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Polarized products of elliptic curves with complex multiplication and field of moduli $\mathbb{Q}$

Fabien Narbonne

TL;DR

The paper establishes an equivalence between polarized abelian varieties over ℂ that are isomorphic to products of CM elliptic curves and integral Hermitian R-lattices, enabling a lattice-theoretic description of these varieties. It then translates the action of the Galois group on CM data into isometries of the associated R-lattices, yielding precise control over fields of moduli and enabling an explicit enumeration of indecomposable principally polarized abelian varieties with field of moduli ℚ in genus 2 and 3. Key contributions include the KQ and QK theorems describing how Galois action manifests on the lattice side, necessary conditions on class groups and Steinitz classes for ℚ as a field of moduli, and unconditional genus-2 and genus-3 classifications supported by Appendix results. The work combines explicit analytic (Weierstrass) identifications, hermitian lattice theory, and computational methods (e.g., Magma/KNRR) to produce complete tables and algorithms for these moduli problems with CM by a fixed maximal order R.

Abstract

Let $R$ be the maximal order in a quadratic imaginary field $K$. We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over $\mathbb{C}$ with complex multiplication (CM) by $R$ and the category of integral hermitian $R$-lattices. Then we apply this equivalence to enumerate all the genus $2$ and $3$ curves with field of moduli $\mathbb{Q}$ and with Jacobian isomorphic to a product of elliptic curves with CM by $R$.

Polarized products of elliptic curves with complex multiplication and field of moduli $\mathbb{Q}$

TL;DR

The paper establishes an equivalence between polarized abelian varieties over ℂ that are isomorphic to products of CM elliptic curves and integral Hermitian R-lattices, enabling a lattice-theoretic description of these varieties. It then translates the action of the Galois group on CM data into isometries of the associated R-lattices, yielding precise control over fields of moduli and enabling an explicit enumeration of indecomposable principally polarized abelian varieties with field of moduli ℚ in genus 2 and 3. Key contributions include the KQ and QK theorems describing how Galois action manifests on the lattice side, necessary conditions on class groups and Steinitz classes for ℚ as a field of moduli, and unconditional genus-2 and genus-3 classifications supported by Appendix results. The work combines explicit analytic (Weierstrass) identifications, hermitian lattice theory, and computational methods (e.g., Magma/KNRR) to produce complete tables and algorithms for these moduli problems with CM by a fixed maximal order R.

Abstract

Let be the maximal order in a quadratic imaginary field . We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over with complex multiplication (CM) by and the category of integral hermitian -lattices. Then we apply this equivalence to enumerate all the genus and curves with field of moduli and with Jacobian isomorphic to a product of elliptic curves with CM by .
Paper Structure (22 sections, 24 theorems, 67 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 22 sections, 24 theorems, 67 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Complex abelian varieties and complex tori
  2. Abelian varieties and polarizable tori
  3. Duality
  4. Polarizations and hermitian forms
  5. Totally split CM abelian varieties and $R$-module structure
  6. Elliptic curve with complex multiplication over $\mathds{C}$
  7. Totally split complex tori and $R$-lattice structure
  8. Polarizable tori and integral lattices
  9. Action of $\mathop{\mathrm{Gal}}\nolimits\left(\overline{\mathds{Q}}/\mathds{Q}\right)$
  10. Positioning of the problem
  11. Action of $\mathop{\mathrm{Gal}}\nolimits\left(\overline{\mathds{Q}}/\mathds{Q}\right)$ on CM elliptic curves
  12. Fixing isomorphisms with Weierstrass $\wp$-functions
  13. Algebraicity of analytic representations for CM elliptic curves
  14. Galois action on polarizations of CM elliptic curves
  15. Product of CM elliptic curves
  16. ...and 7 more sections

Key Result

Theorem 1

Let $R$ be an order in a quadratic imaginary field $K$. There is an equivalence of categories between the category $\mathbfcal{A}_R$ of abelian varieties isomorphic to a product of elliptic curves with CM by $R$, and the category $\mathbfcal{L}_R$ of $R$-lattices given by on objects and, for arrows, $f\colon A(\mathds{C})\simeq V/\Gamma\rightarrow A'(\mathds{C})\simeq V'/\Gamma',$ given by $\math

Figures (4)

  • Figure 1: The bracket isomorphism
  • Figure 2: Analytic representation of an isogeny
  • Figure 3: Factorization of isogenies
  • Figure 4: Analytic representation of polarizations

Theorems & Definitions (49)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • Theorem 3: Description of the action of $\mathop{\mathrm{Gal}}\nolimits(K/\mathds{Q})$
  • Theorem 4: Description of the action of $\mathop{\mathrm{Gal}}\nolimits\left(\overline{\mathds{Q}}/K\right)$
  • Lemma 2
  • ...and 39 more