Adele

An adèle: a real number $r$ beside a profinite number $z \in \hat{\mathbb{Q}}$. Adele(q) for rational $q$ is the principal adèle, $q$ at every place.

Adele(r, z)the real $r$ beside the finite part $z$ (a ProfiniteNumber)

SageMathAdeles(QQ)([r], z)

Adele(q)the principal adèle of a rational $q$: $q$ at the real place and every finite one

SageMathAdeles(QQ)(q)

Adele(idele)the underlying adèle of an Idele

SageMathAdeles(QQ)(idele)

Typed signatures

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

Domain: Adèles and idèles

Details

• The real part is any closed-form real constant compute-engine can evaluate numerically, not only a rational.
• Add, Multiply, Negate, Divide, Power, Equal, NotEqual thread over an adèle, real part and finite part separately.
• An Idele is an adèle together with the extra data an idèle carries (a finite set of distinguished units); Adele(idele) forgets that and keeps only the value.

Examples

15 more kept as data, checked against the oracles

Basic #

the principal adèle: 5 at every place

live ↗

In$\mathrm{Adele}(5)$Out$\mathrm{Adele}(5, 5)$

live ↗

In$3\mathrm{Adele}(2, \mathrm{ProfiniteNumber}(1, 6))$Out$\mathrm{Adele}(6, \mathrm{ProfiniteNumber}(3, 18))$

live ↗

In$\mathrm{Adele}(5)+\mathrm{Adele}(π, \mathrm{ProfiniteNumber}(1, 6))$Out$\mathrm{Adele}(5+π, \mathrm{ProfiniteNumber}(0, 6))$

equal adèles need not share a modulus

live ↗

In$\mathrm{Adele}(1, \mathrm{ProfiniteNumber}(2, 6))=\mathrm{Adele}(1, \mathrm{ProfiniteNumber}(5, 9))$Out$\mathrm{True}$

live ↗

In$\mathrm{Adele}(2, \mathrm{ProfiniteNumber}(2, 6))=\mathrm{Adele}(1, \mathrm{ProfiniteNumber}(5, 9))$Out$\mathrm{False}$

Scope #

the adèle underlying an idèle

live ↗

In$\mathrm{Adele}(\mathrm{Idele}(1, 1, \bigl\lbrack\mathrm{AdicNumeral}(3, 2, 2)\bigr\rbrack))$Out$\mathrm{Adele}(1, \mathrm{ProfiniteNumber}(11, 18))$

Possible issues #

an adèle and a bare profinite number do not combine

live ↗

In$\mathrm{Adele}(1, \mathrm{ProfiniteNumber}(2, 6))+\mathrm{ProfiniteNumber}(1, 6)$Out$\mathrm{ProfiniteNumber}(1, 6)+\mathrm{Adele}(1, \mathrm{ProfiniteNumber}(2, 6))$

About

Examples	22
Checked against	SageMath 15 agree
Crosswalk	SageMath ×2

Implementation

mappedsagepackages/oracle/src/mappings.ts

Hertogh's adeles package; coerces a rational (the principal adele) or an idele (forgetting its idele structure) -- Sage's own coercion dispatches on the argument's type, like ours.

mappedsagepackages/oracle/src/mappings.ts

Hertogh's adeles package; real $1 beside the finite part $2 (a ProfiniteNumber).

See also: ProfiniteNumber, Idele