p-adic absolute CM-periods I
Tomokazu Kashio, Hiroyuki Yoshida 2005 September 27th and 28th in Hakuba
1. The first version of our conjecture.
1.0. Introduction.
The purpose of this section is to formulate a p-adic analogue of Yoshida’s conjecture.
In his paper [Yo1], he formulated an equality between CM-periods and the derivatives of L-functions ats= 0. Furthermore he refined this formula and expressed each CM-period in terms of special values of multiple Γ-functions [Yo2,3]. His approach is to ”factorize” the value of the derivative of the L-function at s= 0 by using Shintani’s formula. Therefore we want to show: thep-adic period can be written by, or at least is connected with special values ofp-adic multiple Γ-functions. The point of this case is also ”factorization” using a p-adic analogue of Shintani’s formula. In fact, as we will show in §2, our conjecture gives a refinement of the Gross’ conjecture on the derivative of the p-adic L-function at s= 0.
In §1.1, we will define the p-adic multiple Γ-function as the derivative of the p-adic multiple ζ-function at s = 0. We will use Cassou Nogu`es’ method of p-adic analysis. In
§1.2, we will give a p-adic analogue of Shintani’s formula. We will express the values of the derivatives of p-adicL-functions at s= 0 in terms of special values ofp-adic multiple Γ-functions. Using this formula, we will define the p-adic absolute CM-period symbol in
§1.3 in the same manner as in Yoshida’s original work.
Our main conjecture states an equality between the Frobenius action on the p-adic period and the p-adic absolute CM-period, so we can consider it as a p-adic analogue of Yoshida’s original conjecture. Since the Frobenius action can be written by CM-theory under a certain condition (which is very close to ”thep-ordinarity condition”), our conjec- ture becomes simpler in such case. It merely states a generalization of the Gross-Koblitz formula. We will formulate this version of conjecture in §1.4. Yoshida will survey the definition of the p-adic period and the general conjecture in this volume.
Notation. Let p be a prime number, Cp the completion of algebraic closure of Qp. Throughout this paper, we fix embeddings Q ֒→ C, Q ֒→ Cp. We denote by | |∞ the usual absolute value on C and by | |p the p-adic absolute value on Cp normalized by
|p|p = 1/p. For any number fieldK, we denote byJK the set of all isomorphismsK ֒→Q and fix an embedding id∈ JK. That means we fix absolute values | |∞,| |p on K, and a prime ideal pK of the ring of integers OK of K, which lies above (p). pK introduces the p-adic topology onK.
We define log, logp on the set of fractional ideals as follows. For each fractional ideal a in K, we take an element Πa ∈ K satisfying the following three properties. 1. There exists a positive integer n such that for each fractional ideal a, an = (Πa) as ideals.
2. Πab = ΠaΠb. 3. For any isomorphism σ : K → Kσ, Πaσ = Πσa. Then we put log(a) := 1nlog(Πa), logp(a) := 1nlogp(Πa). Here logp is Iwasawa’sp-adic log function [Iw].
These functions depend on the choice of Πa. We can take such elements Πa as follows.
We denote by Ke the normal closure of K. It is only necessary to take Πp for any prime ideal p of Ke satisfying properties 1,3. For each prime number p, fix a prime ideal p of Ke which divides (p). Put G := Gal(K/Q) ande Gp := {σ ∈ G | pσ = p}. Let h be the ideal class number ofK. We choose a generator Π of the principal ideale ph. Then we take n :=h|G|, Πpτ := (Q
σ∈GpΠστ)|Gp||G| for τ ∈ G. Actually, we need not define log, logp on all fractional ideals. It is only necessary to define them on a finite set of ideals which are used to define the p-adic absolute CM-period.
1.1. p-adic multiple Γ-functions.
First we recall the definition of Barnes’ multiple Γ-function [Ba]. We define the mul- tiple ζ-function ζr(s, v, z) for z, v = (v1, . . . , vr) with z, vi >0 by
(1) ζr(s, v, z) :=
X∞
n1,...,nr=0
(z+n1v1+· · ·+nrvr)−s.
This can be continued meromorphically to whole s-plane and is analytic ats = 0. Then we define the multiple Γ-function:
(2) LΓr(z, v) := log
Γ(z, v) ρ(v)
:=ζr′(0, v, z).
Cassou-Nogu`es constructed thep-adic multipleζ-functionζp,r(s, v, z) fors ∈Zp which is the p-adic interpolation function [Ca1]. To construct this function, we prepare the p-adic integral in the manner of Robert [Ro]. Namely forf :Zrp →Cp we put
(3)
Z
Zrp
f(x)dx:= lim
l1,...,lr→∞
1 pl1+···+lr
pl1−1,...,pXlr−1
x1=0,...,xr=0
f(x), x= (x1, . . . , xr).
Then the p-adic multiple ζ-function is for z ∈Qp, vi ∈Qp× (4) ζp,r(s, v, z) :=
R
Zrp(z+x1v1+· · ·+xrvr)rhz+x1v1+· · ·+xrvri−sdx (s−1)(s−2). . .(s−r)v1. . . vr
.
We define the functionh i−sas follows. If |z|p <1 we puthzi:= 0. If|z|p ≥1,hzidenotes the unique element ∈ Qp which satisfies that |hzi −1|p < 1 and that z/(pordpzhzi) is a root of unity whose order is prime to p(see also [Iw, §4]). For 1 +z with |z|p <1 we put (1 +z)s :=P∞
k=0 s k
zk. We can show the following properties:
(5) ζp,r(s, v, z) is p-adic analytic at s = 0.
If z, vi ∈Q× satisfyz, vi >0 and |z−1|p,|vi|p <1 then
(6) ζr(−k, v, z) =ζp,r(−k, v, z)∈Q for 0≤k ∈Z.
Now we define the p-adic multiple Γ-function LΓp,r(z, v) by (7) LΓp,r(z, v) :=ζp,r′ (0, v, z).
We can show that:
(8) LΓp,r(z, v) is a continuous function ofz and vi.
(9) LΓp,r(z, v)−LΓp,r(z+vi, v) =LΓp,r−1(z,(v1, . . . , vi−1, vi+1, . . . , vr)) ifr ≥2.
(10) LΓp,1(z,(v1))−LΓp,1(z+v1,(v1)) =−logp(z).
(11) LΓp,1(z,(1)) = logp(Γp(z)) forz ∈Zp.
Here Γp is Morita’s p-adic Γ-function [Mo]. (cf. For z > 0, LΓ1(z,(1)) = log(Γ(z))−
1
2log(2π).)
(12) LΓp,r(z, v) = (−1)rR
Zrpf(x)dx r!v1. . . vr
,
where f(x) = (z+x1v1 +· · ·+xrvr)r(1 + 12 +· · ·+ 1r −logp(z +x1v1 +· · ·+xrvr)) if
|z+x1v1+· · ·+xrvr|p ≥1,f(x) = 0 if |z+x1v1+· · ·+xrvr|p <1. In particular, we can compute special values of p-adic multiple Γ-functions numerically in Cp.
Remark. Note that we slightly changed the definition of the p-adic ζ-function from those of her paper [Ca1] or my papers [Ka1,2]. One problem is how to continue the p- adic multiple ζ-function defined by (6) to any z, v canonically. For example, what is the canonical value of ζp,1(s,(1/p),1/p)? If we assume ζp,1(s,(tv), tz) =hti−sζp,1(s,(v), z) for suitable t by analogy with the property of the classical ζ-function, then ζp,1(s,(1/p),1/p)
= h1/pi−sζp,1(s,(1),1) = ζp,1(s,(1),1). On the other hand, if we assume ζp,1(s,(v), z) = Pn−1
k=0ζp,1(s,(nv), z + kv) for any positive integer n, then we get ζp,1(s,(1/p),1/p) = Pp−1
k=0ζp,1(s,(1),(k + 1)/p) = ζp,1(s,(1),1) +Pp−1
k=1ζp,1(s,(p), k) = 2ζp,1(s,(1),1). We choose the latter definition here and my papers [Ka1,2] were written using the former definition.
1.2. A p-adic analogue of Shintani’s formula.
Shintani’s formula is as follows.
Theorem. (Shintani [Sh2].) Let F be a totally real field of degree n, f an integral ideal of F and Cf the ideal class group of conductor f∞1. . .∞n where {∞1, . . . ,∞n} is
the set of all real places. For each ideal class c∈ Cf, we define the partial ζ function by ζF(s,c) := P
a∈c, integral idealsN(a)−s. Then we get:
(13) ζF′ (0,c) = X
σ∈JF
X
j∈J
X
z∈R(c,j)
LΓr(j)(zσ, vjσ)−log (N(aµf))ζF(0,c) +T(c,{vj},{aµ}).
Here vj = (vj,1, . . . , vj,r(j)) are r(j)-row vectors whose components are in OF and totally positive, R(c, j) is a finite subset of F, J is a finite set of subscript and T(c,{vj},{aµ}) is the sum of correction terms. These depend on the choices of a representative set {aµ} of the narrow ideal class group C(1) and vectors {vj} which induce a cone decomposition R+n = F
j∈J
F
ǫ∈O×F ǫC(vj). Here F
denotes the disjoint union and C(vj) := {t1vj,1 +
· · ·+tr(j)vj,r(j) | (t1, . . . , tr(j)) ∈ R+r(j)} ⊂ R+n. Note that we embed F ֒→ R+n by z 7→ (zσ)σ∈JF. We take an element aµ in the representative set so that aµf = c in C(1). Then we put R(c, j) = {z =Pr(j)
k=1xkvj,k ∈(aµf)−1∩C(vj)|0< xi ≤1,(z)aµf∈c}. We introduce Yoshida’s research work on this formula. It is necessary for canonical factorization of the derivatives of partial ζ-functions at s= 0.
Lemma. (Yoshida [Yo3].) There exist ai ∈F, bi ∈OF× such that (14) T(c,{vj},{aµ}) = X
σ∈JF
X
i∈I
aσi log(bσi).
HereI is a finite set of subscript.
By using this factorization, Yoshida defined a new invariant attached to each ideal class.
Definition. For σ∈JF we put (15)
Xσ(c) := Xσ(c;{vj},{aµ}) = X
j∈J
X
z∈R(c,j)
LΓr(j)(zσ, vjσ)−log(aµf)ζF(0,c) +X
i∈I
aσi log(bσi),
X(c) :=Xid(c).
Remark. In [Yo2,3], Yoshida defined invariants X(c) := G(c) +W(c) +V(c) and W(c) := −1nlog (N(aµf))ζF(0,c). In this paper, we change this part and put W(c) :=
−log(aµf)ζF(0,c) to simplify the formula (26) in our main conjecture.
The p-adic counterpart is as follows. We take an integral ideal:
(16) (p)0 :=
(Q
prime idealsp⊂OF, p|(p)p if p6= 2, (2)Q
prime idealsp⊂OF,p|(2)p if p= 2.
We assume that (p)0 divides f. Then for each ideal class c ∈ Cf, there exists the p-adic partial ζ function ζp,F(s,c) which is p-adic analytic at s = 0 and satisfies the p-adic
interpolation propertyζp,F(−k,c) =ω(c)−kζF(−k,c) for 0 ≤k∈Z. Hereω is a character which is the composite mapping of the Teichm¨uller character and the ideal norm map.
Using Cassou-Nogu´es’ method [Ca2,3], we get a p-adic analogue of Shintani’s formula, which gives the factorization of the derivatives of p-adic partialζ functions at s = 0.
Theorem. (Kashio [Ka1,2].) We assume that (p)0 divides f. Let the notation be as in (13), (14). Then we get
(17)
ζp,F′ (0,c) = X
σ∈JF
X
j∈J
X
z∈R(c,j)
LΓp,r(j)(zσ, vjσ)−logp(N(aµf))ζp,F(0,c) + X
σ∈JF
X
i∈I
aσi logp(bσi).
Note thatai, bi are the same elements as in (14).
Therefore we define a p-adic invariant similarly to the above.
Definition. For σ∈JF we put (18)
Xpσ(c) :=Xpσ(c;{vj},{aµ}) =X
j∈J
X
z∈R(c,j)
LΓp,r(j)(zσ, vσj)−logp(aµf)ζp,F(0,c)+X
i∈I
aσi logp(bσi), Xp(c) :=Xpid(c).
One consequence of Theorem (17) is:
Corollary. Assume that (f,(p)) = 1. For any character χ of Cfwe get (19) ords=0Lp(s, χω)≥2 ifr(χ) := #{p|(p), χ(p) = 1} ≥2.
HereLp(s, χω) :=P
c∈cf(p)0χ(c)ζp,F(s,c) is the p-adicL function.
This corollary is a partial result of Gross’ conjecture [Gr] which is a p-adic analogue of Stark-Tate’s result.
Conjecture. (Gross. See also §2.) If (f,(p)) = 1 and χis primitive, then (20)
Lp(s, χω)∼ ”an algebraic number” ∗ ”the p-adic regulator” ∗sr(χ)+O(sr(χ)+1) (s7→0).
1.3. The p-adic absolute CM-period symbol.
Yoshida’s original conjecture [Yo2,3] is as follows.
Definition. LetK be a CM-field, F a totally real field such that K/F is an abelian extension. For τ ∈G:= Gal(K/F) we put
(21)
gK/F(id, τ) :=gK/F(id, τ;{vj},{aµ}) :=π−µ(τ)/2exp
1
|G| X
χ∈Gˆ−
χ(τ)P
c∈Cfχχ(c)X(c) L(0, χ)
,
and call gK/F the absolute CM-period symbol. Here ˆG− denotes the set of all odd char- acters ofG, fχdenotes the conductor ofχ. We put µ(τ) := 1,−1,0 forτ = id, ρ (complex conjugate) and otherwise, respectively. We can show that gK/F(id, τ) mod OF× ⊗Z Q does not depend on the choices of{vj},{aµ}. Here we denote by OF×⊗ZQ the subgroup {ǫ∈Q×| there exists an integern such thatǫn ∈OF×} ⊂Q×.
Conjecture. (Yoshida [Yo2,3].) For τ ∈G:= Gal(K/F) (22) pK(id, τ)≡gK/F(id, τ) mod Q×.
Here pK denotes Shimura’s CM-period symbol which is defined by factorizing the values of geometric periods of Abelian varieties with complex multiplication by K. For details, see [S].
We consider a p-adic analogue of this symbol.
Definition. For τ ∈G:= Gal(K/F) we put lgp,K/F(id, τ) : =lgp,K/F(id, τ;{vj},{aµ})
= −µ(τ)
2 logp(pF) + 1
|G| X
χ∈Gˆ−
χ(τ)P
c∈Cfp,χχ(c)Xp(c)
L(0, χ) ,
(23)
and call lgp,K/F the (logarithmic)p-adic absolute CM-period symbol. Here we putfp,χ :=
fχ if pF divides fχ, otherwise we put fp,χ := fχpF. We can show that lgp,K/F(id, τ) is uniquely determined mod Qlogp(O×F).
1.4. A generalization of the Gross-Koblitz formula.
We will formulate a version of our conjecture. If K is an imaginary quadratic field, it corresponds with the Gross-Koblitz formula. First we recall this formula.
Theorem. (Gross-Koblitz [GK].) Let K be an imaginary quadratic field with con- ductor −d which corresponds to the Dirichlet character χ. Assume that p splits in K.
Then
(24) logp
pρK pK
= wK
2hK d−1
X
a=1
χ(a) logp Γ(a
d) .
HerewK := #{roots of unity ∈K}.
The condition ”psplits inK” in the Gross-Koblitz formula is generalized and becomes the condition ”pF splits completely in K.” We formulate our main conjecture under this condition.
Conjecture. (Kashio-Yoshida.) Let F, K be a totally real number field and a CM field such thatK/F is an abelian extension. Assume the condition ”pF splits completely in K.” Then
(25) 1
2logp pρK pK
τ−1!
≡lgp,K/F(id, τ) modQlogp(O×F).
More precisely (26) 1
2logp pρK pK
τ−1!
=lgp,K/F(id, τ;{vj},{aµ}) + logp
gK/F(id, τ;{vj},{aµ}) gK/F(id, τ;{vj},{aµpF})
.
Remark. We can show that the absolute CM-period symbol gK/F modulo OF×⊗ZQ does not depend on the choices of {vj},{aµ}. That means ggK/F(id,τ;{vj},{aµ})
K/F(id,τ;{vj},{aµpF}) = ǫx with a unit ǫ ∈ OF× and a rational number x ∈ Q. So logp(ggK/F(id,τ;{vj},{aµ})
K/F(id,τ;{vj},{aµpF})) = xlogp(ǫ) is well-defined. We can show the right hand side of the formula (26) does not depend on the choices of {vj},{aµ}. Although we took elements {Πa} to define log, logp on the set of fractional ideals (See Notation), both sides of (26) do not depend on the choices of them neither.
Remark. We can express the left hand side (= 12logp((ppρK
K)τ−1)) in terms of the p-adic period arising from a comparison isomorphism between certain cohomologies. Therefore our conjecture gives a relational expression between the p-adic period and the p-adic absolute CM-period. We can formulate this expression without the condition ”pF splits completely in K.” For details, see Yoshida’s survey in this volume.
Numerical examples. Put F :=Q(√
5), K := Q(
q
13+√ 5
2 i). (This is the situation in Example 1 of [Yo2].) Then hF = hK = 1 and the conductor of the extension K/F is (13+2√5). Let p = 11,19,29. Then (p) splits in F and one prime ideal in OF lying above (p) splits in K/F and another one remains prime in K. If p = 59, (p) splits completely in K/Q. Let σ, ρ denote elements ∈ JK defined by σ(
q13+√ 5 2 i) =
q13−√ 5 2 i, ρ(
q13+√ 5
2 i) = −
q13+√ 5
2 i and ΠF, ΠK, ΠKσ denote generators of prime ideals pF, pK, pKσ respectively. Then
(27)
K (ΠK) (ΠρK) (ΠσF) (ΠK) (ΠρK) (ΠσK−1σ) (ΠσK−1σρ)
| \ / / \ / \ /
F (ΠF) (ΠσF) (ΠF) (ΠσF)
| \ / \ /
Q p= 11,19,29 p= 59.
We can find:
(28) (ΠF,ΠK) =
(4 +√
5,−√25 − 1+4√5
q13+√ 5
2 i) if p= 11, (9+2√5,−1+34√5 + 12
q13+√ 5
2 i) if p= 19, (11+2√5,3+34√5 +−1+4√5
q13+√ 5
2 i) if p= 29, (8 +√
5,1+2√5 −
q13+√ 5
2 i) if p= 59.
Then we get numerically:
(29) lgp,K/F(id,id)− 1 2logp
ΠρK ΠK
=
−23
4∗41∗p logp(ǫ) if p= 11,
−175
4∗41∗p logp(ǫ) if p= 19,
−2087
4∗41∗p logp(ǫ) if p= 29,
2178
4∗41∗p logp(ǫ) if p= 59.
where ǫ= 3+2√5 ∈O×F, and 41 is the norm of the conductor of K/F.
Now, letζ7 be a primitive 7-th root of unity,ω1 =ζ7+ζ7−1,δ= 8+ω1−ω12,F =Q(ω1), K =Q(√
δi). (This is the situation in Example 11 of [Yo2].) If p = 13,29,41, (p) splits in F/Q. When p = 29, only one of prime ideals lying above (p) splits in K/F. When p= 13,41, two of them split. If we take a suitable embedding ofK, we can find:
(30) (ΠF,ΠK) =
(ω12+ 1,ω221 + −ω122+ω1√
δi) if p= 13,
(ω12+ω1+ 2,−2ω21+1 +−ω21+ω2 1+1√
δi) if p= 29, (3ω12−ω1,ω12−ω21+1 +−2ω1√
δi) if p= 41.
We get numerically:
(31) lgp,K/F(id,id)−1 2logp
ΠρK ΠK
=
−9900811 logp(ǫ)−9252485 logp(η)
23∗1672∗p∗32 if p= 13,
−265801098 logp(ǫ)−308408783 logp(η)
23∗1672∗p2∗3 if p= 29,
951906971 logp(ǫ)−1056605375 logp(η)
23∗1672∗p∗32 if p= 41.
where ǫ, η = 2 +ω1, ω12 ∈ OF×, 167 = N(δ) is the norm of the conductor of K/F. These values are consistent with our conjecture (26).
2. The construction of class fields.
2.0. Introduction.
We will show some applications of our conjecture, concerning Stark’s conjecture, Gross’
conjecture, and the construction of class fields. Stark’s conjecture is a refinement of Stark- Tate’s result: Let S be a finite set of places in F which contains all ramified primes in a Galois extension K/F. For each character χ of Gal(K/F), Stark defined ”the regulator”
and showed:
(32)
LS(s, χ)∼ ”an algebraic number” ∗ ”the regulator” ∗ sr(χ,S)+O(sr(χ,S)+1), (s→0).
HereLS(s, χ) := P
(a,p)=1 for allp∈Sχ(a)N(a)−s and r(χ, S) := #{p∈S |χ(p) = 1}. Gross conjectured a p-adic analogue of this formula (32): Assume that S contains all primes lying above (p). Gross defined ”the p-adic regulator” and conjectured:
(33)
Lp,S(s, χω)∼”an algebraic number”∗”the p-adic regulator”∗sr(χ,S)+O(sr(χ,S)+1),(s→0).
Here ω is a character which is the composite mapping of the Teichm¨uller character and the ideal norm map. Gross conjectured a more precise formula in the first order zero case, which states a p-adic analogue of Stark’s conjecture.
The invariants X(c), Xp(c) involve these conjectures deeply, since these are defined by factorizations of ζ′(0,c), ζp′(0,c). In fact, using X(c), we can construct the Stark- Shintani units [Yo3]. In the p-adic case, we will prove our conjecture is stronger than Gross’ conjecture, in other words,
(34) our conjecture (26) is a refinement of Gross’ conjecture (38).
A direct consequence of Stark’s conjecture or Gross’ conjecture is the construction of class fields. Similarly, we will give a way of the construction of class fields by using the formula (26) which we conjectured.
2.1. Stark’s conjecture in the first order zero case.
Let K/F be an abelian extension with G = Gal(K/F), S a set of places of F such that|S| ≥2,S ∋any infinite place and any ramified place inK/F. First we assume that there exists a place v0 ∈S which splits completely in K. In this case, Stark’s conjecture states:
Conjecture. (Stark. [St3]) There exists an element ǫ ∈ K satisfying the following properties.
(35) ǫ is
(S-unit if |S|= 2, v0-unit if |S|>2.
For any placeω of K which divides v0 and any σ ∈G (36) log(kǫσ kω) =−wKζS′(0, σ).
Here we put wK := #{roots of unity ∈ K}, ζS(s, σ) := P
aN(a)−s where a runs over all integral ideals which satisfy a is prime to any finite place p ∈ S and the image of a under the Artin map =σ. We define k xkω:=|x|2∞,|x|∞, N(ω)−ordω(x) for complex, real and finite places ω respectively.
2.2. Gross’ conjecture.
Let the notation be as above. Additionally we assume that F is totally real, K is CM-field, S contains all places which divide (p) and the splitting place v0 is pF. In this situation we can show that there exist a v0-unit ǫ ∈ K and an integer M such that for any σ ∈G
(37) log(kǫσ kpK) =−MζS′(0, σ).
Therefore Stark’s conjecture merely states we can take M =wK in this case. Note that if we fix M, the v0-unit ǫ satisfying (37) is uniquely determined up to the multiplication by roots of unity.
Conjecture. (Gross [Gr].) Take F, K, S, ǫ, M as above. Then for any σ ∈ G = Gal(K/F)
(38) logp NKpK/Qp(ǫσ)
=−Mζp,S′ (0, σ).
Here ζp,S(s, σ) is the p-adic partial ζ-function which is the p-adic interpolation function of ζS(s, σ) and is p-adic analytic at s= 0. KpK is the completion of K at pK.
We can regard our conjecture (26) as a refinement of Gross’ conjecture (38). That means:
Theorem. (Kashio-Yoshida.) If our conjecture is true, Gross’ conjecture is true.
Proof. We may assume that S is the smallest one, that is, S consists of all ramified places, infinite places and places lying above (p). Then we can show that Gross’ conjecture is equivalent to the statement:
∀χ∈Gˆ−, P
σ∈JF
P
c∈Cfχ(p)0χ(c)Xpσ(c)
L(0, χ) =
Q
p|(p),p6=pF(1−χ(p)) 2
×X
τ∈G
χ(τ) logp
NKp
K/Qp
pρK pK
τ . (39)
On the other hand, our conjecture is equivalent to the statement:
(40)
∀χ∈Gˆ−, P
c∈CfχpF χ(c)Xp(c)
L(0, χ) = [P
c∈CfχpF χ(c)X(c)]p
L(0, χ) +1 2
X
τ∈G
χ(τ) logp
pρK pK
τ .
Here we define [ ]p as follows. We can show that there exist algebraic numbersαi, βi such that P
c∈CfχpF χ(c)X(c) =P
iαilog(βi), so we put [P
c∈Cfp,χχ(c)X(c)]p :=P
iαilogp(βi).
Note that this definition does not depend on the choices ofαi, βi. Since the conductors of the ideal classes appearing on these formulas (39, 40) are different, we prepare the next lemma.
Lemma. Let f be an integral ideal, pa prime ideal, χ a character of Cf and σ ∈JF.
Assume that (f,p) = 1. Then X
c∈Cfp
χ(c)Xσ(c;{vj},{aµ}) = X
c∈Cf
χ(c)Xσ(c;{vj},{aµp})−χ(p)X
c∈Cf
χ(c)Xσ(c;{vj},{aµ}) +χ(p)Y
q|f
(1−χ(q))L(0, χ).
(41)
Additionally assume that pFσ divides fσ. Then X
c∈Cfp
χ(c)Xpσ(c;{vj},{aµ}) = X
c∈Cf
χ(c)Xpσ(c;{vj},{aµp})−χ(p)X
c∈Cf
χ(c)Xpσ(c;{vj},{aµ}) +χ(p)Y
q|f
(1−χ(q))L(0, χ).
(42)
Assume that our conjecture is true. By this lemma and (40), we get for σ ∈ G satisfying (pF)σ =pFσ
P
c∈Cfχ(p)0χ(c)Xpσ(c)
L(0, χ) = [P
c∈Cfχ(p)0χ(c)Xσ(c)]p
L(0, χ)
+ Q
p|(p),p6=pF(1−χ(p)) 2
X
τ∈G
χ(τ) logp pρ
p τ σ
. (43)
If (pF)σ 6=pFσ, we can show (44)
P
c∈Cfχ(p)0χ(c)Xpσ(c)
L(0, χ) = [P
c∈Cfχ(p)0χ(c)Xσ(c)]p
L(0, χ) . Since
[P
σ∈JF
P
c∈Cfχ(p)0χ(c)Xσ(c)]p
L(0, χ) =
Q
p|(p),p6=pF(1−χ(p))
L(0, χ) logp(p)
L(0, χ) = 0,
(45)
the assertion is clear.
Remark. By (39, 40), we can show that Gross’ conjecture is equivalent to our con- jecture if and only if r(χ) := #{p|(p), χ(p) = 1} = 1 and KpK = Qp. Furthermore, if r(χ)>1 then Gross’ conjecture merely states L′p(0, χω) = 0, (we proved it as a corollary (19) of the p-adic analogue of Shintani’s formula) and does not state anything about the values of Xp(c), lgp,K/F(id, τ), although our conjecture predicts them. (See numerical examples in §1.4.)
Remark. By (40), our conjecture states: for τ ∈G
(46) − 1
wK
logp p
P
σ∈GwKζ(0,σ)σ−1 K
τ
=Xp(τ,fK/FpF)−[X(τ,fK/FpF)]p.
Here we denote by fK/F the conductor of K/F, by σc the image of c in G under the Artin map. We put X(τ,fK/FpF) (resp. Xp(τ,fK/FpF)) := P
c∈CfK/FpF,σc=τX(c) (resp.
Xp(c)),ζ(s, τ) = P
c∈CfK/F,σc=τN(a)−s. Note that the Brumer-Stark conjecture [Ta] states p
P
σ∈GwKζ(0,σ)σ−1
K is a principal ideal.
2.3. The construction of class fields.
Class field theory states: For a number field F, assume ”we know everything about F.” Then we can see the Galois groupG:= Gal(K/F) and the splitting patterns of prime ideals in K/F for any class field K. A problem is: Can we see K itself? Or equivalently:
Can we find a polynomial f(X) ∈ F[X] defining the extension K/F? If our conjecture is true, this problem can be solved for all totally real number fields F and CM-fields K.
Below is how to get f(X) defining K/F, assuming our conjecture is true.
1. Find a prime ideal p⊂ OF which splits completely in K/F. Let pbe the positive generator of p∩Z and fix an embedding of F so that pF =p.
2. By (46), there exists an integer N such that (47) z0 := expp N Xp(id,fK/FpF)−[X(id,fK/FpF)]p
= Π−
P
σ∈G N
hKζ(0,σ)σ−1
pK ∈K.
Here we denote by ΠpK a generator of (pK)hK with the class number hK of K. We can compute z0 numerically in Cp. Take an integer M so that
(48) z :=pMz0 ∈OK.
Then K =F(z). Besides we can compute zτ for each τ ∈ G by (46). Then we define a polynomial
(49) f(X) := X|G|+α|G|−1X|G|−1+· · ·+α1X+α0 := Y
τ∈G
(X−zτ)∈OF[X].
This polynomial is what we want.
3. We can compute each coefficient αi, that is, we get f(X) numerically. But it is not enough. For example, although X2 + 1 is very close to X2 + 1− p1000, Q(i) 6= Q(p
p1000−1) ⊂ R. Since |zσ|∞ = pM for any σ ∈ JK, the absolute values (at ∞) of all conjugates of each αi are bounded. Such elements ∈OF are finite, so we can find out αi ∈OF. That means, we get f(X).
Examples. First let F := Q(√
35). Then the class number hF = 2 and the narrow class number hF,0 = 4. Then there exists a class field K of degree 4 over F such that the Galois group Gal(K/F) is equal to the narrow ideal class group C(1). We can show that K is totally complex and contains the Hilbert class field of F which is totally real, so K is a CM-field. We would like to construct K. Choose p = 3 which remains prime in F and splits completely in K/F. Then we get
f(X) = X4−2X3−32∗149X2+ 36∗2X+ 312. (50)
We can show that f(X) define the extension K/F easily.
Now we put F := Q(√
79). Then hF = 3, hF,0 = 6. We want to get the class field K such that Gal(K/F) ∼= C(1). Let p = 11. Then (p) remains prime in F and splits completely in K/F. In this case we get
f(X) =X6+ 23684126X5+ 116∗38858607X4−1112∗1575649852X3 + 1120∗38858607X2+ 1128∗23684126X+ 1142.
(51)
We can show that f(X) defines K/F.
Remark. We can construct some class fields over totally real number fields, even using Stark’s conjecture or Gross’ conjecture similarly to the above. We fix a totally real number field F and compare these methods to construct its class fields.
1. If we assume that our conjecture is true then we can construct any CM-field which is abelian over F as above.
2. Assume that Stark’s conjecture is true. By computing the Stark-Shintani unit, we can get a unit of any class field in which only one real place ofF splits. That means, ifK is a CM-field and F 6=Q then this method does not work. But we can show that for any CM-field K, there exist a finite set of number fields {Ki} satisfying that K is contained in the compositum of{Ki}and that there exists only one real place ofF splitting in each Ki. In this sense, we may construct any CM-field as a subfield of the composite field, but some informations about K/F become complicated.
3. Assume that Gross’ conjecture is true. Since ζp,S′ (0, σ) have to be non-zero, we should assume the condition ”there exists a prime number p such that only one prime ideal inOF lying above (p) splits completely inK/F” to construct K. We can not always take such p. For example, let F :=Q(√
5,√
29), K :=Q(p 4±√
5i,p 6±√
29i). Then K is not abelian over any proper subfield ofF, so we can not constructK over such fields.
Since K is normal over Q, any prime number p satisfying the above condition must not split in F/Q. Additionally F/Q is not a cyclic extension, sopshould ramify, that means p= 5 or 29. Here (5) splits inQ(√
29)/Q, (29) splits in Q(√
5)/Q. Therefore we can not construct K using Gross’ conjecture for any p.
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