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Inseparable endomorphisms and rank-2 sublattices of the Gross lattice

Yves Aubry, Christelle Vincent

TL;DR

The work reframes Love's question about isogenies $E\to E^{(p)}$ of degree $\ell$ as a study of inseparable endomorphisms, linking degree and trace to lattice data inside the Gross lattice $\operatorname{End}(E)^T$. The authors prove a precise criterion: for a supersingular $E$, an endomorphism of degree $\ell p$ with $\operatorname{trd}=0$ exists if and only if $\operatorname{End}(E)^T$ contains a rank-2 sublattice of determinant $4\ell p$, and this trace-zero condition is essential. The method blends quaternion algebra $B_{p,\infty}$, maximal orders, the commutator ideal $[\mathcal{O},\mathcal{O}]$, and explicit constructions $\alpha=\tfrac12\gamma_1\overline{\gamma}_2-\tfrac14\operatorname{trd}(\gamma_1\overline{\gamma}_2)$ to relate endomorphisms and lattice data; it also gives an explicit $\mathbb{Z}$-basis for $[\mathcal{O},\mathcal{O}]$. Beyond the positive result, the paper presents concrete counterexamples showing the necessity of the trace-zero hypothesis, enriching the understanding of when lattice conditions capture endomorphism existence. Overall, it sharpens the connection between endomorphism theory of supersingular curves and lattice theory in the Gross lattice, with explicit constructions and bases that may aid further arithmetic investigations.

Abstract

We answer a question posed by Love asking about a correspondence between isogenies from a supersingular elliptic curve to its Frobenius base-change and rank-2 sublattices of its Gross lattice. We recast the question as one about the inseparable endomorphisms of the curve, and show that the correspondence holds when the trace of the endomorphism is zero, and may not hold otherwise.

Inseparable endomorphisms and rank-2 sublattices of the Gross lattice

TL;DR

The work reframes Love's question about isogenies of degree as a study of inseparable endomorphisms, linking degree and trace to lattice data inside the Gross lattice . The authors prove a precise criterion: for a supersingular , an endomorphism of degree with exists if and only if contains a rank-2 sublattice of determinant , and this trace-zero condition is essential. The method blends quaternion algebra , maximal orders, the commutator ideal , and explicit constructions to relate endomorphisms and lattice data; it also gives an explicit -basis for . Beyond the positive result, the paper presents concrete counterexamples showing the necessity of the trace-zero hypothesis, enriching the understanding of when lattice conditions capture endomorphism existence. Overall, it sharpens the connection between endomorphism theory of supersingular curves and lattice theory in the Gross lattice, with explicit constructions and bases that may aid further arithmetic investigations.

Abstract

We answer a question posed by Love asking about a correspondence between isogenies from a supersingular elliptic curve to its Frobenius base-change and rank-2 sublattices of its Gross lattice. We recast the question as one about the inseparable endomorphisms of the curve, and show that the correspondence holds when the trace of the endomorphism is zero, and may not hold otherwise.
Paper Structure (9 sections, 6 theorems, 39 equations)

This paper contains 9 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background
  4. Existence of an inseparable endomorphism
  5. Existence of a sublattice of rank $2$
  6. Counter-examples when the trace is not zero
  7. Counter-example when $\ell = p$
  8. Counter-example when $\ell \neq p$ and $\ell$ is prime
  9. Counter-example when $\ell \neq p$ and $\ell$ is composite

Key Result

Theorem 1.1

Let $p$ be a prime and $E$ be a supersingular elliptic curve defined over $\overline{\mathbb{F}}_p$. Then for any positive integer $\ell$, there exists an endomorphism of $E$ of degree $\ell p$ and trace zero if and only if $\operatorname{End}(E)^T$ contains a rank-2 sublattice of determinant $4\ell

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • ...and 1 more