the same field as . For the period lattice corresponding to an isomorphism class of elliptic curves, we produce a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of on the upper halfplane. In the case of CM elliptic curves, we prove that all elliptic curves generated by this sequence are isogenous to each other and produce bounds on the degree of isogeny. Finally, we produce a counting estimate for the planar lattices with a prescribed deep hole lattice.
Paper Structure (4 sections, 11 theorems, 63 equations, 2 figures)

This paper contains 4 sections, 11 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometry of deep hole lattices
  3. Deep hole lattices in the fundamental strip
  4. Counting similarity classes with a prescribed deep hole

Key Result

Theorem 1.1

Let $L$ be a lattice in the plane with the angle $\theta \in [\pi/3,\pi/2]$ and successive minima $\lambda_1$ and $\lambda_2 = \alpha \lambda_1$ for some $\alpha \geq 1$. Let $H(L)$ be the deep hole lattice of $L$. The following statements hold:

Figures (2)

  • Figure 1: Space of lattices in ${\mathbb R}^2$ with WR and semi-stable subregions marked by colors.
  • Figure 2: Similarity classes with a prescribed deep hole. Pink lines are radii of the circle centered at $\tau_0$. The brown line $y = \frac{b}{a} x$ intersects the green arc at a point $\tau = a+bi$ defined over $K$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 3.1
  • Lemma 4.1
  • ...and 1 more