The local Euler characteristic on the Grothendieck group of a finite Galois quotient #
Let V be an open normal subgroup of the absolute Galois group G_F of a nonarchimedean local
field F, and let n be invertible in F. A finite-dimensional ZMod n-representation of the
finite group G_F ⧸ V is a finite smooth discrete Galois representation through
fdGalRepOfQuotient, the restriction of galRepOfQuotient to finite-dimensional
representations. By the multiplicativity of the local Euler characteristic χ_F and of the
normalized absolute value φ_F of the order in short exact sequences, both invariants of these
inflated representations descend to homomorphisms
G₀((ZMod n)[G_F ⧸ V]) → Additive ℚ^×_{>0}
through the universal property TauCeti.liftFDRepK0. As every finite smooth discrete Galois
representation is inflated from some finite quotient (exists_galRepOfQuotient_iso), Tate's
local Euler characteristic formula χ_F = φ_F holds exactly when these two homomorphisms agree
for every V. Since ℚ^×_{>0} is torsion-free (Units.instHasUniqueRootsPosSubgroup), that
agreement may be checked on a positive multiple of each class, such as the one the modular Artin
theorem writes as a sum of classes induced from cyclic subgroups.
Main definitions #
TauCeti.ClassFieldTheory.fdGalRepOfQuotient: inflation of finite-dimensional representations ofG_F ⧸ Vto Galois representations.TauCeti.ClassFieldTheory.localEulerCharacteristicK0: the descended Euler characteristic.TauCeti.ClassFieldTheory.localCardNormK0: the descended normalized absolute value of the order.
Main results #
TauCeti.ClassFieldTheory.localEulerCharacteristicK0_of,TauCeti.ClassFieldTheory.localCardNormK0_of: the values on the class of a representation.TauCeti.ClassFieldTheory.localEulerCharacteristicK0_eq_localCardNormK0_iff: the two homomorphisms forVagree exactly whenχ_F = φ_Fon every finite smooth discrete representation on whichVacts trivially.TauCeti.ClassFieldTheory.forall_localEulerCharacteristic_eq_localCardNorm_iff:χ_F = φ_Fon every finite smooth discrete representation exactly when the two homomorphisms agree for everyV.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, §VII.3, proof of (7.3.1).
- J. S. Milne, Arithmetic Duality Theorems, I, proof of Theorem 2.8.
The local Euler characteristic on G₀ of a finite Galois quotient. For n invertible in
F, the local Euler characteristic of inflated representations descends to the Grothendieck group
of finite (ZMod n)[G_F ⧸ V]-modules, by its multiplicativity in short exact sequences
(localEulerCharacteristic_mul_of_exact).
Instances For
The descended Euler characteristic takes the value χ_F of the inflated representation on the
class of a finite-dimensional representation.
The normalized absolute value of the order on G₀ of a finite Galois quotient. The
invariant φ_F of inflated representations descends to the Grothendieck group of finite
(ZMod n)[G_F ⧸ V]-modules, by its multiplicativity in short exact sequences
(localCardNorm_mul_of_exact).
Instances For
The descended normalized absolute value takes the value φ_F of the inflated representation
on the class of a finite-dimensional representation.
The Euler characteristic formula for V on G₀. The descended Euler characteristic and
the descended normalized absolute value of the order agree on the Grothendieck group of
(ZMod n)[G_F ⧸ V] exactly when χ_F(X) = φ_F(X) for every finite smooth discrete Galois
representation X on which V acts trivially.
The Euler characteristic formula and G₀. Tate's local Euler characteristic formula
χ_F(X) = φ_F(X) holds for every finite smooth discrete Galois representation X exactly when, for
every open normal subgroup V of G_F, the descended Euler characteristic and the descended
normalized absolute value of the order agree on the Grothendieck group of (ZMod n)[G_F ⧸ V].