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Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field

Pip Goodman

TL;DR

This work investigates how a large cyclic $2$-torsion field $K(A[2])$ restricts the endomorphism algebra of an abelian variety $A$ defined over a number field $K$. By exploiting Faltings' Isogeny Theorem and $\ell$-adic representations, the authors bound $End^0(A)$ and relate it to the endomorphism field $L$; in particular, when $K=\mathbb{Q}$ and $p=2g+1$ is prime with $\mathrm{Gal}(\mathbb{Q}(A[2])/\mathbb{Q})\cong C_p$, they show there are only finitely many possibilities for $End^0(A)$, often forcing $End^0(A)\subseteq\mathbb{Q}(\zeta_p)$, with concrete CM-type exceptions. They provide a detailed criterion ensuring $L\subseteq K(A[2])$ under suitable hypotheses and deduce a precise finite classification in the cyclic $2$-torsion setting, including the notable case $g=2$ where $End^0(A)\in\{\mathbb{Q},\mathbb{Q}(\sqrt{5})\}$. The paper further advances a quaternionic analogue: for abelian surfaces with quaternion multiplication by a hereditary order in an indefinite quaternion algebra, the endomorphism structure is tightly constrained, and $L/K$ can be only trivial, $C_2$, or $C_2\times C_2$, with explicit End$^0_K(A)$ types and examples illustrating the necessity of heredity. Together, these results yield finite, structurally explicit possibilities for endomorphism algebras in families with large $2$-torsion fields and highlight the interplay between endomorphism fields, CM theory, and QM theory in arithmetic geometry.

Abstract

In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be defined over $K(A[2])$, the field generated by its 2-torsion. When $K= \mathbb{Q}$ and $\mathrm{Gal}(\mathbb{Q}(A[2])/\mathbb{Q})$ is cyclic of prime order $p = 2 \dim(A) +1$, we prove that there are only finitely many possibilities for the geometric endomorphism algebra $\mathrm{End}(A) \otimes \mathbb{Q}$.In fact, when $\dim (A) \not \in \{3,5,9,21,33,81\}$, we show $\mathrm{End}(A) \otimes \mathbb{Q}$ is a proper subfield of the $p$-th cyclotomic field. In particular, when $g=2$, $\mathrm{End}(A) \otimes \mathbb{Q}$ is isomorphic to either $\mathbb{Q}$ or $\mathbb{Q}(\sqrt{5})$.

Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field

TL;DR

This work investigates how a large cyclic -torsion field restricts the endomorphism algebra of an abelian variety defined over a number field . By exploiting Faltings' Isogeny Theorem and -adic representations, the authors bound and relate it to the endomorphism field ; in particular, when and is prime with , they show there are only finitely many possibilities for , often forcing , with concrete CM-type exceptions. They provide a detailed criterion ensuring under suitable hypotheses and deduce a precise finite classification in the cyclic -torsion setting, including the notable case where . The paper further advances a quaternionic analogue: for abelian surfaces with quaternion multiplication by a hereditary order in an indefinite quaternion algebra, the endomorphism structure is tightly constrained, and can be only trivial, , or , with explicit End types and examples illustrating the necessity of heredity. Together, these results yield finite, structurally explicit possibilities for endomorphism algebras in families with large -torsion fields and highlight the interplay between endomorphism fields, CM theory, and QM theory in arithmetic geometry.

Abstract

In this article we study the endomorphism algebras of abelian varieties defined over a given number field with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of to be defined over , the field generated by its 2-torsion. When and is cyclic of prime order , we prove that there are only finitely many possibilities for the geometric endomorphism algebra .In fact, when , we show is a proper subfield of the -th cyclotomic field. In particular, when , is isomorphic to either or .

Paper Structure

This paper contains 3 sections, 16 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Proof of the main theorems
  3. An analogue of Theorem \ref{['thm:intro:endo_field_in_2_torsion']} for QM surfaces

Key Result

Theorem 1.1

Let $A/K$ be an abelian variety of dimension $g \geq 2$ with $p=2g+1$ prime. Suppose $\mathop{\mathrm{\rm Gal}}\nolimits(K(A[2])/K)$ has order $p$ and either $K=\mathbb{Q}$ or $K$ is an imaginary quadratic field such that $p \nmid \# \mathrm{Cl}(K)$. Then either In particular there are only finitely many possibilities for $\mathop{\mathrm{\rm End}}\nolimits^0(A)$.

Theorems & Definitions (36)

  • Theorem 1.1: $\subset$ Corollary \ref{['thm_Cp_field']} + Theorem \ref{['thm_Cp_over_imag_quad']}
  • Theorem 1.2: $=$ Theorem \ref{['thm:endo_field_contained_in_2_torsion_field_when_endos_are_totally_inert_at_2']}
  • Corollary 1.3: $=$ Corollary \ref{['cor:genus_two_C5_over_Q']}
  • Theorem 2.1
  • proof
  • Example 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 26 more