Idele

An idèle: a unit of the adèle ring — a real $r$ together with $s \in \mathbb{Q}^\times$ principal everywhere except at a listed finite set of primes, where distinguished units override it.

Idele(r, s, units)real $r$, principal value $s$ at every prime not named in units

Idele(q)the principal idèle of a non-zero rational $q$: $q$ at every place

SageMathIdeles(QQ)([q], q)

Typed signatures

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

Domain: Adèles and idèles

Details

• Multiply on idèles multiplies real parts, principal values and any shared or distinguished unit parts; an idèle is invertible whenever $r \ne 0$ and $s \ne 0$.
• units is a list of AdicNumeral values at distinct primes, each overriding the principal value $s$ there.
• Adele(idele) forgets the idèle structure and keeps only its value as an adèle.

Examples

11 more kept as data, checked against the oracles

Basic #

the principal idèle: 7 at every place

live ↗

In$\mathrm{Idele}(7)$Out$\mathrm{Idele}(7, 7)$

7 and its inverse cancel everywhere

live ↗

In$\mathrm{Idele}(7)\mathrm{Idele}(\frac{1}{7})$Out$\mathrm{Idele}(1, 1)$

live ↗

In$\mathrm{Idele}(0, 1)$Out$\mathrm{Idele}(0, 1)$

Scope #

a listed component carries its valuation into the scale

live ↗

In$\mathrm{Idele}(1, 1, \bigl\lbrack\mathrm{AdicNumeral}(5, \frac{15}{7}, 4)\bigr\rbrack)$Out$\mathrm{Idele}(1, 5, \bigl\lbrack\mathrm{AdicNumeral}(5, 54, 3)\bigr\rbrack)$

Possible issues #

at 2 a unit known mod 2 says nothing

live ↗

In$\mathrm{Idele}(1, 1, \bigl\lbrack\mathrm{AdicNumeral}(2, 3, 1)\bigr\rbrack)$Out$\mathrm{Idele}(1, 1, \bigl\lbrack\bigr\rbrack)$

About

Examples	16
Checked against	SageMath 11 agree
Crosswalk	SageMath ×2

Implementation

mappedsagepackages/oracle/src/mappings.ts

Hertogh's adeles package; the principal idele of a rational, $1 at every place.

mappedsagepackages/oracle/src/mappings.ts

Hertogh's adeles package; real $1 beside a principal finite value $2. The units-list call form (arity 3) has no row -- it needs AdicNumeral, which is unmapped for sage.

See also: Adele, ProfiniteNumber