Genus characters and narrow-equivalent ideals #
This file proves the cancellation step needed to descend genus characters from norms of ideals to
the narrow class group of a quadratic field. Suppose two nonzero integral ideals I and J are
related by
(x) I = (y) J,
where x * y is totally positive. If N(x) and N(y) are coprime to the modulus of a genus
character, then that character takes the same value on N(I) and N(J).
The proof uses two facts. Total positivity makes N(x) and N(y) have the same sign. The
genus-character relation for element norms says that the character of N(xy) is 1, so the
character values of N(x) and N(y) agree. Taking ideal norms in (x) I = (y) J then cancels
that common value.
This is deliberately not yet packaged as a character of the narrow class group. That construction also needs strong approximation in the form that every narrow class, and every comparison between two representatives, can be chosen coprime to the discriminant. The theorem here isolates the arithmetic cancellation that such a construction consumes.
The genus-character argument is classical; see D. A. Cox, Primes of the Form x² + ny², §3.B, and F. Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein, §2.2.
Main result #
TauCeti.Multiquadratic.genusCharFun_absNorm_eq_of_span_mul_eq_span_mul: genus characters agree on the norms of two ideals related by a coprime, totally positive principal ratio.TauCeti.Multiquadratic.genusCharFunCoprimeIdealHom_eq_one_of_eq_span_singleton: the unit-valued character is trivial on a totally positive coprime principal ideal.TauCeti.Multiquadratic.genusCharFunCoprimeIdealHom_eq_of_span_mul_eq_span_mul: the same character is invariant under a coprime narrow comparison.
Genus characters are invariant under a coprime narrow-principal comparison.
Let K = ℚ(√d), with a prime-discriminant factorization indexed by s, and let t ⊆ s index a
genus character. Suppose nonzero integral ideals I and J satisfy (x) I = (y) J, where x * y
is totally positive. If the element norms of x and y are coprime to the product of the factors
in t, then the genus character has the same value on N(I) and N(J).
The nonvanishing hypotheses on x and y are separate because total positivity is vacuous over
totally complex fields, where it does not exclude zero.
By NumberField.NarrowClassGroup.mk0_eq_mk0_iff, the displayed ideal equality and positivity are
exactly the data witnessing equality of the narrow classes of I and J. The extra coprimality
conditions are the strong-approximation input still needed to package this result as a character
of the entire narrow class group.
The unit-valued genus character is invariant under a coprime narrow-principal comparison.
Let I and J be
nonzero integral ideals whose absolute norms are coprime to the modulus of a genus character. If
they are related by a narrow comparison (x) I = (y) J, and the principal factor y is coprime
to the modulus, then the corresponding values of
genusCharFunCoprimeIdealHom agree.
The explicit coprimality condition on one principal factor is the remaining strong-approximation input needed to remove the coprimality restriction when constructing a character on the narrow class group.
A principal ideal with a totally positive generator, coprime to the modulus, has trivial
genus character. Let I be a nonzero integral ideal whose absolute norm is coprime to the
modulus of a genus character. If I = (x) for a totally positive x, then the coprime-ideal genus
character of I is 1; the corresponding properties of x follow from those of I.
This is the kernel calculation needed before the character can descend through the narrow class group: a totally positive principal ideal is narrow-trivial, and this theorem handles the finite coprimality restriction of the arithmetic character.