The infinity type of a Hecke character #
Let χ be a Hecke character of a number field K. Restricting χ along the embedding of the
units of an archimedean completion K_w into the idele class group gives a continuous character
of K_wˣ, the component of χ at w. Transporting it along K_w ≃ ℝ at a real place and
along K_w ≃ ℂ at a complex place (the identification extending the embedding w.embedding
chosen by Mathlib), the classification of continuous characters of ℝˣ and ℂˣ writes these
components uniquely as
x ↦ |x| ^ s_w * sgn(x) ^ ε_wat a real place, withs_w : ℂandε_w : ZMod 2, andz ↦ |z| ^ s_w * (z / |z|) ^ k_wat a complex place, withs_w : ℂandk_w : ℤ.
These parameters form the infinity type χ.infinityType : ContinuousInfinityType K. The
exponents are complex: the norm character ‖·‖ ^ s has exponent s at every real place and
2 * s at every complex place, because the normalized absolute value at a complex place is the
square of the usual one. In particular ‖·‖ ^ s has finite order only for s = 0, while a Hecke
character of finite order has vanishing exponents and angular frequencies, so its infinity type is
a FiniteOrderInfinityType: signs at the real places and nothing else.
Main definitions #
TauCeti.GlobalNumberFields.HeckeCharacter.infiniteComponent: the character ofK_wˣobtained by restricting a Hecke character.TauCeti.GlobalNumberFields.HeckeCharacter.realComponent,TauCeti.GlobalNumberFields.HeckeCharacter.complexComponent: the components at real and complex places as characters ofℝˣandℂˣ.TauCeti.GlobalNumberFields.HeckeCharacter.infinityType: the continuous infinity type of a Hecke character.
Main results #
TauCeti.GlobalNumberFields.HeckeCharacter.infinityType_eq_iff: the infinity type is the unique continuous infinity type whose parameters describe every real and complex component.TauCeti.GlobalNumberFields.HeckeCharacter.infiniteComponent_neg_one_eq_one_iff: a Hecke character is trivial on the sign-1at a real place exactly when its real parity there is zero.TauCeti.GlobalNumberFields.HeckeCharacter.infinityType_mul: multiplying Hecke characters adds their infinity types.TauCeti.GlobalNumberFields.HeckeCharacter.infinityType_normPow: the infinity type of the norm character‖·‖ ^ s.TauCeti.GlobalNumberFields.HeckeCharacter.exists_finiteOrderInfinityType: the infinity type of a finite-order Hecke character comes from a finite-order infinity type.TauCeti.GlobalNumberFields.HeckeCharacter.infiniteComponent_eq_one_iff: a finite-order Hecke character is trivial at a real place exactly when its real parity there is zero.HeckeCharacter.infiniteComponent_eq_one_of_mem_range_ofRayClassCharacter: a Hecke character coming from a ray class character of𝔪is trivial at every infinite place outside𝔪.TauCeti.GlobalNumberFields.HeckeCharacter.mem_range_ofRayClassCharacter_erase: a Hecke character coming from𝔪and trivial at a real placewof𝔪comes from𝔪withwremoved.TauCeti.GlobalNumberFields.HeckeCharacter.isFiniteOrder_normPow_iff:‖·‖ ^ shas finite order exactly whens = 0.TauCeti.GlobalNumberFields.HeckeCharacter.re_realExponent_infinityType,TauCeti.GlobalNumberFields.HeckeCharacter.re_complexExponent_infinityType: the real parts of the modulus exponents are the shiftσofχat every real place and2 * σat every complex place, since|χ| = ‖·‖ ^ σ.
References #
- A. Weil, Basic Number Theory, Chapter VII, §3.
- J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, in J. W. S. Cassels and A. Fröhlich, eds., Algebraic Number Theory, §2.3 and §3.
Archimedean components #
The component of a Hecke character χ at an infinite place w: the continuous character
of K_wˣ obtained by composing χ with the embedding of K_wˣ into the idele class group.
Instances For
The component of χ at w evaluates χ on the idele class concentrated at w.
The trivial Hecke character has trivial components.
Components of a product of Hecke characters are the products of the components.
Components of the inverse of a Hecke character are the inverses of the components.
Components of a power of a Hecke character are the powers of the components.
The component of a Hecke character at a real place w, as a character of ℝˣ: the
component at w transported along the isomorphism ℝ ≃ K_w inverse to the real extension
embedding.
Instances For
The component of χ at a real place w evaluates the component of χ at w on the unit of
K_w corresponding to a real unit.
The component of a Hecke character at a complex place w, as a character of ℂˣ: the
component at w transported along the isomorphism ℂ ≃ K_w inverse to the extension of
w.embedding.
Instances For
The component of χ at a complex place w evaluates the component of χ at w on the unit
of K_w corresponding to a complex unit.
The trivial Hecke character has trivial real components.
Real components of a product of Hecke characters are the products of the real components.
Real components of the inverse of a Hecke character are the inverses of the real components.
Real components of a power of a Hecke character are the powers of the real components.
The trivial Hecke character has trivial complex components.
Complex components of a product of Hecke characters are the products of the complex components.
Complex components of the inverse of a Hecke character are the inverses of the complex components.
Complex components of a power of a Hecke character are the powers of the complex components.
The infinity type #
The infinity type of a Hecke character χ: at a real place w the parameters (s, ε)
with χ.realComponent w = |·| ^ s * sgn ^ ε, and at a complex place w the parameters (s, k)
with χ.complexComponent w = |·| ^ s * (· / |·|) ^ k. The exponents s are complex.
Instances For
The component of χ at a real place w is x ↦ |x| ^ s * sgn(x) ^ ε, for the real
modulus exponent s and parity ε of the infinity type of χ at w.
The component of χ at a complex place w is z ↦ |z| ^ s * (z / |z|) ^ k, for the
complex modulus exponent s and angular frequency k of the infinity type of χ at w.
Characterization of the infinity type. A continuous infinity type t is the infinity
type of χ exactly when its parameters describe every real and every complex component of
χ.
The real parity reads off the sign. At a real place w, a Hecke character takes the
value (-1) ^ ε on the idele that is -1 at w and 1 elsewhere, where ε is the real parity
of its infinity type at w.
A Hecke character is trivial on the idele that is -1 at a real place w and 1 elsewhere
exactly when the real parity of its infinity type at w vanishes. Unlike
infiniteComponent_eq_one_iff, this needs no finite-order hypothesis.
The trivial Hecke character has the zero infinity type.
Multiplying Hecke characters adds their infinity types.
The infinity type of a power of a Hecke character is the corresponding multiple.
The infinity type of the inverse of a Hecke character is the negated infinity type.
Characters of finite order #
The infinity type of a finite-order Hecke character. Its modulus exponents and complex angular frequencies vanish, so it is the continuous infinity type of its real parities.
The infinity type of a Hecke character of finite order is a finite-order infinity type: signs at the real places, with no modulus exponents and no complex angular frequencies.
A finite-order Hecke character is trivial at a real place exactly when its real parity there
is zero. Its modulus exponents vanish, so its component at w is x ↦ sgn(x) ^ ε_w.
Characters coming from ray class characters #
A Hecke character coming from a ray class character of 𝔪 is trivial at every infinite
place outside 𝔪. The idele classes concentrated at such a place lie in raySubgroup 𝔪.
Removing a real place at which a character is trivial. A Hecke character coming from 𝔪
and trivial at a real place w comes from the modulus with the same finite part and with w
removed from the infinite part: dividing an idele congruent to one modulo the smaller modulus by its
own component at w makes it congruent to one modulo 𝔪.
The norm characters #
The component of the norm character ‖·‖ ^ s at an infinite place w sends u to
|u|_w ^ s, for the normalized absolute value |·|_w of K_w.
The component of the norm character ‖·‖ ^ s at a real place is x ↦ |x| ^ s.
The component of the norm character ‖·‖ ^ s at a complex place is z ↦ |z| ^ (2 * s),
since the normalized absolute value of a complex place is the square of the usual one.
The infinity type of the norm character ‖·‖ ^ s: modulus exponent s and parity 0 at
every real place, and modulus exponent 2 * s and angular frequency 0 at every complex place,
since the normalized absolute value of a complex place is the square of the usual one.
The norm character ‖·‖ ^ s has finite order exactly when s = 0. For s ≠ 0 its
modulus exponents at the infinite places are nonzero, although for purely imaginary s its
shift vanishes.
Absolute values of the components and the shift #
The absolute value of an infinite component is the shift-th power of the normalized absolute value.
The absolute value of a real component is the shift-th power of |x|. At a real place
w, the component of χ satisfies |χ_w(x)| = |x| ^ σ, where σ is the shift of χ.
The absolute value of a complex component is the power of ‖z‖ with exponent twice the
shift. At a complex place w, the component of χ satisfies
|χ_w(z)| = |z| ^ (2 * σ), where σ is the shift of χ, since the normalized absolute value of
a complex place is the square of the usual one.
The real part of a real modulus exponent is the shift. The infinity type of χ has a
complex exponent s_w at each real place w; its real part is the shift of χ, independently
of w.
The real part of a complex modulus exponent is twice the shift. The infinity type of
χ has a complex exponent s_w at each complex place w; its real part is 2 * σ for the shift
σ of χ, independently of w.