[Paper Review] Elementary divisor theory for the modular group over quadratic field extensions and quaternion algebras

Martin Raum|arXiv (Cornell University)|Jul 16, 2009

Advanced Algebra and Geometry11 references3 citations

TL;DR

This paper establishes an elementary divisor theory for the modular and unimodular groups over maximal orders in quadratic field extensions and quaternion algebras over number fields. It proves a one-to-one correspondence between modular double cosets and unimodular matrices under a class group triviality condition, showing that symplectic structure compensates for dimension doubling, and provides explicit obstructions to certain elementary divisor configurations via local invariants and ideal-theoretic constraints.

ABSTRACT

We develop an elementary divisor theory for the unimodular and the modular group over quadratic field extensions and quaternion algebras. In particular, we investigate which sets of elementary divisors can occur. Under an additional hypothesis we establish a correspondence of unimodular and modular double cosets.

Motivation & Objective

• To develop a comprehensive elementary divisor theory for the modular and unimodular groups over maximal orders in quadratic field extensions and quaternion algebras.
• To determine which sets of elementary divisors can occur in the modular and unimodular cases under number field conditions.
• To establish a correspondence between modular double cosets and unimodular equivalence classes when the ring of integers of the base field has trivial class group.
• To investigate obstructions to the existence of matrices with prescribed elementary divisors, particularly in non-principal ideal cases.
• To provide explicit constructions and local invariants that classify equivalence classes via ideal-theoretic and valuation-theoretic methods.

Proposed method

• Define local elementary divisors $ e_{\mathfrak{p},i}(M) \in \Lambda_{\mathfrak{p}} $ for finite places $ \mathfrak{p} $ of the maximal order $ \Lambda $.
• Use the symplectic structure via the matrix $ J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} $ and the involution $ \imath $ to define the monoid $ \Delta_n(\Omega_{\mathbb{A}}, m) $ of similitude matrices satisfying $ M^* J M = m J $.
• Characterize the extended symplectic group $ \mathrm{Sp}_n(\Lambda_{\mathbb{A}}) $ as the union of $ \Delta_n(\Lambda_{\mathbb{A}}, m) $ over $ m \in H = \mathrm{Fix}_\imath(\Lambda_{\mathbb{A}}^\times) $.
• Apply the Smith normal form and elementary row/column operations over localizations to reduce matrices to block-diagonal forms that preserve elementary divisors.
• Use the normalized exponential valuation $ \nu_{\mathfrak{p}} $ and the norm $ N(a) = a\imath(a) $ to analyze global constraints on elementary divisors.
• Leverage the triviality of the class group $ \mathrm{Cl}(\mathfrak{o}_{\mathbb{K}}) $ to establish a bijection between modular double cosets and unimodular matrices satisfying $ \nu_{\mathfrak{p}}(N(e_n(M))) \leq \nu_{\mathfrak{p}}(m) $ for all finite places $ \mathfrak{p} $.

Experimental results

Research questions

• RQ1 Which sets of elementary divisors can occur for matrices in the modular group $ \mathrm{Sp}_n(\Lambda) $ over a maximal order in a quadratic field extension or quaternion algebra?
• RQ2 Under what conditions does a matrix with prescribed elementary divisors exist in the modular group, particularly when the ideals involved are non-principal?
• RQ3 Is there a canonical correspondence between unimodular and modular double coset classes under global class group conditions?
• RQ4 Can the decomposition of double cosets into local components (e.g., 2- and 3-adic parts) be achieved, or are there obstructions due to non-principal ideals?
• RQ5 What role does the norm condition $ \nu_{\mathfrak{p}}(N(e_n(M))) \leq \nu_{\mathfrak{p}}(m) $ play in determining the realizability of elementary divisor configurations?

Key findings

• A one-to-one correspondence is established between the modular double coset space $ \Delta_n(\Lambda, m)/\sim_m $ and the unimodular equivalence classes of matrices $ M \in \mathrm{Inv}_n(\Lambda) $ satisfying $ \nu_{\mathfrak{p}}(N(e_n(M))) \leq \nu_{\mathfrak{p}}(m) $ for all finite places $ \mathfrak{p} $, under the assumption that $ \mathrm{Cl}(\mathfrak{o}_{\mathbb{K}}) $ is trivial.
• The loss of degrees of freedom due to the symplectic structure is compensated by the doubling of dimension from $ n $ to $ 2n $, as reflected in the correspondence with $ n $-dimensional unimodular modules.
• There exist obstructions to the existence of matrices with certain elementary divisor configurations: for example, no matrix exists with elementary divisors $ \mathfrak{m}_2 $ and $ 2 $ in a quaternion algebra ramified at 17 and $ \infty $, where $ \mathfrak{m}_2 $ is a non-principal maximal left ideal over 2.
• In the quaternionic case, the matrix $ \mathrm{diag}(M, 12(M^*)^{-1}) $ with $ M = \begin{pmatrix} -2 & 6 \\ -h_2 & 2h_2 \end{pmatrix} $ has elementary divisors $ (\mathfrak{m}_2, 2) $ and $ (1, \mathfrak{m}_3) $, but its double coset does not decompose into 2- and 3-adic components due to non-principal ideals.
• Local elementary divisors are computed via $ e_{\mathfrak{p},i}(M) \in \Lambda_{\mathfrak{p}} $, and the global elementary divisor $ e_n(M) $ is the product over all $ \mathfrak{p} $, with the norm condition $ \nu_{\mathfrak{p}}(N(e_n(M))) \leq \nu_{\mathfrak{p}}(m) $ being necessary and sufficient for realizability under the class group assumption.
• For a matrix $ T \in \mathrm{M}_2(\Lambda) $ over $ \mathbb{K} = \mathbb{Q}(\sqrt{-6}) $, the local elementary divisors are $ (1, \mathfrak{m}_{4\rho-1}) $, $ (\mathfrak{m}_3, 3\mathfrak{m}_3) $, and $ (2\mathfrak{m}_\rho, 2\rho) $, computed via the minimal $ m $ such that $ mT^{-1} \in \mathrm{M}_2(\Lambda) $.