Isogenies of Polarised Abelian Varieties

• Isogenies of Polarised Abelian Varieties are morphisms that respect the polarisation, enabling the decomposition of abelian varieties into simpler CM components.
• Explicit construction via optimal embeddings and idempotents provides sharp degree estimates and arithmetic control, essential for moduli space analysis.
• Applications include the study of Shimura varieties, deformation theory, and class field constructions by reducing noncommutative complexities to tractable CM factors.

Isogenies of Polarised Abelian Varieties are morphisms of abelian varieties that are compatible with the polarisation structure, often with additional arithmetic or geometric constraints. Such isogenies play a central role in the theory of moduli spaces of abelian varieties, the paper of special points on Shimura varieties, and the explicit classification of abelian varieties with extra endomorphisms. Rigorous understanding of these isogenies involves explicit algebraic constructions via endomorphism algebras, the use of idempotents, degree bounds based on norm computations in number fields, and their application to the arithmetic geometry of moduli spaces and deformation theories.

1. QM+CM Type and the Structure of Isogenies

The setting of (Ufer, 2012) concerns principally polarised, even-dimensional abelian varieties $A$ defined over a field (typically $\mathbb{C}$ or a finite field), equipped with an action of a maximal order $\mathcal{O}_D$ in an indefinite quaternion algebra $D$ over a totally real number field $K$ (QM), together with a compatible embedding of a CM field $L$ (a totally imaginary quadratic extension of $K$) into $\operatorname{End}^0(A)$. The polarisation is typically assumed to be principal.

The existence of both QM and CM structures (QM+CM type) ensures that $\operatorname{End}^0(A)$ contains a large commutative subalgebra, namely $L$. The center of the endomorphism algebra is an order $\mathcal{O}_{L,c}\subset L$ of conductor $c$. This additional endomorphism structure enables explicit decompositions and constructions of isogenies, reducing the paper of such abelian varieties to more tractable, lower-dimensional CM constituents.

2. Explicit Construction via Optimal Embeddings and Idempotents

A core technique is the use of an optimal embedding $$\varphi: \mathcal{O}_{L,c} \hookrightarrow \mathcal{O}_D$$. Optimality is defined by the condition $\varphi^{-1}(\mathcal{O}_D) = \mathcal{O}_{L,c}$. This embedding allows for the explicit construction of non-trivial idempotent elements in the algebra $D \otimes_K L$.

Fix $a\in \mathcal{O}_{L,c}$ with $\operatorname{Tr}_{L/K}(a) = 0$ (i.e., a trace-zero element). Then, following Proposition 2.1, one constructs an idempotent as: $e = \frac{1}{2} \left(1 + a^{-1} a \right)$ with complement $\bar{e} = 1 - e$. These idempotents correspond, via their action on $A$, to projections onto abelian subvarieties $A_e = e(A)$ and $A_{\bar{e}} = \bar{e}(A)$, each of which is, under suitable choices, an abelian subvariety of dimension $g/2$ inheriting a CM structure by (at least) $\mathcal{O}_{L,c}$.

3. Isogeny and Degree Estimates

The explicit isogeny is inherently determined by the action of the idempotents: $$\psi_e : A \to A_e \times A_{\bar{e}} \,\,,\,\, \psi_e(P) = (e\cdot P, \bar{e}\cdot P)$$ Alternatively, for elements $a$ as above (possibly multiplied by scalars, e.g., $n=2a$ for normalisation), the isogeny can be written $\psi_{(a)} : A \to A_e \times A_{\bar{e}}$.

A pivotal quantitative result is the degree estimate: $\deg(\psi_e) \mid N_{L/\mathbb{Q}}(a)^2$ This ties the geometric complexity of the isogeny to the reduced norm of the trace-zero element $a$ in the CM field. For abelian surfaces ($g=2$, $L\simeq \mathbb{Q}(\sqrt{-d})$ and $a = 2c\sqrt{-d}$), this yields: $\deg(\psi_e) \mid (4c^2 d)^2$ The degree divisibility mirrors the interaction of $a$ with the ambient endomorphism algebra, and relates the structure of the isogeny to field arithmetic invariants.

4. Decomposition of the Endomorphism Algebra and Moduli Spaces

The quaternion algebra $D$ splits over $L$, providing an explicit description: $D \cong L \oplus L u$, with $u^2\in K^\times$ and $u \ell = \bar{\ell} u$ for all $\ell\in L$, reflecting that $L$ is a splitting field for $D$, i.e., $D \otimes_K L \simeq M_2(L)$. This structure is essential for building the idempotent operators and controlling the descent of CM structure to the projected factors.

CM points in the Shimura moduli space of (principally) polarised abelian varieties with QM correspond to abelian varieties with QM+CM. The explicit decomposition via $\psi_e$ provides a splitting of moduli points, arithmetic control on isogeny degrees, and a mechanism to paper the behavior of CM points under $p$-adic deformation (comparison between centers of endomorphism algebras for $A$ and its deformations $A_y$).

5. Impact and Applications of Explicit Decomposition

The construction and degree bounds described have several profound implications:

• Endomorphism Structure Reduction: By decomposing $A$ as $A_e \times A_{\bar{e}}$, one reduces the possible noncommutative complexity coming from QM to a pair of (commutative) CM factors. This allows for transfer of many results from the theory of CM abelian varieties to the more general QM+CM setting.
• Sharp Quantitative Control: The degree divisibility by a norm square provides arithmetic control, essential for applications in field-of-moduli computations, explicit class field constructions from CM points, and $p$-adic or deformation-theoretic analyses.
• Special Cases: In the fake elliptic curve case ($g=2$), the factors $A_e$ and $A_{\bar{e}}$ (typically elliptic curves with CM) acquire isomorphic endomorphism rings, connecting the decomposition with classical theory for elliptic curves.
• Moduli and Class Field Theory: The explicit isogeny and its degree properties underpin the construction of Shimura curves, the description of Hecke orbits and CM points, and have implications for unlikely intersections, as well as the construction of class fields via moduli of abelian varieties with additional endomorphism structure.

6. Summary of Key Formulas and Structural Results

Construction	Formula / Property	Context
Idempotent (QM+CM setting)	$e = \frac{1}{2}(1 + a^{-1} a)$, $a \in \mathcal{O}_{L,c}$, $\operatorname{Tr}_{L/K}(a) = 0$	Enables splitting $A$
Isogeny decomposition	$\psi_e(P) = (e\cdot P, (1-e)\cdot P)$	$A \to A_e \times A_{\bar{e}}$
Degree estimate	$\deg(\psi_e) \mid N_{L/\mathbb{Q}}(a)^2$	Degree controlled by norms
Quaternion algebra description	$D \cong L \oplus L u$, $u^2\in K^\times$, $u\ell = \bar{\ell} u$	Splitting structure

These structural devices underlie both explicit decompositions of polarised abelian varieties with QM+CM and the sharp control over isogeny degrees.

7. Implications for Deformation Theory and Moduli Arithmetic

The described decomposition is crucial for understanding moduli of abelian varieties with additional endomorphisms (Shimura varieties). The explicit isogenies allow one to:

• Understand the behavior of CM points in families and formal deformation spaces.
• Analyze how centers of endomorphism algebras behave in characteristic $p$ lifts and deformations, with consequences for $p$-adic uniformisation of Shimura curves.
• Relate the specialisation and extension of endomorphism structures to the geometry of moduli spaces.

These results bridge the concrete arithmetic of quaternion and CM multiplication with the abstract properties of moduli spaces and their points, providing tools for both explicit and theoretical investigations into the landscape of polarised abelian varieties and their isogenies.