The Herbrand quotient of the S-ideles #
Let L/K be an extension of number fields and S a finite set of finite places of K. An
S-idele of L is an idele whose finite part is a finite S-idele, with arbitrary infinite
part, so the S-ideles split as I_{L,S} = L_∞ˣ × I_{L,S}^f. As representations of Aut(L/K)
this is the short exact sequence
0 → L_∞ˣ → I_{L,S} → I_{L,S}^f → 0
(sIdelesInfiniteShortComplex_shortExact), given by the infinite ideles and the finite part. It
is split by the infinite part I_L → L_∞ˣ of an idele (idelesToInfiniteIdeles) and the idele
(1, a) of a finite S-idele a (finiteSIdelesToSIdeles): an S-idele is the product of its
infinite part and its finite part (sIdeles_hom_decomp).
For a cyclic extension unramified outside S, the Herbrand quotient is multiplicative along it,
so the archimedean computation herbrandQuotient_infiniteIdelesRep and the finite one
herbrandQuotient_finiteSIdelesRep combine to
h(I_{L,S}) = ∏_{v ∣ ∞} [L_w : K_v] · ∏_{v ∈ S} [L_w : K_v]
(herbrandQuotient_sIdelesRep). This is also the Herbrand quotient of the permutation lattice on
the infinite places of L and the primes above S
(herbrandQuotient_sIdelesRep_eq_herbrandQuotient_ofMulAction), the lattice with which the
logarithmic embedding compares the S-units. In the formula
h(C_L) = h(I_{L,S}) / h(U_{L,S}) of herbrandQuotient_ideleClassRep, it is the idelic factor
of the computation h(C_L) = [L : K], from which the first fundamental inequality for cyclic
extensions follows.
Main definitions #
TauCeti.ClassFieldTheory.sIdelesInfiniteShortComplex L S: the sequenceL_∞ˣ → I_{L,S} → I_{L,S}^f.TauCeti.ClassFieldTheory.idelesToInfiniteIdeles K L: the infinite partI_L → L_∞ˣ.TauCeti.ClassFieldTheory.finiteSIdelesToSIdeles L S: the sectionI_{L,S}^f → I_{L,S}of the finite part.
Main results #
TauCeti.ClassFieldTheory.sIdelesInfiniteShortComplex_shortExact: the sequence is short exact.TauCeti.ClassFieldTheory.sIdeles_hom_decomp: anS-idele is the product of its infinite part and its finite part.TauCeti.ClassFieldTheory.herbrandQuotient_sIdelesRep: forL/Kcyclic and unramified outsideS,h(I_{L,S})is the product of the local degrees at the infinite places and atS.TauCeti.ClassFieldTheory.herbrandQuotient_sIdelesRep_eq_herbrandQuotient_ofMulAction: it is the Herbrand quotient of the permutation lattice on the infinite places and the primes aboveS.TauCeti.ClassFieldTheory.finite_tateCohomology_sIdelesRep: the Tate cohomology of theS-ideles is finite in degrees0and-1.
References #
- J. S. Milne, Class Field Theory, Chapter VII, §2 and the proof of Theorem 4.3.
- J. Tate, Global class field theory, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter VII, §§6–7.
The infinite ideles and the finite part #
The inclusion L_∞ˣ → I_{L,S} of the infinite ideles, with all finite components 1, as a
morphism of representations of Aut(L/K).
Instances For
infiniteIdelesToSIdeles sends an infinite idele x to the idele (x, 1).
The finite part I_{L,S} → I_{L,S}^f of an S-idele, as a morphism of representations of
Aut(L/K).
Instances For
sIdelesToFiniteSIdeles sends an S-idele to its finite part.
The sequence L_∞ˣ → I_{L,S} → I_{L,S}^f of representations of Aut(L/K), given by the
infinite ideles and the finite part of an S-idele.
Instances For
The sequence 0 → L_∞ˣ → I_{L,S} → I_{L,S}^f → 0 is exact: an S-idele is determined
by its infinite and finite parts, and those parts can be prescribed independently.
The splitting of the S-ideles #
The infinite part I_L → L_∞ˣ of an idele, as a morphism of representations of
Aut(L/K).
Instances For
idelesToInfiniteIdeles takes the infinite part of an idele.
The idele (1, a) of a finite S-idele a, as a morphism of representations of Aut(L/K),
a section of sIdelesToFiniteSIdeles.
Instances For
finiteSIdelesToSIdeles sends a finite S-idele a to the idele (1, a).
An S-idele is the product of its infinite part and its finite part: the identity of the
S-ideles is the sum of the composites through the infinite ideles and through the finite
S-ideles.
The finite part of the idele (1, a) of a finite S-idele a is a.
The finite part of the idele (1, a) of a finite S-idele a is a.
The infinite part of the idele (1, a) of a finite S-idele a is trivial.
The infinite part of the idele (1, a) of a finite S-idele a is trivial.
The infinite part of the idele (x, 1) of an infinite idele x is x.
The infinite part of the idele (x, 1) of an infinite idele x is x.
The finite part of the idele (x, 1) of an infinite idele x is trivial.
The finite part of the idele (x, 1) of an infinite idele x is trivial.
The Herbrand quotient #
The Herbrand quotient of the S-ideles. For a cyclic extension L/K of number fields and
a finite set S of finite places of K outside which L/K is unramified, the S-ideles of L
have Herbrand quotient ∏_{v ∣ ∞} [L_w : K_v] · ∏_{v ∈ S} [L_w : K_v] as a representation of
Gal(L/K), the first product running over the infinite places of K. The places w above each
v may be chosen arbitrarily.
The S-ideles and the place lattice have the same Herbrand quotient: for a cyclic
extension L/K of number fields unramified outside a finite set S of finite places of K,
h(I_{L,S}) is the Herbrand quotient of the permutation lattice on the infinite places of L and
the primes of L above S (herbrandQuotient_ofMulAction_places).
The Tate cohomology of the S-ideles is finite in degrees 0 and -1, for a cyclic
extension L/K of number fields unramified outside a finite set S of finite places of K: the
Herbrand quotient of the S-ideles is nonzero.