[Paper Review] Quaternions, polarizations and class numbers

Víctor Rotger|ArXiv.org|Nov 6, 2002

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes a number-theoretic criterion for the existence and count of principal polarizations on abelian varieties with quaternionic multiplication over totally real fields, using Eichler's theory of optimal embeddings and relative class numbers of CM fields. It proves that in even dimensions, the number of non-isomorphic principal polarizations can grow without bound, while in odd square-free dimensions, it is uniformly bounded by $2^{g-1}$, resolving a long-standing question about the finiteness of such polarizations.

ABSTRACT

We study abelian varieties $A$ with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms. Moreover, we give an expression for the number $π_0(A)$ of isomorphism classes of principal polarizations on $A$ in terms of relative class numbers of CM fields by means of Eichler's theory of optimal embeddings. As a consequence, we exhibit simple abelian varieties of any even dimension admitting arbitrarily many non-isomorphic principal polarizations. On the other hand, we prove that $π_0(A)$ is uniformly bounded for simple abelian varieties of odd square-free dimension.

Motivation & Objective

• To determine the existence and count of principal polarizations on abelian varieties with multiplication by a totally indefinite quaternion algebra over a totally real number field.
• To express the number of isomorphism classes of principal polarizations, $\pi_0(A)$, in terms of relative class numbers of CM fields via Eichler's theory of optimal embeddings.
• To resolve whether simple abelian varieties of dimension $g$ can admit arbitrarily many non-isomorphic principal polarizations.
• To establish uniform bounds on $\pi_0(A)$ for simple abelian varieties of odd, square-free dimension.

Proposed method

• The authors use Eichler's theory of optimal embeddings to translate the geometric problem of counting principal polarizations into an arithmetic problem involving orders in CM fields.
• They express $\pi_0(A)$ as a sum over orders $S$ in the CM field $F(\sqrt{-D})$ containing $R_F[\sqrt{-D}]$, weighted by their class numbers $h(S)$.
• The key formula is $\pi_0(A) = \frac{1}{2}\sum_S h(S)$, where $D$ is a totally positive generator of the reduced discriminant ideal of the maximal order $\mathcal{O}$.
• Analytical tools from class number theory—particularly asymptotic bounds and estimates for relative class numbers of CM fields—are applied via the Brauer-Siegel Theorem.
• The proof leverages Čebotarev's Density Theorem to construct families of totally positive ideals $D_j$ with coprime discriminants and differentials, ensuring the existence of corresponding quaternion algebras and abelian varieties.
• For the odd dimension case, the argument adapts Lange’s method using the structure of unit groups and norms in CM fields to bound $\pi_0(A)$ by $2^{g-1}$.

Experimental results

Research questions

• RQ1 Can abelian varieties of even dimension with quaternionic multiplication admit arbitrarily many non-isomorphic principal polarizations?
• RQ2 Is the number of principal polarizations on a simple abelian variety of odd, square-free dimension uniformly bounded?
• RQ3 What is the precise arithmetic formula for the number of isomorphism classes of principal polarizations on such abelian varieties?
• RQ4 How do relative class numbers of CM fields relate to the geometry of principal polarizations on quaternionic abelian varieties?

Key findings

• For a complex abelian variety $A$ of dimension $2n$ with maximal order in a totally indefinite quaternion algebra over a totally real field $F$, $\pi_0(A) = \frac{1}{2}\sum_S h(S)$, where $S$ runs over orders in $F(\sqrt{-D})$ containing $R_F[\sqrt{-D}]$.
• In the case of abelian surfaces ($g=2$), $\pi_0(A) = \frac{h(-4D)+h(-D)}{2}$ if $D \equiv 3 \pmod{4}$, and $\frac{h(-4D)}{2}$ otherwise.
• For even dimension $g$, there exist simple abelian varieties with $\pi_0(A)$ arbitrarily large, as shown by constructing families with $|\mathrm{N}_{F/\mathbb{Q}}(D)| \to \infty$ and applying asymptotic class number estimates.
• For odd, square-free dimension $g$, $\pi_0(A) \leq 2^{g-1}$, with equality possible only under specific unit group conditions.
• The logarithmic growth of $\pi_0(A)$ satisfies $\log \pi_0(A) \sim \log \sqrt{|\mathrm{N}_{F/\mathbb{Q}}(D)| \cdot D_F}$, derived from the Brauer-Siegel Theorem on class numbers.
• The bound in odd square-free dimensions is sharp and follows from analyzing the unit group structure and norms in CM fields, extending Lange’s earlier bound to the quaternionic case.