symmetric monoidal (∞,1)-category of spectra
The group of ideles is the group of units in the ring of adeles :
In classical algebraic number theory one embeds a number field into the Cartesian product of its completions at its archimedean absolute values. This embedding is very useful in the proofs of several fundamental theorems. However, it was noticed by Claude Chevalley and André Weil that the situation was improved somewhat if the number field is embedded in a certain restricted product of its formal completions at all of its absolute values. These objects are known as the adeles, and the units of this ring are called the ideles.
When considering the adeles and ideles, it is their topology as much as their algebraic structure that is of interest. Many important results in number theory translate into simple statements about the topologies of the adeles and ideles. For example, the finiteness of the ideal class group and the Dirichlet unit theorem are equivalent to a certain quotient of the ideles being compact and discrete.
The group of units of the ring of adeles is called the group of ideles
It is a topological group via identification with the set , seen as a subspace of .
The topology on is strictly finer than the subspace topology inherited from . For example, the set is a neighborhood of in , but not in the subspace topology. Cf. the discussion here. Note: multiplicative inversion is not continuous in the subspace topology.
The same definition holds for the ring of adeles of any other global field , here one writes
or similar. The notation is also common.
The idele class group, def. , appears prominently in the description of the moduli space of line bundles over the arithmetic curve on which is the rational functions. From there it appears in the abelian Langlands correspondence and in the abelian case of Tamagawa measures.
The idele class group is a key object in class field theory.
Recall the p-adic norm on for a prime number, given by
for coprime to . The usual absolute value norm one writes
and associates with the “prime at infinity”. When an index runs over the set of all primes (“finite primes”) union with the “prime at infinity” one usually writes it “” instead of .
This induces:
The idele norm
is the function given by
Notice that by construction there is a diagonal map .
(product formula)
The idele norm, def. , is trivial on the diagonal of inside the ideles, in that
The product formula, prop. , says that the idele norm descends to the idele class group, def. .
(e.g. Garrett 11, section 1)
(e.g. Goldfeld-Hundley 11, prop. 1.4.5 and below (2.2.7))
This implies that the ring of adeles may be decomposed into a rational and an idele class factor as:
(e.g. Goldfeld-Hundley 11, prop. 1.4.6 and below (2.2.7))
This decomposition is crucial in the discussion of the Riemann zeta function (see there) as an adelic integral.
The automorphic forms of the idele group are essentially Dirichlet characters in disguise (Goldfeld-Hundley 11, below def. 2.1.4)
Via the function field analogy one may understand any number field or function field as being the rational functions on an arithmetic curve . Under this identification the ring of adeles of has the interpretation of being the ring of functions on all punctured formal disks around all points of , such that only finitely many of them do not extend to the given point. (Frenkel 05, section 3.2).
This means for instance that the general linear group with coefficients in the ring of adeles has the interpretation as being the Cech cocycles for algebraic vector bundles of rank on an algebraic curve with respect to any cover of that curve by the complement of a finite number of points together with the formal disks around these points. Here for then is the group of ideles.
This is part of a standard construction of the moduli stack of bundles on algebraic curves, see at Moduli space of bundles and the Langlands correspondence.
Basics are recalled in
Adeles pdf
Pete Clark, Adeles and Ideles (pdf)
Erwin Dassen , Adeles & Ideles (pdf)
Tom Weston, The idelic approach to number theory (pdf)
Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)
Paul Garrett, Iwasawa-Tate on ζ-functions and L-functions, 2011 (pdf
Discussion in the context of the geometric Langlands correspondence is in
Last revised on September 11, 2024 at 18:48:16. See the history of this page for a list of all contributions to it.