[Paper Review] Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

Qun Li, Jiangwei Xue+1 more|arXiv (Cornell University)|Jul 12, 2018

Algebraic Geometry and Number Theory32 references3 citations

TL;DR

This paper establishes a refined class number and type number formula for maximal orders in totally definite quaternion algebras over real quadratic fields by analyzing the unit groups modulo the center. It provides explicit formulas for the number of conjugacy classes of maximal orders with reduced unit group isomorphic to any finite noncyclic group G, and applies this to prove the existence of superspecial abelian surfaces over finite fields of characteristic p ≠ 1 mod 24 with endomorphism algebra ℚ(√p).

ABSTRACT

We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $\mathbb{Q}(\sqrt{p})$ in all positive characteristic $p ot\equiv 1\pmod{24}$.

Motivation & Objective

• To develop a refined class number and type number formula for maximal orders in totally definite quaternion algebras over real quadratic fields by incorporating the structure of unit groups modulo the center.
• To classify all conjugacy classes of maximal orders whose reduced unit groups are isomorphic to a given finite noncyclic group G.
• To derive explicit formulas for the number of such conjugacy classes and provide representatives for each.
• To apply the refined formulas to prove the existence of superspecial abelian surfaces with endomorphism algebra ℚ(√p) in positive characteristic p ≢ 1 mod 24.

Proposed method

• Use Eichler's trace formula and the classification of finite subgroups of SO₃(ℝ) to analyze the structure of reduced unit groups.
• Define and compute the Vignéras unit index and relate it to the fundamental unit of the real quadratic field F = ℚ(√d).
• Construct minimal G-orders for noncyclic finite groups G and analyze their normalizers to classify maximal orders.
• Study CM-extensions K/F and their unit groups to determine the possible reduced unit groups of maximal orders.
• Analyze quadratic OF-orders in CM-fields, particularly F(√−1), F(√−3), and F(√−ε), to classify unit group structures.
• Apply the results to the case F = ℚ(√p) and H = H∞₁,∞₂ to compute refined type numbers and prove existence results for superspecial abelian surfaces.

Experimental results

Research questions

• RQ1 For a given finite noncyclic group G, how many conjugacy classes of maximal orders in a totally definite quaternion algebra over a real quadratic field have reduced unit group isomorphic to G?
• RQ2 What is an explicit representative for each conjugacy class of maximal orders with a given reduced unit group G?
• RQ3 Under what conditions on the prime p does there exist a maximal order in H∞₁,∞₂ over ℚ(√p) with a non-abelian reduced unit group?
• RQ4 When does the endomorphism algebra of a superspecial abelian surface over a finite field of characteristic p equal ℚ(√p)?
• RQ5 How can the refined class number and type number formulas be used to construct abelian varieties over function fields with prescribed endomorphism algebras?

Key findings

• For any finite noncyclic group G, the paper provides an explicit formula for the number of conjugacy classes of maximal orders in a totally definite quaternion algebra over a real quadratic field whose reduced unit group is isomorphic to G.
• The paper constructs a representative for each conjugacy class of maximal orders with a given reduced unit group G, enabling a complete classification.
• The reduced unit group of a maximal order is non-abelian if and only if p ≢ 1 mod 24, where F = ℚ(√p) is the base field.
• There exists a maximal order in H∞₁,∞₂ over ℚ(√p) with reduced unit group isomorphic to S₄ or A₅ if and only if p ≢ 1 mod 24.
• For every prime p ≢ 1 mod 24, there exists a superspecial abelian surface over a field of characteristic p whose endomorphism algebra is exactly ℚ(√p).
• Over the field K = 𝔽p^ℓ^∞(t) for ℓ ≥ 1, there exist infinitely many K-forms of a superspecial abelian surface X₀ with endomorphism algebra isomorphic to ℚ(√p) when p ≢ 1 mod 24.