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Doubly isogenous curves of genus two with a rational action of $D_6$

Jeremy Booher, Everett W. Howe, Andrew V. Sutherland, José Felipe Voloch

TL;DR

The paper investigates doubly isogenous genus-$2$ curves with a $D_6$ automorphism action, revealing an overabundance explained by multiple geometric factors arising from unramified covers and Prym varieties. By decomposing Jacobians into elliptic factors via 2-, 3-, and 4-cover data and analyzing special and general triple covers, the authors develop refined heuristics, identify extraordinary phenomena in characteristic zero, and, conditional on the Zilber--Pink conjecture, prove finiteness results. They also construct an algorithm, based on Buium’s differential criterion, to detect simultaneous isogeny correspondences and demonstrate how higher-degree covers can distinguish nonisomorphic doubly isogenous curves. Finally, they connect these geometric insights to a deterministic polynomial-time factoring approach over finite fields, proposing a practically efficient candidate using the D$_6$-family elliptic factors. The work combines explicit algebraic geometry of genus-2 curves with arithmetic statistics and algorithmic number theory to illuminate isogeny phenomena and potential cryptanalytic applications.

Abstract

Let $C$ and $C'$ be curves over a finite field $K$, provided with embeddings $ι$ and $ι'$ into their Jacobian varieties. Let $D\to C$ and $D'\to C'$ be the pullbacks (via these embeddings) of the multiplication-by-$2$ maps on the Jacobians. We say that $(C,ι)$ and $(C',ι')$ are \emph{doubly isogenous} if $\mathrm{Jac}(C)$ and $\mathrm{Jac}(C')$ are isogenous over $K$ and $\mathrm{Jac}(D)$ and $\mathrm{Jac}(D')$ are isogenous over~$K$. When we restrict attention to the case where $C$ and $C'$ are curves of genus $2$ whose groups of $K$-rational automorphisms are isomorphic to the dihedral group $D_6$ of order $12$, we find many more doubly isogenous pairs than one would expect from reasonable heuristics. Our analysis of this overabundance of doubly isogenous curves over finite fields leads to the construction of a pair of doubly isogenous curves over a number field. That such a global example exists seems extremely surprising. We show that the Zilber--Pink conjecture implies that there can only be finitely many such examples. When we exclude reductions of this pair of global curves in our counts, we find that the data for the remaining curves is consistent with our original heuristic. Computationally, we find that doubly isogenous curves in our family of $D_6$ curves can be distinguished from one another by considering the isogeny classes of the Prym varieties of certain unramified covers of exponent $3$ and $4$. We discuss how our family of curves can be potentially be used to obtain a deterministic polynomial-time algorithm to factor univariate polynomials over finite fields via an argument of Kayal and Poonen.

Doubly isogenous curves of genus two with a rational action of $D_6$

TL;DR

The paper investigates doubly isogenous genus- curves with a automorphism action, revealing an overabundance explained by multiple geometric factors arising from unramified covers and Prym varieties. By decomposing Jacobians into elliptic factors via 2-, 3-, and 4-cover data and analyzing special and general triple covers, the authors develop refined heuristics, identify extraordinary phenomena in characteristic zero, and, conditional on the Zilber--Pink conjecture, prove finiteness results. They also construct an algorithm, based on Buium’s differential criterion, to detect simultaneous isogeny correspondences and demonstrate how higher-degree covers can distinguish nonisomorphic doubly isogenous curves. Finally, they connect these geometric insights to a deterministic polynomial-time factoring approach over finite fields, proposing a practically efficient candidate using the D-family elliptic factors. The work combines explicit algebraic geometry of genus-2 curves with arithmetic statistics and algorithmic number theory to illuminate isogeny phenomena and potential cryptanalytic applications.

Abstract

Let and be curves over a finite field , provided with embeddings and into their Jacobian varieties. Let and be the pullbacks (via these embeddings) of the multiplication-by- maps on the Jacobians. We say that and are \emph{doubly isogenous} if and are isogenous over and and are isogenous over~. When we restrict attention to the case where and are curves of genus whose groups of -rational automorphisms are isomorphic to the dihedral group of order , we find many more doubly isogenous pairs than one would expect from reasonable heuristics. Our analysis of this overabundance of doubly isogenous curves over finite fields leads to the construction of a pair of doubly isogenous curves over a number field. That such a global example exists seems extremely surprising. We show that the Zilber--Pink conjecture implies that there can only be finitely many such examples. When we exclude reductions of this pair of global curves in our counts, we find that the data for the remaining curves is consistent with our original heuristic. Computationally, we find that doubly isogenous curves in our family of curves can be distinguished from one another by considering the isogeny classes of the Prym varieties of certain unramified covers of exponent and . We discuss how our family of curves can be potentially be used to obtain a deterministic polynomial-time algorithm to factor univariate polynomials over finite fields via an argument of Kayal and Poonen.
Paper Structure (20 sections, 18 theorems, 92 equations, 5 tables)

This paper contains 20 sections, 18 theorems, 92 equations, 5 tables.

Table of Contents

  1. Introduction
  2. The family of genus-two curves with automorphism group D6
  3. The 2-torsion and unramified elementary abelian 2-covers
  4. Unramified cyclic 4-covers
  5. Unramified cyclic 3-covers
  6. The special triple covers
  7. The general elliptic triple covers
  8. Unramified abelian 3-covers in characteristic 3
  9. Initial heuristics and data
  10. Unexpected sources for doubly isogenous curves
  11. Doubly isogenous twists
  12. Doubly isogenous nontwists
  13. An unlikely intersection in characteristic zero
  14. Isogeny correspondences
  15. The Zilber--Pink conjecture
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $K = {\mathbb Q}(\sqrt{29})$. Let $L$ be the degree-$6$ Galois extension of $K$ obtained by adjoining the roots of $x^3 + x^2 + 2$ and $x^2 + 1$. There exist $D_6$ curves $C_1$ and $C_2$ defined over $K$, with Weierstrass points defined over $L$, such that $C_1$ and $C_2$ are doubly isogenous ov

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2: cf. Theorem \ref{['T:extraordinary']}
  • Remark 1.3
  • Definition 2.1
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 51 more