1 Introduction
An algebraic torus T over a number field k is a connected linear algebraic group over k such that
$T\otimes _k \bar k$
isomorphic to
$({{\mathbb {G}}_{\mathrm {m}}})^d \otimes _k \bar k$
over the algebraic closure
$\bar k$
of k for some integer
$d\ge 1$
. The class number,
$h(T)$
, of T is by definition, the cardinality of
$T(k)\backslash T(\mathbb {A}_{k,f})/U_T$
, where
$\mathbb {A}_{k,f}$
is the finite adele ring of k and
$U_T$
is the maximal open compact subgroup of
$T(\mathbb {A}_{k,f})$
. As a natural generalization for the class number of a number field, Ono [Reference Ono18, Reference Ono19] studied the class numbers of algebraic tori. Let
$K/k$
be a finite extension and let
$R_{K/k}$
denote the Weil restriction of scalars form K to k, then we have the following exact sequence of tori defined over k
where
$R^{(1)}_{K/k}(\mathbb {G}_{\mathrm {m},K})$
is the kernel of the norm map
$N: R_{K/k}(\mathbb {G}_{\mathrm {m},K})\longrightarrow \mathbb {G}_{\mathrm {m},k}$
. It is easy to see that
$h(R_{K/k}(\mathbb {G}_{\mathrm {m},K}))$
and
$h(\mathbb {G}_{\mathrm {m},k})$
coincide with the class numbers
$h_K$
and
$h_k$
of K and k, respectively. In order to compute the class number
$h(R^{(1)}_{K/k}(\mathbb {G}_{\mathrm {m},K}))$
, Ono [Reference Ono21] introduced the arithmetic invariant
and expressed it in terms of certain cohomological invariants when
$K/k$
is Galois. In [Reference Katayama10], Katayama proved a formula for
$E(K/k)$
for any finite extension
$K/k$
. He also studied its dual arithmetic invariant
$E'(K/k)$
and gave a similar formula. The latter gives a formula for the class number of the quotient torus
$R_{K/k}(\mathbb {G}_{\mathrm {m},K})/\mathbb {G}_{\mathrm {m},k}$
. The class numbers of general tori have been investigated by Shyr [Reference Shyr23, Theorem 1], Morishita [Reference Morishita16], González-Avilés [Reference González-Avilés7, Reference González-Avilés8], and Tran [Reference Tran24].
Besides the class number
$h(T)$
of an algebraic torus T, another important arithmetic invariant is the Tamagawa number
$\tau (T)$
. Roughly speaking, the Tamagawa number is the volume of a suitable fundamental domain. More precisely, for any connected semi-simple algebraic group G over k, one associates the group
$G(\mathbb {A}_k)$
of adelic points on G, where
$\mathbb {A}_k$
is the adele ring of k. As
$G(\mathbb {A}_k)$
is a unimodular locally compact group, it admits a unique Haar measure up to a scalar. Tamagawa defined a canonical Haar measure on
$G(\mathbb {A}_k)$
now called the Tamagawa measure. The Tamagawa number
$\tau (G)$
is then defined as the volume of the quotient space
$G(k)\backslash G(\mathbb {A}_{k})$
(or a fundamental domain of it) with respect to the Tamagawa measure. Similar to the case of class numbers, the calculation of the Tamagawa number is usually difficult. A celebrated conjecture of Weil states that any semi-simple simply connected algebraic group has Tamagawa number
$1$
. The Weil conjecture has been proved in many cases by many people (Weil, Ono, Langlands, Lai, and others) and it is finally proved by Kottwitz [Reference Kottwitz11].
For a more general linear algebraic group G, the quotient space
$G(k)\backslash G(\mathbb {A}_k)$
may not have finite volume. This occurs precisely when G has nontrivial characters defined over k, that is also the case for tori. For this reason, the necessity of introducing convergence factors in a canonical way leads to the emergence of Artin L-functions. We shall recall the definition of the Tamagawa number
$\tau (T)$
for any algebraic torus T. Then the famous analytic class number formula can be reformulated by the statement
$\tau (\mathbb {G}_{\mathrm {m},k})=1$
.
In this paper, we investigate the class numbers of CM tori. Let
$K=\prod _{i=1}^r K_i$
be a CM algebra, where each
$K_i$
is a CM field. The subalgebra
$K^+$
of elements in K fixed by the canonical involution is the product
$K^+=\prod _{i=1}^rK_i^+$
of the maximal totally real subfield
$K_i^+$
of
$K_i$
. Denote
$N_i$
as the norm map from
$K_i$
to
$K^+_i$
and
$N_{K/K^+}=\prod _{i=1}^r N_i:K\to K^+$
the norm map.
Now, we put
$T^K=\prod _{i=1}^r T^{K_i}$
with
$T^{K_i}=R_{K_i/\mathbb {Q}}(\mathbb {G}_{\mathrm {m},K_i})$
, and
$T^{K^+_i}=R_{K^+_i/\mathbb {Q}}(\mathbb {G}_{\mathrm {m},K_i})$
. We denote
where
$Q_i=Q_{K_i}:= [O^\times _{K_i}:\mu _{K_i}O^\times _{K^+_i}]$
is the Hasse unit index of the CM extension
$K_i/K_i^+$
and
$\mu _{K_i}$
is the torsion subgroup of
$O^{\times }_{K_i}$
. One has
$h(T^K)=h(K)$
and
$h(T^{K^+})=h(K^+)$
. It is known that
$Q_i\in \{1,2\}$
. Finally, we let
$t=\sum ^r_{i=1}t_i$
, where
$t_i$
is the number of primes in
$K^+_i$
ramified in
$K_i$
. Then we have the following exact sequence of algebraic tori defined over
$\mathbb {Q}$
where
$T^K_1:=\mathrm{ker} (N_{K/K^+})$
, which is the product of norm one subtori
$T^{K_i}_1:=\{x\in T^{K_i}\mid N_i(x)=1\}$
. We regard
${{\mathbb {G}}_{\mathrm {m}}}$
as a
$\mathbb {Q}$
-subtorus of
$T^{K^+}$
via the diagonal embedding. Let
$T^{K,\mathbb {Q}}$
denote the preimage of
${{\mathbb {G}}_{\mathrm {m}}}$
in
$T^K$
under the map
. We have the second exact sequence of algebraic tori over
$\mathbb {Q}$
as follows. Here, for brevity, we write N for
.
The purpose of this paper is concerned with the class number and the Tamagawa number of
$T^{K,\mathbb {Q}}$
. Let
$T({\mathbb {Z}}_p)$
denote the unique maximal open compact subgroup of
$T({\mathbb {Q}}_p)$
.
Theorem 1.1. Let
$T^{K,\mathbb {Q}}$
denote the preimage of
${{\mathbb {G}}_{\mathrm {m}}}$
in
$T^K$
under the map
as in (1.2).
-
(1) We have
where r is the number of components of K.
-
(2) We have
where
$e_{T, p}:=[\mathbb {Z}^\times _p: N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))]$
and
$S_{K/K^+}$
is the set of primes p such that there exists a place
$v|p$
of
$K^+$
ramified in K.
To make the formulas in Theorem 1.1 more explicit, one needs to calculate the indices
$e_{T,p}$
and
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
. We determine the index
$e_{T,p}$
for all primes p; the description in the case where
$p=2$
requires local norm residue symbols. For the global index
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
, we could only compute some special cases including the biquadratic fields and therefore obtain a clean formula for these CM fields. For example, if
$K=\mathbb {Q}(\sqrt {p},\sqrt {-1})$
with prime p, then
where
$h(-p):=h(\mathbb {Q}(\sqrt {-p}))$
. The global index
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
may serve another invariant which measures the complexity of CM fields and it requires further investigation. Nevertheless, the indices
$e_{T,p}$
and
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
are all powers of
$2$
(in fact
$e_{T,p}\in \{1,2\}$
if
$p\neq 2$
). Then from Theorem 1.1, we deduce
where t and r are as in Theorem 1.1, e is an integer with
$0\le e\le {e(K/K^+,\mathbb {Q}),}$
and
$e(K/K^+,\mathbb {Q})$
is the invariant defined in (4.4). In particular, we conclude that
$h(T^{K,\mathbb {Q}})$
is equal to
$h_{K}/h_{K^+}$
only up to
$2$
-power.
It is well known that the double coset space
$T^{K,\mathbb {Q}}(\mathbb {Q})\backslash T^{K,\mathbb {Q}}(\mathbb {A}_f)/T^{K,\mathbb {Q}}(\widehat {\mathbb {Z}})$
parameterizes CM abelian varieties with additional structures and conditions. Thus, Theorem 1.1 counts such CM abelian varieties and yields a upper bound for CM points of Siegel modular varieties. There are several investigations on CM points in the literature which have interesting applications and we mention a few for the reader’s information. Ullmo and Yafaev [Reference Ullmo and Yafaev25] give a lower bound for Galois orbits of CM points in a Shimura variety. This plays an important role toward the proof of the André-Oort conjecture under the Generalized Riemann hypothesis. Under the same assumption, Daw [Reference Daw4] proves an upper bound of n-torsion of the class group of a CM torus, motivated from a conjecture of Zhang [Reference Zhang32].
On the other hand, one can also express the number of connected components of a complex unitary Shimura variety
$\mathrm {Sh}_U(G,X)_{\mathbb {C}}$
as a class number of
$T^{K,\mathbb {Q}}$
or
$T^K_1$
. Thus, our result also gives an explicit formula for
$|\pi _0(\mathrm {Sh}_U(G,X)_{\mathbb {C}})|$
. This information is especially useful when the Shimura variety
$\mathrm {Sh}_U(G,X)$
(over the reflex field) has good reduction modulo p. Indeed, by the existence of a smooth toroidal compactification due to Lan [Reference Lan13], the geometric special fiber
$Sh/\overline {\mathbb {F}}_p$
of
$\mathrm {Sh}_U(G,X)$
has the same number of connected components of
$\mathrm {Sh}_U(G,X)_{\mathbb {C}}$
. In some special cases, one may be able to show that an stratum (e.g., Newton, EO, or leaves) in the special fiber is “as irreducible as possible,” namely, the intersection with each connected component of
$Sh/\overline {\mathbb {F}}_p$
is irreducible. In that case, the stratum then has the same number of irreducible components as those of
$\mathrm {Sh}_U(G,X)_{\mathbb {C}}$
.
In [Reference Achter, Altug, Gordon, Li and Rüd2], Achter studies the geometry of the reduction modulo a prime p of the unitary Shimura variety associated to
$GU(1,n-1)$
, extending the work of Bültel and Wedhorn [Reference Bültel and Wedhorn3] (in fact Achter considers one variant of moduli spaces). Though the main result asserts the irreducibility of each nonsupersingular Newton stratum in the special fiber
$Sh/\overline {\mathbb {F}}_p$
, the proof actually shows the “relative irreducibility.” That is, every nonsupersingular Newton stratum
$\mathcal {W}$
in each connected component of
$Sh/\overline {\mathbb {F}}_p$
is irreducible (and nonempty). Thus,
$\mathcal {W}$
has
$|\pi _0(Sh/\overline {\mathbb {F}}_p)|$
irreducible components and we give an explicit formula for the number of its irreducible components.
There is also a connection of class numbers of CM tori with the polarized abelian varieties over finite fields. Indeed, the set of polarized abelian varieties within a fixed isogeny class can be decomposed into certain orbits which are the analogue of genera of the lattices in a Hermitian space. When the common endomorphism algebra of these abelian varieties is commutative, each orbit is isomorphic to the double coset space associated to either
$T^{K,\mathbb {Q}}$
or
$T^K_1$
(see Section 6). Marseglia [Reference Marseglia15] gives an algorithm to compute isomorphism classes of square-free polarized ordinary abelian varieties defined over a finite field. Achter et al. [Reference Achter1] also study principally polarized ordinary abelian varieties within an isogeny class over a finite field from a different approach. They utilize the Langlands–Kottwitz counting method and express the number of abelian varieties in terms of discriminants and a product of certain local density factors, reminiscent of the Smith–Minkowski Siegel formula (cf. [Reference Gan and Yu6, Section 10]).
This paper is organized as follows. Section 2 recalls the definition of the Tamagawa number of an algebraic torus. The proof of Theorem 1.1 is given in Section 3. In Section 4, we compute the local and global indices appearing in Theorem 1.1 and give an improvement and a second proof. We calculate the class number of the CM torus associated to any biquadratic CM field in Section 5. In the last section, we discuss applications of Theorem 1.1 to polarized CM abelian varieties, connected components of unitary Shimura varieties, and polarized abelian varieties with commutative endomorphism algebras over finite fields.
2 Tamagawa numbers of algebraic tori
Following [Reference Ono18], we recall the definition of Tamagawa number of an algebraic torus T over a number field k. Fix the natural Haar measure
$dx_v$
on
$k_v$
for each place v such that it has measure
$1$
on the ring of integers
$\mathfrak {o}_v$
in the nonarchimedean case, measure
$1$
on
$\mathbb {R}/\mathbb {Z}$
in the real place case, and measure
$2$
on
$\mathbb {C}/\mathbb {Z}[i]$
in the complex place case. Let
$\omega $
be a nonzero invariant differential form of T of highest degree defined over k. To each place v, one associates a Haar measure
$\omega _v$
on
$T(k_v)$
. We say that the product of the Haar measures
converges absolutely if the product
converges absolutely, where
$T(\mathfrak {o}_v)\subset T(k_v)$
is the maximal open compact subgroup. In this case, one defines a Haar measure
$\omega _{\mathbb {A}}$
on the locally compact topological group
$T(\mathbb {A}_k)$
. Since the space of invariant differential forms is a one-dimensional k-vector space, by the product formula, the Haar measure
$\omega _{\mathbb {A}}$
does not depend on the choice of
$\omega $
, which is called the canonical measure.
However, the measure (2.1) does not converge if T admits a nontrivial rational character. Thus, we must modify the local measures by suitable convergence factors
$\lambda _v$
for each v so that the product
${\prod \limits _v} (\lambda _v \cdot \omega _v)$
is absolutely convergent on
$T(\mathbb {A}_k)$
. Such a collection
$\lambda =\left \{\lambda _v\right \}$
is called a set of convergence factors for
$\omega $
; the resulting measure is denoted by
$\omega _{\mathbb {A},\lambda }$
.
Suppose T splits over a Galois extension
$K/k$
with Galois group
$\mathfrak {g}$
. The group
of characters is a finite free
$\mathbb {Z}$
-module with a continuous action of
$\mathfrak {g}$
. Let
$\chi _T:\mathfrak {g}\to {\mathbb {C}}$
be the character associated to the representation
$\widehat T\otimes {\mathbb {Q}}$
of
$\mathfrak {g}$
.
Let
$\chi _i,\ 1\leq i\leq h,$
be all the irreducible characters of
$\mathfrak {g}$
and we denote
$\chi _1$
as the trivial character. Express
$\chi _T=\sum ^h_{i=1}a_i\chi _i$
as the sum of irreducible characters
$\chi _i$
with non-negative integral coefficients. Note that
$a_1$
is the rank of the group
$(\widehat T)^{\mathfrak {g}}$
of rational characters. The Artin L-function of
$\chi _T$
with respect to the field extension
$K/k$
is equal to
We define the number
$\rho (T)$
to be the nonzero number
$\lim _{s\rightarrow 1}(s-1)^{a_1}L(s,\chi _T,K/k)$
, that is,
On the other hand, note that there exists a finite set S of places of k such that
$T\otimes k_v$
admits a smooth model over
$\mathfrak {o}_v$
for each finite place v outside S. For such v, the reduction map
$ T(\mathfrak {o}_v)\rightarrow T(k(v)) $
is surjective, where
$k(v)\simeq \mathbb {F}_{q_v}$
is the residue field of
$\mathfrak {o}_v$
. Let
$T^{(1)}(\mathfrak {o}_v)$
be the kernel of the reduction map. By [Reference Weil27, Theorem 2.2.5] and [Reference Ono18, (3.3.2)], we have
where d is the dimension of T and
$L_v(s,\chi _T,K/k)$
is the local factor of the Artin L-function at v. We now choose the set of convergence factors
$\left \{\lambda _v\right \}$
such that
$\lambda _v$
is equal to
$1$
if v is archimedean and is equal to
otherwise, and hence define a measure
$\omega _{\mathbb {A},\lambda }$
on
$T(\mathbb {A}_k)$
.
Let
$\xi _i, i=1,\dots , a_1$
, be a basis of
$(\widehat T)^{\mathfrak {g}}$
. Define
where
$\mathbb {R}_+:=\{x>0\in \mathbb {R}\}$
. Let
$T(\mathbb {A}_k)^1$
denote the kernel of
$\xi $
; one has an isomorphism
$T(\mathbb {A}_k)/T(\mathbb {A}_k)^1 \simeq \mathbb {R}^{a_1}_{+} \subset (\mathbb {R}^\times )^{a_1}$
. Let
$d^\times t:=\prod _{i=1}^{a_1} dt_i/t_i$
be the canonical measure on
$\mathbb {R}^{a_1}_{+}$
. Let
$\omega ^1_{\mathbb {A},\lambda }$
be the unique Haar measure on
$T(\mathbb {A}_k)^1$
such that
$\omega _{\mathbb {A},\lambda }=\omega ^1_{\mathbb {A},\lambda }\cdot d^\times t$
, that is, for any measurable function F on
$T(\mathbb {A}_k)$
one has
By a well-known theorem of Borel and Harish-Chandra [Reference Platonov and Rapinchuk22, Theorem 5.6], the quotient space
$T(\mathbb {A}_k)^1/T(k)$
has finite volume with respect to a Haar measure. The Tamagawa number of T is then defined by
where
$d_k$
is the discriminant of the field k.
3 Proof of Theorem 1.1
3.1 q-Symbols and relative class numbers
Suppose
$\alpha : G \rightarrow G'$
is a homomorphism of abelian groups such that
$\ker \alpha $
and
are finite. Following Tate, the q-symbol of
$\alpha $
is defined by
It is easy to see whenever both G and
$G'$
are finite, one has
$q(\alpha )=\vert G'\vert /\vert G\vert $
. Let
denote the Galois group of
$\mathbb {Q}$
. For any isogeny
$\lambda :T\rightarrow T' $
of algebraic tori defined over
$\mathbb {Q}$
, we have the following induced maps:
Thus, we have the corresponding q-symbols. Note that
$T(\mathbb {Z})=T(\mathbb {Q})\cap [T(\mathbb {R}) \times \prod _{p<\infty }T(\mathbb {Z}_p)]$
.
Shyr [Reference Shyr23] showed that these q-symbols play a role in the connection between the ratios of Tamagawa numbers and class numbers of T and
$T'$
as follows.
Theorem 3.1. Let
$\lambda : T \rightarrow T'$
be an isogeny of algebraic tori defined over
$\mathbb {Q}$
. Then
Proof. See [Reference Shyr23, Theorem 2].
For any exact sequence
of algebraic tori defined over
$\mathbb {Q}$
, we associate a number to the exact sequence
$(E)$
by [Reference Ono19, Section 4]
Theorem 3.2. Let
and
$\hat {\iota }^\Gamma : {\hat {T}}^\Gamma \rightarrow \widehat {T'}^\Gamma $
be the maps derived from the exact sequence
$(E)$
. Then the subgroups
and
$\ker \mu $
are finite, and we have
Proof. See [Reference Ono19, Section 4.3 and Theorem 4.2.1].
Now we let
Since
$x^2N(x)^{-1}$
is of norm
$1$
, the map
$\lambda $
defined by
is an isogeny. Applying Theorem 3.1 to this
$\lambda $
and Theorem 3.2 to the exact sequence (1.2) together, we have
As
$h({{\mathbb {G}}_{\mathrm {m}}})=1$
, we obtain
We shall determine each term in (3.4).
3.2 Calculation of cokernel
Lemma 3.3. The cardinality of
is
$1$
.
Proof. Taking the character groups of (1.2), we have
Thus, it suffices to show
$(\widehat {T^K_1})^\Gamma =0$
. Again from (1.2), we have
Recall that
$T^K=\prod _{i=1}^r T^{K_i}$
with
$T^{K_i}=R_{K_i/\mathbb {Q}}\mathbb {G}_{\mathrm {m},K_i}$
. Let
. Also, note that
The norm map
$N_i$
sends x to
$x \bar {x}$
, where
$x \in K_i$
and
$\bar {x}$
is the complex conjugate of x. Therefore,
${\widehat {N_i}}(\chi _i)=\chi _i+ \bar {\chi _i}$
for
$\chi _i \in \widehat {T^{K_i^+}}$
. This shows
$(\widehat {T^{K^+_i}})^\Gamma \stackrel {\sim }{\longrightarrow }(\widehat {T^{K_i}})^\Gamma $
. Note that the left exact sequence (
$*$
)
$\otimes _{\mathbb {Z}} \mathbb {Q}$
is also right exact. Thus,
$(\widehat {T^{K_i}_1})^\Gamma \otimes \mathbb {Q}=0$
and
$(\widehat {T^{K_i}_1})^\Gamma $
is a torsion
$\mathbb {Z}$
-module. It follows that
$(\widehat {T^{K_i}_1})^\Gamma =0$
, because it is a submodule of a finite free
$\mathbb {Z}$
-module
$(\widehat {T^{K_i}_1})$
. It follows that
.
We remark that Lemma 3.3 also follows from
. The proof will be also used in Lemma 3.8.
3.3 Calculation of indices of rational points
Recall that for any commutative
$\mathbb {Q}$
-algebra R, the groups of R-points of
$T^K$
and
$T^{K,\mathbb {Q}}$
are
respectively. For
$v \in V_K$
, the union of the sets
$V_{K_i}$
of places of
$K^+_i$
for
$1\leq i\leq r$
, we put
$K_v=(K_i)_v$
if
$v\in V_{K_i}$
. For any prime p, let
$S_p$
be the set of places of
$K^+$
lying over p. We have
and
Note that
$x_p$
is uniquely determined by
$(x_v)_v$
and we may also represent an element x in
$T^{K,\mathbb {Q}}({\mathbb {Q}}_p)$
by
$(x_v)_v$
.
Lemma 3.4. We have
$N(T^{K,\mathbb {Q}}(\mathbb {Q}_p))\cap \mathbb {Z}_p^\times =N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))$
for every prime number p.
Proof. Clearly,
$N(T^{K,\mathbb {Q}}(\mathbb {Q}_p))\cap \mathbb {Z}_p^\times \supset N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))$
. We must prove the other inclusion
$N(T^{K,\mathbb {Q}}(\mathbb {Q}_p))\cap \mathbb {Z}_p^\times \subset N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))$
. Suppose
$x=(x_v)\in T^{K,\mathbb {Q}}({\mathbb {Q}}_p)\cap N^{-1}({\mathbb {Z}}_p^\times )$
. We will find an element
$x^{\prime }=(x^{\prime }_v)\in T^{K,\mathbb {Q}}({\mathbb {Z}}_p)$
such that
$N(x)=N(x^{\prime })$
.
Suppose v is inert or ramified in K and let w be the unique place of K over v. Then
$x_v\in K_v=K_w$
and
$N(x_v)\in {\mathbb {Z}}_p^\times $
. We have
and hence
$x_v\in O_{K_w}^\times $
.
Suppose
$v=w \bar w$
splits in K. Then
$x_v=(x_w, x_{\bar w})\in K_v=K_w\times K_{\bar w}=K^+_v\times K^+_v$
and
$N(x_v)=x_w x_{\bar w}\in {\mathbb {Z}}_p^\times $
. We have
for some
$a_v\in \mathbb {Z}$
. Put
$x_v^{\prime }:=(\varpi _v^{-a_v} x_w, \varpi _v^{a_v} x_{\bar w})$
, where
$\varpi _v$
is a uniformizer of
$K^+_v$
. Clearly,
$x^{\prime }_v\in O_{K_v}^\times $
and
$N(x_v)=N(x^{\prime }_v)$
.
Now suppose
$y\in N(T^{K,\mathbb {Q}}({\mathbb {Q}}_p))\cap \mathbb {Z}_p^\times $
and
$N(x)=y$
for some
$x\in T^{K, \mathbb {Q}}({\mathbb {Q}}_p)$
. Set
$x^{\prime }:=(x^{\prime }_v)$
with
$x^{\prime }_v= x_v$
if v is inert or ramified in K, and
$x^{\prime }_v$
as above if v splits in K. Then
$y=N(x)=N(x^{\prime })\in N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))$
. This proves
$N(T^{K,\mathbb {Q}}(\mathbb {Q}_p))\cap \mathbb {Z}_p^\times \subset N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))$
.
Lemma 3.5. We have
$[N(T^{K,\mathbb {Q}}(\mathbb {A}))\cap {{\mathbb {G}}_{\mathrm {m}}}(\mathbb {Q}): N(T^{K,\mathbb {Q}}(\mathbb {Q}))]=1$
and
Proof. Since
$T^{K,\mathbb {Q}}(\mathbb {A})=\{x\in \mathbb {A}_K^\times \mid N(x)\in \mathbb {A}^\times \}$
, we have
$N(T^{K,\mathbb {Q}}(\mathbb {A}))=N(\mathbb {A}_K^\times )\cap \mathbb {A}^\times $
. Applying the norm theorem [Reference Ono19, Theorem 6.1.1], we have
$N(\mathbb {A}_K^\times )\cap (K^+)^\times =N(K^\times )$
. Hence
This proves the first statement. The second statement then follows from Lemma 3.3 and Theorem 3.2.
3.4 Calculation of q-symbols
We are going to evaluate each q-symbol in (3.4). Recall the isogeny in (3.2), namely
Note that
$\ker \lambda =\{x\in T^{K,\mathbb {Q}}\mid N(x)=1,\ x^2=1\}=\{x\in T^K_1|\ x^2=1\}$
. Hence, we have
$\ker \lambda = \ker \mathrm {Sq}_{T^K_1},$
where
$\mathrm {Sq}_{T^K_1}: T^K_1 \rightarrow T^K_1,\ x \mapsto x^2$
is the squared map.
Lemma 3.6. Suppose
$d=[K^+:\mathbb {Q}]$
. The q-symbol of
$\lambda _\infty $
is equal to
$2^{-d+1}$
.
Proof. Since
$K=\prod _{i=1}^r K_i$
is a CM algebra, we have
$T^K(\mathbb {R})=(K\otimes _{\mathbb {Q}} \mathbb {R})^\times =(\mathbb {C}^d)^\times $
. According to (3.8), we have
and hence the exact sequence
Since
$N(\mathbb {C}^\times )=\mathbb {R}^\times _{+}$
is connected, the image of
$\lambda _\infty $
is
$(S^1)^d\times \mathbb {R}^\times _+$
and
. Therefore,
$q(\lambda _\infty )=2/\vert \{\pm 1\}^d\vert =2^{-d+1}.$
Lemma 3.7. The q-symbol of
$\lambda _{\mathbb {Z}}$
is equal to
$2$
.
Proof. It is clear that
$N(x)=1$
for
$x\in T^{K,\mathbb {Q}}(\mathbb {Z})$
. Note that any element
$x\in O_K^\times $
with
$x\bar x=1$
is a root of unity. Then
where
$\mu _{K_i}$
is the group of roots of unity in
$K_i$
. Since
$T^{K,\mathbb {Q}}(\mathbb {Z})$
and
$(T^K_1\times {{\mathbb {G}}_{\mathrm {m}}}) (\mathbb {Z})$
are finite, we have
Lemma 3.8. The q-symbol of
$\hat {\lambda }^\Gamma $
is equal to
$1$
.
Proof. In the proof of Lemma 3.3, we have showed that
$(\widehat {T^K_1})^\Gamma =0$
. Therefore, the map
$\hat {\lambda }:(\widehat {{{\mathbb {G}}_{\mathrm {m}}}})^\Gamma \times (\widehat {T^K_1})^\Gamma \rightarrow (\widehat {T^{K,\mathbb {Q}}})^\Gamma $
is just given by
$\hat {N}:(\widehat {{{\mathbb {G}}_{\mathrm {m}}}})^\Gamma \rightarrow (\widehat {T^{K,\mathbb {Q}}})^\Gamma .$
The map
$\hat {N}$
is in fact an isomorphism from (3.5). Therefore,
$q(\hat {\lambda }^\Gamma )=1$
.
Lemma 3.9. Let
$A \xrightarrow {\alpha } B \xrightarrow {\beta } C$
be group homomorphisms of abelian groups with finite
$\ker \beta \alpha $
and
. Then the cardinality of
is equal to
Proof. The first equality is obvious and the second equality follows by applying the snake lemma to the following diagram
Lemma 3.10. We have
where
$e_{T,p}=[\mathbb {Z}_p^\times : N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))]$
and
$d=[K^+:\mathbb {Q}]$
.
Proof. By (3.7), every element
$x\in T^{K, \mathbb {Q}}(\mathbb {Z}_p)$
is of the form
$((x_v)_{v \in S_p}, x_p)$
with
$N(x_v)=x_p$
for all
$ v \in S_p$
. Consider the homomorphisms
where
and recall that
. It is easy to see that the composition
$\lambda _p^c \circ m$
is the squared map
$\mathrm {Sq}:(y,y^{\prime })\mapsto (y^2,(y^{\prime })^2)$
. Therefore, by Lemma 3.9, we have
First,
. Suppose
$y=((y_v)_v, y_p) \in \ker \mathrm {Sq}$
. Then
$y_p=\pm 1$
, and
$y_v^2=1 \ \forall v$
. If v is inert or ramified in K, then
$y_v=\pm 1$
. If v splits in K, then
$y_v=(y_w, y_{\bar {w}})$
, and
$y_w=\pm 1$
,
$y_{\bar {w}}= \pm 1$
. Since
$N(y_v)=1$
,
$y_v=(1, 1)$
or
$(-1, -1)$
, that is,
$y_v=\pm 1$
. We conclude that
$\ker \mathrm {Sq} =\{\pm 1\}^{S_p} \times \{\pm 1\}$
.
On the other hand, since
$\ker N \subset {\rm im} m$
, we have
Recall that
${\rm im} m=\{((x_v x_p)_{v \in S_p}, x_p^2)\}$
, one has
$N({\rm im} m)=\{x_p^2 \mid x_p \in \mathbb {Z}_p^\times \}$
. Therefore,
. Clearly,
$\ker m=\{\pm 1\}.$
Now,
The lemma then follows from Lemma 3.12.
For proving Lemma 3.12, we recall the structure theorem of p-adic local units.
Proposition 3.11. Let
$k/{\mathbb {Q}}_p$
be a finite extension of degree d with ring of integers
$O_k$
and residue field
${\mathbb {F}}_q$
. Then
and
where
$p^a=\vert \mu _{p^\infty }(O_k)\vert .$
Proof. See [Reference Neukirch17, Proposition 5.7, p. 140].
Lemma 3.12. We have
and
where
$d=[K^+:\mathbb {Q}]$
and
$S_p$
is the set of places of
$K^+$
lying over p.
Proof. Note that
$T^K_1(\mathbb {Z}_p)= \prod _{v\in S_p} O_{K_v}^{(1)}$
, where
$O_{K_v}^{(1)}$
consists of norm one elements in
$O_{K_v}^\times $
. We need to calculate
$[O_{K_v}^{(1)}:(O_{K_v}^{(1)})^2]$
for
$v\in S_p$
.
Let
$d_v=[(K^+)_v:\mathbb {Q}_p]$
and consider the exact sequence
Note that
$\hbox{rank}_{\mathbb {Z}_p}O^\times _{K_{v}}=2d_v$
. Since
$[O_{K^+_v}^\times :N(O_{K_v}^\times )]$
is finite, we have
$\hbox{rank}_{\mathbb {Z}_p}N(O_{K_v}^\times )=\hbox{rank}_{\mathbb {Z}_p}O_{K^+_v}^\times =d_v$
and hence
$\hbox{rank}_{\mathbb {Z}_p}O^{(1)}_{K_v}=d_v$
.
By Proposition 3.11, we have
$O^{(1)}_{K_v} \simeq A \oplus B \oplus \mathbb {Z}_p^{d_v}$
, where A is a finite cyclic group of prime-to-p order and B is a finite cyclic group of p-power order.
Suppose p is odd. Then
$O^{(1)}_{K_v}/(O^{(1)}_{K_v})^2 \simeq A/ 2A$
. Since
$O^{(1)}_{K_v}$
contains
$-1$
, we have
$A/2A=\mathbb {Z}/2\mathbb {Z}$
. Thus,
$[O^{(1)}_{K_v}:(O^{(1)}_{K_v})^2]=2$
and
$[T^K_1(\mathbb {Z}_p): T^K_1(\mathbb {Z}_p)^2]= \prod _{v\in S_p}[O^{(1)}_{K_v}:(O^{(1)}_{K_v})^2]=2^{\vert S_p \vert }$
.
Suppose p is even. The group
$O^{(1)}_{K_v}$
contains
$-1$
; Therefore,
$B/2B=\mathbb {Z}/2\mathbb {Z}$
. We have
$O^{(1)}_{K_v}/(O^{(1)}_{K_v})^2 \simeq \mathbb {Z}/2\mathbb {Z} \oplus (\mathbb {Z}/2 \mathbb {Z})^{d_v}.$
Therefore,
$[O^{(1)}_{K_v}:(O^{(1)}_{K_v})^2]=2^{1+d_v}$
and
$[T^K_1(\mathbb {Z}_p): T^K_1(\mathbb {Z}_p)^2]= \prod _{v\in S_p}[O^{(1)}_{K_v}:(O^{(1)}_{K_v})^2]=2^{\vert S_p\vert +d}$
as
$\sum _{v \in S_p}d_v=d$
. This proves the first result.
The second result follows from the first result and
$\vert \ker \mathrm {Sq}_{T^K_1(\mathbb {Z}_p)}\vert = \vert \{\pm 1\}^{S_p}\vert =2 ^{\vert S_p\vert }$
(see the proof of Lemma 3.10).
3.5 Proof of Theorem 1.1
-
(1) By [Reference Ono20, Remark, p. 128], we have
$\tau ({{\mathbb {G}}_{\mathrm {m}}})=1$
, and
$\tau (T^{K_i}_1)=2$
for each iand we conclude from (3.1), Theorem 3.2 and Lemma 3.5. -
(2) By (3.4) and Lemmas 3.3, 3.5, 3.6, 3.7, 3.8, and 3.10, we obtain
It is known (see [Reference Shyr23, (16), p. 375]) that
where
$Q_i$
is the Hasse unit index of
$K_i/K^+_i$
, and
$t_i$
is the number of primes of
$K^+_i$
ramified in
$K_i$
. Thus,
This completes the proof of the theorem.
4 Local and global indices
4.1 Local indices
Keep the notation of the previous section. Denote by
$f_v$
, the inertia degree of a finite place v of
$K^+$
. Let
$N_v:=N(O_{K_v}^\times )$
and
$H_v:={\mathbb {Z}}_p^\times \cap N_v $
, if
$v|p$
. We define an integer
$e(K/K^+,\mathbb {Q}, p)$
, where p is a prime, as follows. For
$p\neq 2$
, let
Suppose
$p=2$
. For any place
$v|2$
of
$K^+$
, let
$\Phi _v:\mathbb {Z}_2^\times \to O_{K^+_v}^\times /(O_{K^+_v}^\times )^2$
be the natural map. Consider the condition
where
$\overline N_v$
is the image of
$N_v$
in
$O_{K^+_v}^\times /(O_{K^+_v}^\times )^2$
. Note that
$[\mathbb {Z}_2^\times :H_v]=2$
if and only if (4.2) holds (see an explanation in the proof of Proposition 4.1(4)). Define
Define
Note that
$e(K/K^+,\mathbb {Q},p)=0$
if
$p\not \in S_{K/K^+}$
. Recall that
$e_{T,p}=[\mathbb {Z}^\times _p:N(T^{K,\mathbb {Q}}(\mathbb {Z}_p))]$
. We shall evaluate
$e_{T,p}$
and interpret
$e_{T,p}$
by using the invariant
$e(K/K^+,\mathbb {Q})$
in the following.
Proposition 4.1.
-
(1) We have
$e_{T,p}=[{\mathbb {Z}}_p^\times : \cap _{v\in S_p} H_v]$
. -
(2) We have
$e_{T,p}|2$
if
$p\neq 2$
, and
$e_{T,2}|4$
. -
(3) Suppose
$p\neq 2$
. Then
$e_{T,p}=2$
if and only if there exists a place
$v\in S_p$
such that v is ramified in K and
$f_v$
is odd. -
(4) Suppose
$p=2$
. Then
$2|e_{T,p}$
if and only if there exists
$v\in S_p$
satisfying (4.2). Moreover,
$e_{T,p}=4$
if and only if there exist two places
$v_1, v_2\in S_p$
satisfying (4.2) and
$H_{v_1}\neq H_{v_2}$
.
Proof.
-
(1) It follows immediately from the description of
$T^{K,\mathbb {Q}}({\mathbb {Z}}_p)$
that
$N(T^{K,\mathbb {Q}}({\mathbb {Z}}_p))=\cap _{v\in S_p} H_v$
. -
(2) Consider the inclusion
${\mathbb {Z}}_p^\times /H_v \hookrightarrow O_{K^+_v}^\times /N(O_{K_v}^\times )$
. The latter group has order dividing
$2$
by the local norm index theorem. Thus,
$H_v\subset {\mathbb {Z}}_p^\times $
is of index
$1$
or
$2$
. In particular
$H_v$
contains
$({\mathbb {Z}}_p^\times )^2$
. It follows that the intersection
$\cap H_v$
contains
$({\mathbb {Z}}_p^\times )^2$
, and that
$e_{T,p}$
divides
$[{\mathbb {Z}}_p^\times :({\mathbb {Z}}_p^\times )^2]$
, which is
$4$
or
$2$
according as
$p=2$
or not. -
(3) As
$p\neq 2$
, we have
$e_{T,p}=2$
if and only if there exists
$v\in S_p$
such that
$[{\mathbb {Z}}_p^\times :H_v]=2$
. We must show that
$[{\mathbb {Z}}_p^\times :H_v]=2$
if and only if v is ramified in K and
$f_v$
is odd. If v is unramified in K, then
$N_v:=N(O_{K_v}^\times )$
is equal to
$O_{K^+_v}^\times $
and
$[{\mathbb {Z}}_p^\times :H_v]=1$
. Thus,
$[{\mathbb {Z}}_p^\times :H_v]=2$
only when v is ramified, and we assume this now. Note that
$N_v\subset O_{K^+_v}^\times $
is the unique subgroup of index
$2$
and it contains the principal unit subgroup
$1+\pi _v O_{K^+_v}$
. Therefore, its image
$\overline N_v$
in the residue field
$\kappa ^\times =\mathbb {F}_{q_v}^\times $
is equal to
$(\mathbb {F}_{q_v}^\times )^2$
. The inclusion
${\mathbb {Z}}_p^\times \hookrightarrow O_{K^+_v}^\times $
induces the inclusion
${\mathbb {F}}_p^\times \hookrightarrow \mathbb {F}_{q_v}^\times $
for which the image
$\overline H_v$
of
$H_v$
is equal to
${\mathbb {F}}_p^\times \cap \overline N_v$
. Since
$\overline N_v$
is the unique subgroup of
$\mathbb {F}_{q_v}^\times $
of index
$2$
, we have Since
$\mathbb {F}_{q_v}^\times $
is cyclic,
${\mathbb {F}}_p^\times \subset (\mathbb {F}_{q_v}^\times )^2$
if and only if
$p-1|(q_v-1)/2$
. The latter is equivalent to
$2|(p^{f_v-1}+\dots +1)$
or that
$f_v$
is even. Thus,
$[{\mathbb {Z}}_p^\times :H_v]=2$
if and only if
$f_v$
is odd. -
(4) Since
$N_v$
contains
$(O_{K^+_v}^\times )^2$
, we have
$[{\mathbb {Z}}_p^\times :H_v]=[\Phi _v({\mathbb {Z}}_p^\times ):\Phi _v({\mathbb {Z}}_p^\times )\cap \overline N_v]$
, which is equal to
$2$
if and only if
$\Phi _v({\mathbb {Z}}_p^\times ) \not \subset \overline N_v$
. This proves the first statement. Clearly,
$e_{T,p}=4$
if and only if there exist two places
$v_1,v_2\in S_p$
such that
$[{\mathbb {Z}}_p^\times :H_{v_1}]=[{\mathbb {Z}}_p^\times :H_{v_2}]=2$
and
$H_{v_1}\neq H_{v_2}$
. Then the second statement follows from what we have just proved.
Corollary 4.2. We have
$e_{T,p}=2^{e(K/K^+,\mathbb {Q},p)}$
for all primes p.
By Proposition 4.1, one has
$e_{T,2}|4$
. We now give an example of
$T=T^{K,\mathbb {Q}}$
such that
$e_{T,2}=4$
. Let E and
$E^{\prime }$
be two imaginary quadratic field such that
$2$
is ramified in both E and
$E^{\prime }$
. We also assume that
$N_{E_2/\mathbb {Q}_2}(O_{E_2}^\times )\neq N_{E^{\prime }_2/\mathbb {Q}_2}(O_{E^{\prime }_{2}}^\times )$
. Put
$K:=E\times E^{\prime }$
, the product of E and
$E^{\prime }$
(not the composite), and then
$K^+=\mathbb {Q}\times \mathbb {Q}$
. Let
$v_1$
and
$v_2$
be the two places of
$K^+$
over
$2$
. We have
$H_{v_1}=N_{E_2/\mathbb {Q}_2}(O_{E_2}^\times )$
and
$H_{v_2}=N_{E^{\prime }_2/\mathbb {Q}_2}(O_{E^{\prime }_{2}}^\times )$
. Then we see that
$[\mathbb {Z}_2^\times :H_{v_1}]=[\mathbb {Z}_2^\times :H_{v_2}]=2$
, and that by Proposition 4.1(1),
$e_{T,2}=[\mathbb {Z}_2^\times :H_{v_1} \cap H_{v_2}]=4$
.
For
$f\in \mathbb {N}$
and
$q=p^f$
, denote by
$\mathbb {Q}_q$
the unique unramified extension of
${\mathbb {Q}}_p$
of degree f and
$\mathbb {Z}_q$
the ring of integers.
Lemma 4.3. Let
$f\in \mathbb {N}$
and
$q=2^f$
.
-
(1)
$\mathbb {Z}_q^\times /(\mathbb {Z}_q^\times )^2\simeq (1+2\mathbb {Z}_q)/(1+2\mathbb {Z}_q)^2\simeq (\mathbb {Z}/2\mathbb {Z})^{f+1}$
. -
(2) We have
$1+8\mathbb {Z}_q\subset (1+2\mathbb {Z}_q)^2 \subset 1+4\mathbb {Z}_q$
. Under the isomorphism
$(1+4\mathbb {Z}_q)/(1+8\mathbb {Z}_q)\simeq {\mathbb {F}}_q (\,[1+4a]\mapsto \bar a\,)$
, the subgroup
$(1+2\mathbb {Z}_q)^2/(1+8\mathbb {Z}_q)$
corresponds to the image
$\varphi ({\mathbb {F}}_q)$
of the Artin–Schreier map
$\varphi (x)=x^2-x: {\mathbb {F}}_q\to {\mathbb {F}}_q$
. -
(3) We have
Proof.
-
(1) The Teichmüller lifting
$\omega $
gives a splitting of the exact sequence
$1\to (1+2\mathbb {Z}_q) \to \mathbb {Z}_q^\times \to {\mathbb {F}}_q^\times \to 1$
. Thus,
$\mathbb {Z}_q^\times ={\mathbb {F}}_q^\times \times (1+2\mathbb {Z}_q)$
and hence
$(\mathbb {Z}_q^\times )^2={\mathbb {F}}_q^\times \times (1+2\mathbb {Z}_q)^2$
because
$q-1$
is odd. This proves the first isomorphism. The second isomorphism follows from
$1+2\mathbb {Z}_q\simeq \{\pm 1 \} \times \mathbb {Z}_2^f$
; see Proposition 3.11. -
(2) For
$a\in \mathbb {Z}_q$
, we have
$(1+2a)^2=1+4(a^2+a)$
. Therefore,
$(1+2\mathbb {Z}_q)^2 \subset 1+4\mathbb {Z}_q$
. On the other hand, for any
$b\in \mathbb {Z}_q$
, the equation
$T^2+T=2b$
has a solution in
$\mathbb {Z}_q$
by Hensel’s lemma. This proves
$1+8\mathbb {Z}_q\subset (1+2\mathbb {Z}_q)^2$
. The image of
$(1+2\mathbb {Z}_q)^2/(1+8\mathbb {Z}_q)$
in
${\mathbb {F}}_q$
consists of elements
$\bar a^2+\bar a=\varphi (\bar a)$
for all
$\bar a\in {\mathbb {F}}_q$
. -
(3) It is clear that
$1+8 \mathbb {Z}_2\subset \mathbb {Z}_2^\times \cap (1+2\mathbb {Z}_q)^2\subset 1+4\mathbb {Z}_2$
. Note that
$1+4 \mathbb {Z}_2\subset (1+2\mathbb {Z}_q)^2$
if and only if
$5\in (1+2\mathbb {Z}_q)^2$
. The latter is also equivalent to that the equation
$1=t^2-t$
is solvable in
${\mathbb {F}}_q$
by (2). Since
$\mathbb {F}_4=\mathbb {F}_2[t]/(t^2+t+1)$
, the previous condition is the same as
$\mathbb {F}_4\subset {\mathbb {F}}_q$
, or equivalently,
$2|f$
.
Lemma 4.4. Let the notation be as in Lemma 4.3.
-
(1) If f is even, then there are
$2(2^f-1)$
(respectively
$2^{f+1}$
) ramified quadratic field extensions
$K/\mathbb {Q}_q$
such that
$\mathbb {Z}_2^\times \subset N(O_K^\times )$
(resp.
$\mathbb {Z}_2^\times \not \subset N(O_K^\times )$
). -
(2) If f is odd, then there are
$2^{f}-2$
(respectively
$2^{f+2}-2^f$
) ramified quadratic field extensions
$K/\mathbb {Q}_q$
such that
$\mathbb {Z}_2^\times \subset N(O_K^\times )$
(respectively
$\mathbb {Z}_2^\times \not \subset N(O_K^\times )$
).
Proof.
-
(1) Since
$\mathbb {Q}_q^\times /(\mathbb {Q}_q^\times )^2\simeq (\mathbb {Z}/2\mathbb {Z})^{f+2}$
, there are
$2^{f+2}-1$
subgroups
$\widetilde N\subset \mathbb {Q}_q^\times $
of index
$2$
. By the local class field theory, there are
$2^{f+2}-1$
quadratic extensions
$K/\mathbb {Q}_q$
, and
$2^{f+2}-2$
of them are ramified. On the other hand, since
$\mathbb {Z}_q^\times /(\mathbb {Z}_q^\times )^2\simeq (\mathbb {Z}/2\mathbb {Z})^{f+1}$
, there are
$2^{f+1}-1$
subgroups
$N\subset \mathbb {Z}_q^\times $
of index
$2$
. It is not hard to see that for each N there are exactly two ramified extensions
$K/\mathbb {Q}_q$
such that
$N=N(O_K^\times )$
. Suppose
$2|f$
, then
$\Phi (\mathbb {Z}_2^\times )$
is a one-dimensional subspace in
$\mathbb {Z}_q^\times /(\mathbb {Z}_q^\times )^2=(\mathbb {Z}/2\mathbb {Z})^{f+1}$
by Lemma 4.3(3). Therefore, there are
$2^f-1$
subspaces
$\overline N$
of co-dimension one containing
$\Phi (\mathbb {Z}_2^\times )$
, and
$2^f$
subspaces
$\overline N$
of co-dimension one not containing
$\Phi (\mathbb {Z}_2^\times )$
. -
(2) Suppose that f is odd. By Lemma 4.3(3),
$\Phi (\mathbb {Z}_2^\times )$
is a two-dimensional subspace in
$\mathbb {Z}_q^\times /(\mathbb {Z}_q^\times )^2=(\mathbb {Z}/2\mathbb {Z})^{f+1}$
. There are
$2^{f-1}-1$
subspaces
$\overline N$
of co-dimension one containing
$\Phi (\mathbb {Z}_2^\times )$
, and the other
$2^{f+1}-2^{f-1}$
subspaces
$\overline N$
not containing
$\Phi (\mathbb {Z}_2^\times )$
. This proves the lemma.
Lemma 4.5. Let
$F/\mathbb {Q}_2$
be a finite extension of
$\mathbb {Q}_2$
, and let
$L/F$
be a quadratic extension of F. Then
$\mathbb {Z}_2^\times \subset N_{L/F}(O_L^\times )$
if and only if the norm residue symbols
$(-1,L/F)=1$
and
$(5,L/F)=1$
.
Proof. This follows directly from the basic fact that
$\mathbb {Z}_2^\times =\{\pm 1\}\times \overline {\langle 5\rangle }$
, where
$\overline {\langle 5\rangle }$
is the closure of the cyclic subgroup
${\langle 5\rangle }$
in
$\mathbb {Z}_2^\times $
.
4.2 Global indices
We obtain the following partial results for the global index
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
.
Lemma 4.6.
-
(1) We have
where
$\mathbb {Q}_+:=\mathbb {Q}^\times \cap \mathbb {R}_+$
.
-
(2) The index
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
divides
$\prod _{p\in S_{K/K^+}} e_{T,p}$
.
Proof.
-
(1) We have
$\mathbb {A}^\times =\mathbb {R}^\times \times \mathbb {A}_f^\times $
and
$N(T^{K,\mathbb {Q}}(\mathbb {A})) =\mathbb {R}_{+}\times N(T^{K,\mathbb {Q}}(\mathbb {A}_f))$
. We use
$\mathbb {Q}^\times $
to reduce
$\mathbb {R}^\times $
to
$\mathbb {R}_{+}$
. Thus, (4.5) -
(2) Since
$\widehat {\mathbb {Z}}^\times \cap N(T^{K,\mathbb {Q}}(\mathbb {A}_f))\cdot \mathbb {Q}_{+} \supset \widehat {\mathbb {Z}}^\times \cap N(T^{K,\mathbb {Q}}(\mathbb {A}_f)),$
the group is a quotient of
$\widehat {\mathbb {Z}}^\times /\left ( \widehat {\mathbb {Z}}^\times \cap N(T^{K,\mathbb {Q}}(\mathbb {A}_f)) \right )$
. On the other hand, we have by Lemma 3.4. Therefore,
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
divides This proves the lemma.
Lemma 4.7. Suppose K is a CM field which contains two distinct imaginary quadratic fields
$E_1$
and
$E_2$
. Then
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]=1$
.
Proof. By the global norm index theorem,
$[\mathbb {A}^\times : N(\mathbb {A}_{E_i}^\times )\cdot \mathbb {Q}^\times ]=2$
for
$i=1,2$
. Since
$E_1\neq E_2$
, the subgroup
$N(\mathbb {A}_{E_1}^\times )\cdot N(\mathbb {A}_{E_2}^\times )\cdot \mathbb {Q}^\times $
of
$\mathbb {A}^\times $
generated by
$N(\mathbb {A}_{E_i}^\times )\cdot \mathbb {Q}^\times $
(
$i=1,2$
) strictly contains
$N(\mathbb {A}_{E_1}^\times )\cdot \mathbb {Q}^\times $
. Thus,
$[\mathbb {A}^\times : N(\mathbb {A}_{E_1}^\times )\cdot N(\mathbb {A}_{E_2}^\times )\cdot \mathbb {Q}^\times ]=1$
. On the other hand, the subgroup
$N(T^{K,\mathbb {Q}}(\mathbb {A}))$
contains
$N(\mathbb {A}_{E_i}^\times )$
for
$i=1,2$
. Therefore,
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]=1$
.
4.3 Consequences and a second proof
Using our computation of the local index
$e_{T,p}$
and
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
, we obtain the following improvement of Theorem 1.1.
Theorem 4.8. Let the notation be as in Theorem 1.1. We have
where e is an integer with
$0\le e\le {e(K/K^+,\mathbb {Q})}$
, where
$e(K/K^+,\mathbb {Q})$
is the invariant defined in (4.4).
Since the Hasse unit index
$Q_{K}$
is a power of
$2$
, we obtain the following result from Theorem 4.8.
Proposition 4.9. The class number
$h(T^{K,\mathbb {Q}})$
is equal to
$h_K/h_{K^+}$
up to a power of
$2$
.
A second proof of Theorem 1.1.
Put
$T:=T^{K,\mathbb {Q}}$
and
$T^{\prime }:=T^K_1$
. We first show that the sequence
is exact. The kernel of N is
If
$t=q u\in T^{\prime }(\mathbb {A}_f)\cap T(\mathbb {Q})\cdot T(\widehat {\mathbb {Z}})$
with
$q\in T(\mathbb {Q})$
and
$u\in T(\widehat {\mathbb {Z}})$
, Then
So
$q\in T^{\prime }(\mathbb {Q})$
and
$u\in T^{\prime }(\widehat {\mathbb {Z}})$
and
$T^{\prime }(\mathbb {A}_f)\cap T(\mathbb {Q})\cdot T(\widehat {\mathbb {Z}})=T^{\prime }(\mathbb {Q})\cdot T^{\prime }(\widehat {\mathbb {Z}})$
. This proves the exactness of (4.6).
We now prove that
Suppose
$t=q u \in N(T(\mathbb {A}_f))\cap N(T(\widehat {\mathbb {Z}}))\cdot \mathbb {Q}_+$
with
$q\in \mathbb {Q}^\times _+$
and
$u\in N(T(\widehat {\mathbb {Z}}))$
. Then q is a local norm everywhere. Thus, there is an element
$x\in K^\times $
such that
$N(x)=q$
by the Hasse principle. By the definition, the element x lies in
$T(\mathbb {Q})$
and hence
$q\in N(T(\mathbb {Q}))$
. This verifies (4.7).
Note that
By (4.5),
$[\mathbb {A}_f^\times : N(T(\mathbb {A}_f))\cdot \mathbb {Q}_+]=[\mathbb {A}^\times : N(T(\mathbb {A}))\cdot \mathbb {Q}^\times ]$
. It is also easy to see
$[\mathbb {A}_f^\times :N(T(\widehat {\mathbb {Z}}))\cdot \mathbb {Q}_+]=\prod _{p} [{\mathbb {Z}}_p^\times : N(T({\mathbb {Z}}_p))][\mathbb {A}_f^\times :N(T(\widehat {\mathbb {Z}}))\cdot \mathbb {Q}^\times _+]$
. Then Theorem 1.1 follows from (4.6) and (4.7).
Question 4.10.
-
(1) Let
$N:\widetilde T\to T$
be a homomorphism of algebraic tori over
$\mathbb {Q}$
such that
$T^{\prime }:=\ker N$
is again an algebraic torus. Then by Lang’s theorem, the map
$N:\widetilde T(\mathbb {A})\to T(\mathbb {A})$
is open and then
$[T(\mathbb {A}): N(\widetilde T(\mathbb {A}))\cdot T(\mathbb {Q})]$
is finite. What is the index
$[T(\mathbb {A}): N(\widetilde T(\mathbb {A}))\cdot T(\mathbb {Q})]$
? When
$\widetilde T=T^K$
,
$T=T^k$
and N is the norm map, where
$K/k$
is a finite extension of number fields, then
$[T(\mathbb {A}): N(\widetilde T(\mathbb {A}))\cdot T(\mathbb {Q})]=[\mathbb {A}_k^\times :k^\times N(\mathbb {A}_K^\times )]$
is nothing but the global norm index and it is equal to the degree
$[K_0:k]$
of the maximal abelian subextension
$K_0$
of k in K [Reference Lang14, IX, Section 5, p. 193]. The global norm index theorem requires deep analytic results. It is also expected that one may equally need deep analytic and arithmetic results for computing
$[T(\mathbb {A}): N(\widetilde T(\mathbb {A}))\cdot T(\mathbb {Q})]$
. -
(2) Suppose
$\lambda :T\to T^{\prime }$
is an isogeny of tori over
$\mathbb {Q}$
of degree d. Is it true that for any prime
$\ell \nmid d$
, the
$\ell $
-primary parts of
$h(T)$
and
$h(T^{\prime })$
are the same? This is inspired by Proposition 4.9.
5 Examples
5.1 Imaginary quadratic fields
Suppose K is an imaginary quadratic field. Then
$K^+=\mathbb {Q}$
and
$T^{K,\mathbb {Q}}=T^K$
. Thus, we have
$h(T^{K,\mathbb {Q}})=h(T^K)=h_K$
without any computation. On the other hand, we use Theorem 1.1 to compute
$h(T^{K,\mathbb {Q}})$
. It is easy to compute that
$Q=1$
,
$e_{T,p}=2$
for each
$p\in S_{K/K^+}$
and we have
by the global norm index theorem. Thus, we have
where t is the number of rational primes ramified in K. This also gives the result
$h(T^{K,\mathbb {Q}})=h_K$
.
5.2 Biquadratic CM fields
Let K be a biquadratic CM field and F the unique real quadratic subfield. Write
$F=\mathbb {Q}(\sqrt {d})$
, where
$d>0$
is the unique square-free positive integer determined by F, and
$K=EF$
, where
$E=\mathbb {Q}(\sqrt {-j})$
for a square-free positive integer j. Finite places of F,
$E,$
and K will be denoted by v,
$u,$
and w, respectively. Recall that
$S_{K/F}$
denotes the set of primes p such that there exists a place
$v|p$
of F which is ramified in K.
Lemma 5.1. Let
$K=EF=\mathbb {Q}(\sqrt {d},\sqrt {-j})$
be a biquadratic CM field over
$\mathbb {Q}$
. Assume that none of primes of
$\mathbb {Q}$
is totally ramified in K.
-
(1) A prime p lies in
$S_{K/F}$
if and only if p is ramified in E and is unramified in F. -
(2) If p is ramified in E and splits in F, then
$e_{T,p}=2$
. -
(3) If p is ramified in E and is inert in F, then
$e_{T,p}=1$
.
Proof.
-
(1) Suppose a prime p is unramified in E. Then every place
$v|p$
of F remains unramified in K (see Lang [Reference Lang14, Chapter II, Section 4, Proposition 8(ii)]) and hence
$p\not \in S_{K/F}$
.Suppose a prime p is both ramified in E and in F. Then the unique place
$v|p$
must be unramified in K, because if v is ramified in K then p is totally ramified in K which contradicts to our assumption. Thus, p lies in
$S_{K/F}$
if and only if it is ramified in E and is unramified in F. -
(2) Let
$v_1,v_2$
be the places of F over p. One has
$F_{v_i}=\mathbb {Q}_p$
,
$K_p=E_u\times E_u$
and
$H_{v_i}=N(O_{E_u}^\times )$
for
$i=1,2$
. Thus,
$e_{T,p}=[\mathbb {Z}^\times _p:N(O^\times _{E_u})]=2$
. -
(3) Suppose first that
$p\neq 2$
. We have
$F_v=\mathbb {Q}_{p^2}$
with inertia degree
$f=2$
. Then
$e_{T,p}=1$
follows from Proposition 4.1(3). Now assume
$p=2$
. By Lemma 4.3(3), one has
$5\in N(O_{K_w}^\times )$
because
$5\in 1+4\mathbb {Z}_2\subset (1+2\mathbb {Z}_4)^2\subset N(O_{K_w}^\times )$
. By Lemma 5.2, we also have
$-1\in N(O_{K_w}^\times )$
. Thus, by Lemma 4.5,
$\mathbb {Z}_2^\times \subset N(O_{K_w}^\times )$
and we obtain
$e_{T,2}=1$
. This proves the lemma.
Lemma 5.2. Let
$\mathsf {E}/{\mathbb {Q}}_2$
be a ramified quadratic extension of
${\mathbb {Q}}_2$
, and let
$\mathsf {L}=\mathsf {E}\cdot {\mathbb {Q}}_4$
be the composite of
$\mathsf {E}$
and
${\mathbb {Q}}_4$
. Then
$-1\in {N}_{\mathsf {L}/{\mathbb {Q}}_4}(O_{\mathsf {L}}^\times )$
.
Proof. Since
$\mathsf {E}/{\mathbb {Q}}_2$
is ramified, we can write
$\mathsf {E}={\mathbb {Q}}_2(\sqrt {d_{\mathsf {E}}})$
and
$O_{\mathsf {E}}=\mathbb {Z}_2[\sqrt {d_{\mathsf {E}}}]$
for some
$d_{\mathsf {E}}\in \{3,7,2,6,10,14\}$
. Put
$j:=-d_{\mathsf {E}}$
and then
$j \mod 8\in \{1,5,2,6\}$
.
Note that
${\mathbb {Q}}_4={\mathbb {Q}}_2[t]$
with
$t^2+t+1=0$
. For each element
$a \in {\mathbb {Q}}_4$
, write
$a=a_0+a_1 t$
with
$a_0,\ a_1 \in {\mathbb {Q}}_2$
. For
$x \in O_{\mathsf {L}}$
, we write
$x=a+b \sqrt {-j}$
where
$a, b \in \mathbb {Z}_4$
and
$a=a_0+a_1 t$
,
$b=b_0+b_1 t$
. Then
Hence for
$x=a+b\sqrt {-j}$
satisfying
$N(x)\in \mathbb {Z}_2$
, the element x must satisfy the condition
Observe
$1+8\mathbb {Z}_4 \subset (1+2\mathbb {Z}_4)^2 \subset N(O_{\mathsf {L}} ^\times )$
. If
$-1 \in N(O_{\mathsf {L}}^\times )/ (1+8\mathbb {Z}_4) \subset (\mathbb {Z}_4/8 \mathbb {Z}_4)^\times $
, then
$-1 \in N(O_{\mathsf {L}}^\times )$
. We may solve the equation
$N(x)\equiv -1 \pmod 8$
, and regard
$a, b \in \mathbb {Z}_4/8 \mathbb {Z}_4$
. For simplicity, let
$j\in \{1,\ 5,\ 2,\ 6\}.$
Case
$\;\mathbf {j}\;$
is even: Since j is even, by (5.1), we have
$2|a_1$
and write
$a_1=2c_1.$
Consider the case
$2|b_1$
. Then we have
$2a_0a_1-a_1^2=4a_0c_1-4c_1^2=0 \pmod 8$
. Hence
$(a_0,\ c_1) \equiv (1,\ 1)$
,
$(0,\ 0),$
or
$(1,\ 0) \pmod 2$
.
We have
$N(x)=[a_0^2+jb_0^2-(a_1^2+jb_1^2)]=a_0^2+jb_0^2-4c_1^2$
. For
$j=2$
, take
$(a_0,\ b_0,\ c_1)\equiv (1,\ 1,\ 1) \pmod 2$
; for
$j=6$
, take
$(a_0,\ b_0,\ c_1)\equiv (1,\ 1,\ 0) \pmod 2.$
Then
$N(x)=-1$
.
Case
$\;\mathbf {j}\;$
is odd: Consider the case
$2 \nmid a_1$
and
$\ 2 \nmid b_1$
. Since j is odd, the condition (5.1) is equivalent to
By (5.2), we require
$a_0-b_0\equiv 1 \pmod 2$
. Suppose
$2|a_0$
and
$2\nmid b_0$
, and write
$a_0=2c_0$
. Then
$N(x)=-1$
gives the equation
$a_0^2+jb_0^2-(a_1+jb_1^2)=4c_0^2+j-1-j=-1 \pmod 8.$
Thus,
$c_0=2 d_0$
for some
$d_0 \in \mathbb {Z}_4/8\mathbb {Z}_4$
. Moreover, substituting
$a_0=4d_0$
into (5.2), we have the condition
$b_0b_1-(1+j)/2\equiv 0 \pmod 4$
. Since
$j \in \{1, 5\}$
, there exists
$b_1$
satisfying this condition. Conclusively, if we take
$(c_0,\ b_0,\ a_1,\ b_1)\equiv (0,\ 1,\ 1,\ 1) \pmod 2$
and
$b_0 b_1=(1+j)/2$
, then
$N(x)=-1$
.
Let
$\zeta _n$
denote a primitive nth root of unity.
Lemma 5.3. Let
$\mathsf {L}$
be a totally ramified biquadratic field extension of
${\mathbb {Q}}_p$
. Then
-
(1)
$p=2$
and
$\mathsf {L}\simeq \mathbb {Q}(\zeta _8)\otimes {\mathbb {Q}}_2={\mathbb {Q}}_2[t]$
with relation
$t^4+1=0$
; -
(2) for any quadratic subextension
$\mathsf {E}$
of
$\mathsf {L}$
over
${\mathbb {Q}}_2$
, one has
$N_{\mathsf {L/\mathsf {E}}} (O_{\mathsf {L}}^\times )\supset \mathbb {Z}_2^\times $
.
Proof.
-
(1) By local class field theory [Reference Lang14],
, where
$I_p$
is the inertia group of p. As
${\mathbb {Z}}_p^\times $
is pro-cyclic for odd prime p, this is possible only when
$p=2$
. Clearly,
$2$
is totally ramified in
$\mathbb {Q}(\zeta _8)$
. Thus, it suffices to show that
${\mathbb {Q}}_2$
has only one totally ramified biquadratic extension, that is, there is only one subgroup
$H\subset \mathbb {Z}_2^\times $
satisfying
$\mathbb {Z}_2^\times /H=\mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/2\mathbb {Z}$
by the existence theorem. Now
$\mathbb {Z}_2^\times \simeq \mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}_2$
and one easily checks that
$H=\{0\}\times 2\mathbb {Z}_2$
is the unique subgroup satisfying
${\mathbb {Z}}_p^\times / H\simeq \mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/2\mathbb {Z}$
. This proves (1). -
(2) Put
$\mathsf {E}_1={\mathbb {Q}}_2(\sqrt {2})={\mathbb {Q}}_2[t-t^3]$
,
$\mathsf {E}_2={\mathbb {Q}}_2(\sqrt {-2})={\mathbb {Q}}_2[t+t^3]$
and
$\mathsf {E}_3={\mathbb {Q}}_2(\sqrt {-1})= {\mathbb {Q}}_2[t^2]$
. The Galois group
, where
$\sigma _1(t)=t^{-1}$
,
$\sigma _2(t)=t^3$
and
$\sigma _3(t)=t^5=-t$
. Then
$\mathsf {E}_i$
is the fixed subfield of the element
$\sigma _i$
for each
$i=1,2,3$
. We choose a uniformizer
$\pi _i$
of
$\mathsf {E}_i$
as
$t-t^3$
,
$t+t^3$
, and
$t^2-1$
for
$i=1,2,3$
, respectively. Thus, we have
$(\mathsf {E},\pi )=(\mathsf {E}_i, \pi _i)$
for some i. Let
$x=a + b t +c t^2+ d t^3 \in O_{\mathsf {L}}$
with
$a,b,c, d\in \mathbb {Z}_2$
. It is well-known that every element in
$1+4 \pi O_{\mathsf {E}}$
is a square, and hence
$N_{\mathsf {L/\mathsf {E}}}(O_{\mathsf {L}}^\times )\supset (O_{\mathsf {E}}^\times )^2 \supset 1+4 \pi O_{\mathsf {E}}$
. To show
$N_{\mathsf {L/\mathsf {E}}}(O_{\mathsf {L}}^\times )\supset \mathbb {Z}_2^\times $
, it suffices to show that the group
$N_{\mathsf {L/\mathsf {E}}}(O_{\mathsf {L}}^\times )$
mod
$4\pi O_{\mathsf {E}}$
contains
$1,3,5,7$
mod
$8$
.For
$i=1$
, we compute Put
$(a,b,c,d)=(1,1,0,1)$
, one has
$N(x) \mod 4\pi _1$
is equal to
$3 \mod 8$
. Put
$(a,b,c,d)=(1,0,2,0)$
, one has
$N(x) \mod 4\pi _1$
is equal to
$7 \mod 8$
. Thus,
$N_{\mathsf {L}/\mathsf {E}_1}(O_{\mathsf {L}}^\times )\supset \mathbb {Z}_2^\times $
,For
$i=2$
, we compute Put
$(a,b,c,d)=(2,0,1,0)$
, one has
$N(x) \mod 4\pi _2$
is equal to
$5 \mod 8$
. Put
$(a,b,c,d)=(0,1,1,1)$
, one has
$N(x) \mod 4\pi _2$
is equal to
$7 \mod 8\,$
. Thus,
$N_{\mathsf {L}/\mathsf {E}_2}(O_{\mathsf {L}}^\times )\supset \mathbb {Z}_2^\times $
,For
$i=3$
, we compute Put
$(a,b,c,d)=(1,1,0,1)$
, one has
$N(x) \mod 4\pi _3$
is equal to
$3 \mod 8$
. Put
$(a,b,c,d)=(2,0,1,0)$
, one has
$N(x) \mod 4\pi _3$
is equal to
$7 \mod 8$
.Thus,
$N_{\mathsf {L}/\mathsf {E}_3}(O_{\mathsf {L}}^\times )\supset \mathbb {Z}_2^\times $
,
Corollary 5.4. Let
$K=EF$
be a biquadratic CM field. If p is a prime totally ramified in K, then
$p=2$
and
$e_{T,2}=1$
.
Proposition 5.5. Let F be a real quadratic field and E an imaginary quadratic field, and let
$K=EF$
. Then
where t is the number of places of F ramified in K, s is the number of primes p that are ramified in E and split in F, and
$Q=Q_{K}$
.
Proof. By Lemma 5.1 and Corollary 5.4,
$\prod _{p\in S_{K/F}} e_{T,p}=2^s$
. Since K contains two distinct imaginary quadratic fields, by Lemma 4.7, we have
$[\mathbb {A}^\times : N(T^{K,\mathbb {Q}}(\mathbb {A}))\cdot \mathbb {Q}^\times ]=1$
. Thus, the formula (5.3) follows from Theorem 1.1.
Note that we may rewrite (5.3) as
Indeed, suppose we let m be the number of primes p that are ramified in E and inert in F. Then
$t=m+2s+\delta $
and
$|S_{K/F}|=m+s+\delta $
, where
$\delta =1$
if
$2$
is totally ramified in K and
$\delta =0$
otherwise. So
$t-s=|S_{K/F}|$
.
If
$K\neq \mathbb {Q}(\sqrt {2},\sqrt {-1})$
, then by Herglotz [Reference Herglotz9] (cf. [Reference Xue, Yang and Yu28, Section 2.10])
where
$E^{\prime }\subset K$
is the other imaginary quadratic field. By (5.4) and (5.5), we have
For
$K=\mathbb {Q}(\sqrt {2},\sqrt {-1})=\mathbb {Q}(\zeta _8)$
, it is known that
$h(\mathbb {Q}(\zeta _8))=1$
, and
$Q_{\mathbb {Q}(\zeta _8)}=1$
as
$8$
is a prime power [Reference Washington26, Corollary 4.13, p. 39]. Moreover,
$S_{K/F}=\{2\}$
and
$e_{T,2}=1$
(Corollary 5.4). Thus,
We specialize to the case where
$K=K_j=FE=\mathbb {Q}(\sqrt {p}, \sqrt {-j})$
, where
$F=\mathbb {Q}(\sqrt {p})$
,
$E=\mathbb {Q}(\sqrt {-j})$
, p is a prime and
$j\in \{1,2,3\}$
. Note that we have
$\mathbb {Q}(\sqrt {2},\sqrt {-2})=\mathbb {Q}(\sqrt {2},\sqrt {-1})=\mathbb {Q}(\zeta _8)$
and
$\mathbb {Q}(\sqrt {3},\sqrt {-3})=\mathbb {Q}(\sqrt {3},\sqrt {-1})=\mathbb {Q}(\zeta _{12})$
. We may assume that
$p\neq 2$
if
$j=2$
and
$p \neq 3$
if
$j=3$
.
The set
$S_{K/F}$
is given as follows:
-
(i)
$S_{K_1/F}=\{2\}$
if
$p \equiv 1 \pmod 4$
, and
$S_{K_1/F}=\emptyset $
otherwise; -
(ii)
$S_{K_2/F}=\{2\}$
if
$p \equiv 1 \pmod 4$
, and
$S_{K_2/F}=\emptyset $
otherwise; -
(iii)
$S_{K_3/F}=\{3\}$
always (recall
$p\neq 3$
).
Thus,
By (5.6) and (5.8), if
$K\neq \mathbb {Q}(\sqrt {2},\sqrt {-1})$
, we have
Remark 5.6. Observe from formula (5.6) that for computing the class number
$h(T^{K,\mathbb {Q}}) $
or
$h(T^{K}_1)$
, one needs not to calculate the Hasse unit index
$Q_K$
. For the case where
$F=\mathbb {Q}(\sqrt {p})$
with prime p, one has
$Q_K=2$
if and only if
$p\equiv 3\pmod 4$
, and either
$K=\mathbb {Q}(\sqrt {p},\sqrt {-1})$
or
$K=\mathbb {Q}(\sqrt {p},\sqrt {-2})$
; see [Reference Xue, Yang and Yu28, Proposition 2.7].
6 Polarized CM abelian varieties and unitary Shimura varieties
6.1 CM points
Let
$(K,O_K, V,\psi , \Lambda ,h)$
be a PEL-datum, where
-
•
$K=\prod _{i=1}^r K_i$
be a product of CM fields
$K_i$
with canonical involution
$\bar { }$
; -
•
$O_K$
the maximal order of K; -
• V is a free K-module of rank one;
-
•
$\psi :V\times V\to \mathbb {Q}$
be a nondegenerate alternating pairing such that -
•
$\Lambda $
be an
$O_K$
-lattice with
$\psi (\Lambda ,\Lambda )\subset \mathbb {Z}$
; -
•
be an
$\mathbb {R}$
-algebra homomorphism such that and that the pairing
$(x,y):=\psi (h(i)x,y)$
is symmetric and positive definite.
Let
$V_{\mathbb {C}}=V^{-1,0}\oplus V^{0,-1}$
be the decomposition into
$\mathbb {C}$
-subspaces such that
$h(z)$
acts by z (respectively
$\bar z$
) on
$V^{-1,0}$
(respectively
$V^{0,-1}$
). Let
$T=T^{K,\mathbb {Q}}$
and
$U\subset T(\mathbb {A}_f)$
be an open compact subgroup. Put
$g=\frac {1}{2} \dim _{\mathbb {Q}} V$
. Let
$M_{(\Lambda ,\psi ),U}$
be the set of isomorphism classes of tuples
$(A,\lambda ,\iota ,\bar \eta )_{\mathbb {C}}$
, where
-
• A is a complex abelian variety of dimension g;
-
•
is a ring monomorphism; -
•
$\lambda :A\to A^t$
is an
$O_K$
-linear polarization, that is, it satisfies
$\lambda \iota (\bar b)=\iota (b)\lambda $
for all
$b\in O_K$
; -
•
$\bar \eta $
is an U-orbit of
$O_K\otimes \widehat {\mathbb {Z}}$
-linear isomorphisms preserving the pairing up to a scalar in
$\widehat {\mathbb {Z}}^\times $
, where
$T_\ell (A)$
is the
$\ell $
-adic Tate module of A such that
-
(a)
$\det (b; V^{-1,0})=\det (b; \hbox{Lie} (A))$
for all
$b\in O_K$
; -
(b) there exists a K-linear isomorphism
(6.1)that preserves the pairings up to a scalar in
$\mathbb {Q}^\times $
, where
$\langle ,\rangle _\lambda $
is the pairing induced by the polarization
$\lambda $
.
Two members
$(A_1,\lambda _1,\iota _1,\bar \eta _1)$
and
$(A_2,\lambda _2,\iota _2,\bar \eta _2)$
are said to be isomorphic if there exists an
$O_K$
-linear isomorphism
$\varphi :A_1\to A_2$
such that
$\varphi ^* \lambda _2=\lambda _1$
and
$\varphi _* \bar \eta _1=\bar \eta _2$
.
Lemma 6.1. Let T be an algebraic torus over
$\mathbb {Q}$
,
$U\subset T(\mathbb {A}_f)$
an open compact subgroup, and
$U_\infty \subset T(\mathbb {R})$
an open subgroup. Then
where
$U_T:=T(\widehat {\mathbb {Z}})$
is the maximal open compact subgroup of
$T(\mathbb {A}_f)$
and
$T(\mathbb {Z})_\infty =T(\mathbb {Z})\cap U_\infty $
.
Proof. Let
$T(\mathbb {Q})_\infty =T(\mathbb {Q})\cap U_\infty $
. One has
$[T(\mathbb {A}):T(\mathbb {Q})U_\infty U]=[T(\mathbb {A}_f):T(\mathbb {Q})_\infty U]$
. Now consider the exact sequence
It is easy to verify
$U\cdot T(\mathbb {Q})_\infty \cap U_T=U \cdot T(\mathbb {Z})_\infty $
. Using the exact sequence (6.3) and the following one
we obtain the first equation of (6.2).
Now we prove the second equality. Consider the exact sequence
where the inclusion
$T(\mathbb {Q})\hookrightarrow T(\mathbb {A})$
is given by the diagonal map, the map
$T(\mathbb {R})\hookrightarrow T(\mathbb {A})=T(\mathbb {R})\times T(\mathbb {A}_f)$
sends
$t_\infty \mapsto (t_\infty ,1)$
, and the intersection
$U_T \cdot T(\mathbb {Q}) \cdot U_\infty \cap T(\mathbb {R})$
is taken in
$T(\mathbb {A})$
. Suppose
$utu_\infty =(tu_\infty , tu)$
is an element in
$U_T \cdot T(\mathbb {Q}) \cdot U_\infty \cap T(\mathbb {R})$
. Then
$tu=1$
and
$t=u^{-1}\in T(\mathbb {Q})\cap U_T=T(\mathbb {Z})$
. Thus,
$U_T \cdot T(\mathbb {Q}) \cdot U_\infty \cap T(\mathbb {R})=T(\mathbb {Z}) \cdot U_\infty \subset T(\mathbb {R})$
and we have
$[T(\mathbb {A}): T(\mathbb {Q})U_\infty U_T]=[T(\mathbb {R}):T(\mathbb {Z}) U_\infty ]\cdot h(T)$
.
By [Reference Deligne5, 4.11], the set
$M_{(\Lambda ,\psi ),U}$
is isomorphic to the Shimura set
$\mathrm {Sh}_U(T,h)\simeq T(\mathbb {Q})\backslash T(\mathbb {A}_f)/U$
. By Lemma 6.1, we have
where
$\mu _K=\prod _{i=1}^r \mu _{K_i}$
and
$T=T^{K,\mathbb {Q}}$
. Using Theorem 1.1, we obtain the following result.
6.2 Connected components of unitary Shimura varieties
In this subsection we consider a PEL-datum
$(K,O_K, V,\psi , \Lambda ,h)$
, where
-
• K is a CM field with canonical involution
$\bar { }$
; -
• V is a free K-module of rank
$n>1$
; -
•
$O_K,\psi ,h$
are as in Section 6.1.
Let
$G=GU_K(V,\psi )$
be the group of unitary similitudes of
$(V,\psi )$
. The kernel of the multiplier homomorphism
$c:G\to {{\mathbb {G}}_{\mathrm {m}}}$
is the unitary group
$U_K(V,\psi )$
associated to
$(V,\psi )$
. Let X be the
$G(\mathbb {R})$
-conjugacy class of h, and
$U\subset G(\mathbb {A}_f)$
an open compact subgroup. The complex Shimura variety associated to the PEL datum is defined by
As in Section 6.1, we define
$M_{(\Lambda ,\psi ),U}$
as the moduli space of complex abelian varieties
$(A,\lambda ,\iota ,\bar \eta )$
with additional structures satisfying the conditions (a) and (b). By [Reference Deligne5, 4.11], one has
$Sh_U(G,X)_{\mathbb {C}}\simeq M_{(\Lambda ,\psi ),U}$
; this provides the modular interpretation of the Shimura variety
$Sh_U(G,X)_{\mathbb {C}}$
. We are interested in the number of the connected components of the moduli space
$M_{(\Lambda ,\psi ),U}$
, or equivalently, those of the Shimura variety
$\mathrm {Sh}_U(G,X)_{\mathbb {C}}$
.
Let
$X^+$
be the connected component of X that contains the base point h, and let
be the stabilizer of
$X^+$
in
$G(\mathbb {R})$
. We have
where
$G(\mathbb {Q})_+:=G(\mathbb {Q})\cap G(\mathbb {R})_+$
. Let
$G^{\mathrm {der}}$
be the derived group of G, and let
$D:=G/G^{\mathrm {der}}$
be the quotient torus. Denote by
$\nu :G\to D$
the natural homomorphism. Note that the derived group
$G^{\mathrm {der}}=SU_K(V,\psi )$
is semi-simple and simply connected.
Theorem 6.3. Assume that
$G^{\mathrm {der}}(\mathbb {R})$
is not compact. Then the complex Shimura variety
$\mathrm {Sh}_U(G,X)_{\mathbb {C}}$
has
connected components, where
$t, Q_K$
and
$e_{T,p}$
are as in Theorem 1.1.
Proof. Using the strong approximation argument and Kneser’s theorem (namely,
$H^1({\mathbb {Q}}_p, G^{\mathrm {der}})=1$
for all primes p), the morphism
$\nu $
induces a bijection
(see [Reference Yu30, Lemma 2.2]). By [Reference Kottwitz12, Section 7, pp. 393 – 394], one has
Using the Hasse principle, one shows that
$\nu (G(\mathbb {Q})_{+})=D(\mathbb {Q})\cap \nu (G(\mathbb {R})_+)$
. One directly checks
As a result, the intersection
$D(\mathbb {Z})_\infty :=D(\mathbb {Z})\cap \nu (G(\mathbb {R})_+)$
is equal to
$\mu _K$
for all n. Applying Lemma 6.1, (6.9), (6.11), and the formula for
$h(D)$
using Theorem 1.1, we obtain the result.
6.3 Polarized abelian varieties over finite fields
In this subsection, we formulate two counting problems for polarized abelian varieties over finite fields in an isogeny class and compute their cardinality using the class number formula of CM tori. Let k be a finite field.
Definition 6.4. Let
$\underline A_1=(A_1,\lambda _1)$
and
$\underline A_2=(A_2,\lambda _2)$
be two polarized abelian varieties over k.
-
(1) They (
$\underline A_1$
and
$\underline A_2$
) are isomorphic, denoted
$\underline A_1\simeq \underline A_2$
, if there exists an isomorphism
$\alpha :A_1\stackrel {\sim }{\to } A_2$
such that
$\alpha ^* \lambda _2=\lambda _1$
. Similarly, their polarized
$\ell $
-divisible groups
$\underline A_1[\ell ^\infty ]$
and
$\underline A_2[\ell ^\infty ]$
are said to be isomorphic, denoted
$\underline A_1[\ell ^\infty ]\simeq \underline A_2[\ell ^\infty ]$
, if there exists an isomorphism
$\alpha _\ell :A_1[\ell ^\infty ]\stackrel {\sim }{\to } A_2[\ell ^\infty ]$
such that
$\alpha _\ell ^* \lambda _2=\lambda _1$
. -
(2) They are said to be in the same isogeny class if there exists a quasi-isogeny
$\alpha : A_1 \to A_2$
(i.e., a multiple of
$\alpha $
by an integer is an isogeny) such that
$\alpha ^* \lambda _2=\lambda _1$
. Denote by
$\mathrm {Isog}(A_1,\lambda _1)$
the set of isomorphism classes of
$(A_2,\lambda _2)$
lying in the same isogeny class of
$(A_1,\lambda _1)$
. -
(3) They are said to be similar, denoted
$\underline A_1\sim \underline A_2$
, if there exists an isomorphism
$\alpha : A_1 \to A_2$
such that
$\alpha ^* \lambda _2=q \lambda _1$
for some
$q\in {\mathbb {Q}}_{>0}$
. Similarly, their polarized
$\ell $
-divisible groups
$\underline A_1[\ell ^\infty ]$
and
$\underline A_2[\ell ^\infty ]$
are said to be similar, denoted
$\underline A_1[\ell ^\infty ]\sim \underline A_2[\ell ^\infty ]$
, if there exists an isomorphism
$\alpha _\ell :A_1[\ell ^\infty ]\stackrel {\sim }{\to } A_2[\ell ^\infty ]$
such that
$\alpha _\ell ^* \lambda _2=q \lambda _1$
, for some
$q\in {\mathbb {Q}}_\ell ^\times $
. -
(4) They are said to be isomorphic locally everywhere, if
for all primes
$\ell $
including the prime
$\text {char } k$
.
-
(5) They are said to be similar locally everywhere, if
for all primes
$\ell $
(also including the prime
$\text {char } k$
).
Now we start with a polarized abelian variety
$(A_0,\lambda _0)$
over k. Assume that the endomorphism algebra
is commutative. So
for some CM algebra K, and
is a CM order. Let
$T:=T^{K,\mathbb {Q}}$
and
$T^1:=T^{K}_1$
. On the other hand, consider
and
Proposition 6.5. Let
$U:=T(\mathbb {A}_f)\cap \widehat R^\times $
and
$U^1:=T^1(\mathbb {A}_f)\cap \widehat R^\times $
. We have
and
Proof. The main point is the following natural bijections
See [Reference Xue and Yu29, Theorem 5.8 and Section 5.4]; also see [Reference Yu31, Theorem 2.2]. Then formulas (6.16) and (6.17) follow from (6.18), Theorem 1.1, and Lemma 6.1.
Acknowledgments
Guo is partially supported by the MoST grants 106-2115-M-002-009MY3. Yu is partially supported by the MoST grants 104-2115-M-001-001MY3, 107-2115-M-001-001-MY2, and 109-2115-M-001-002-MY3. The authors thank the referee for a careful reading and helpful comments which improve the exposition of this paper.
