ProfiniteDecomposition

$\{b, a\}$ with $m = b \cdot a$, $b \in GL_n(\hat{\mathbb{Z}})$ and $a \in GL_n^+(\mathbb{Q})$ upper triangular — strong approximation for a matrix over $\hat{\mathbb{Q}}$.

ProfiniteDecomposition(m)$m$ a square matrix (list of lists) of ProfiniteNumber or rational entries

ProfiniteDecomposition(m, d)with $d$ = $\det m$ supplied explicitly rather than computed from $m$'s entries

SageMathenumeratio_profinite_decomposition(m, d)

Typed signatures

Type	Package	Applies
(list, number?) -> list	adeles	always

Domain: Adèles and idèles

Details

• Hertogh's Algorithm 8.4: $GL_n(\hat{\mathbb{Q}}) = GL_n(\hat{\mathbb{Z}}) \cdot GL_n^+(\mathbb{Q})$, strong approximation for $GL_n$ over $\mathbb{Q}$.
• $b$'s entries are profinite numbers; $a$'s are rationals. Built on HermiteDecomposition of the denominators.

Examples

14 more kept as data, checked against the oracles

Basic #

already in $GL_n^+(\mathbb{Q})$ upper-triangular form: $b$ is the identity

live ↗

In$\mathrm{ProfiniteDecomposition}(\bigl\lbrack\bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack0, 3\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack0, 3\bigr\rbrack\bigr\rbrack\bigr\rbrack$

About

Examples	15
Checked against	SageMath 14 agree
Crosswalk	SageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

Hertogh's Algorithm 8.4 (factor_GLQhat); no one-liner, so run.ts's SAGE_PREAMBLE carries the matrix-building helper. The arity-1 form (det computed from $1) has no row -- Sage's generic matrix determinant does not work over the Qhat ring (checked; TypeError coercing a ProfiniteNumber to an integer).

See also: HermiteDecomposition, ProfiniteNumber