Absolute CM-periods
Hiroyuki Yoshida
This paper reproduces my talk at the Hakuba conference faithfully. Its purpose is to give a brief survey on absolute CM-periods, as preparations for p-adic theory. I also stressed the importance of the factorization process involved in the construction of the absolute CM-period symbol. Such a fac- torization idea has been employed efficiently in number theory (cf. §4) and will play an important role in the future research.
For full details of absolute CM-periods in the complex case, see my book [Y5].
§1. The Chowla–Selberg formula
A prototype of the absolute CM-period is the Chowla–Selberg formula.
Let K be an imaginary quadratic field of discriminant −d. Let E be an elliptic curve defined over Q with complex multiplication by K and ̟ be a period of E. The Chowla–Selberg formula states
(1) exp(L′(0, χ) L(0, χ)) = 1
d
d−1Y
a=1
Γ(a
d)wχ(a)/2h ∼π−1̟ ∼πpK(id,id)2.
Here h and w denote the class number and the number of roots of unity of K respectively; χ denotes the Dirichlet character corresponding to K; pK
denotes Shimura’s period symbol. For a, b ∈C, we writea ∼b if b6= 0 and a/b ∈Q.
The absolute CM-period gives a conjectural generalization of (1) using the multiple gamma function in place of the gamma function.
§2. Shimura’s period symbol
LetM be an algebraic number field. We understand thatM is a subfield ofC. The ring of integers ofM is denoted byOM. ByhM andRM, we denote the class number and the regulator ofM respectively. By JM, we denote the set of all isomorphisms of M into C and IM denotes the free abelian group generated by JM.
LetK be a CM-field. Shimura’s period symbol pK is a bilinear mapping pK :IK×IK −→C×/Q×.
For notational convenience, we regard pK taking values in C× which are well defined modulo Q×. The relation of pK with the geometric period is given as follows. Let A be an abelian variety defined over Q with complex multiplication by K. Let Φ be the equivalence class of the representation of K on the space of holomorphic 1-forms. For every σ ∈ Φ, we can find a holomorphic 1-form ωσ rational over Q such that
a∗ωσ =aσωσ, a∈K∩End(A).
Then we have (2)
Z
c
ωσ ∼πpK(σ,Φ), ∀c∈H1(A,Z).
For full details of the period symbol pK, see [S3], Chapter VII, §32.
Example (Weil). Letl be an odd prime and a be an integer such that 1≦a≦(l−1)/2. We consider an algebraic curve
C :yl=xa(1−x)
andJ be the jacobian variety ofC. The genus ofCis (l−1)/2 and dim(J) = (l−1)/2. J has complex multiplication by K =Q(ζ), ζ =e2πi/l. For t ∈Z not divisible byl, letσ(t) denote the automorphism ofKsuch thatζσ(t) =ζt. For x ∈ R, let hxi denote the decimal part of x, i.e., 0 ≦ hxi < 1, x ≡ hxi mod Z. Put
Ta ={t |1≦t ≦l−1, hat l i+ht
li<1}, Φa ={σ(t)|t∈Ta}.
A formula of Weil [W], p. 815 and (2) imply that (3) πpK(σ(t),Φa)∼ Γ(hatl i+htli)
Γ(hatli)Γ(htli), t∈Ta.
§3. The logarithmic derivative of the Artin L-function at s = 0 The formula (3) contains sufficient information on pK for K =Q(e2πi/l).
In fact, using the bilinearity ofpK, we can expresspK in terms of the gamma function. For general cyclotomic fields, we have to use the Fermat curve xn+yn= 1. 1 The result can be summarized by the following theorem.
Theorem (Anderson [A]). LetK be a CM-field abelian over Q. Put G= Gal(K/Q). Then we have
pK(id, τ)∼π−µ(τ)/2 Y
χ∈Gˆ−
exp(χ(τ)
|G|
L′(0, χ)
L(0, χ)), τ ∈G.
Here Gb− denotes the set of all odd characters of G and
µ(τ) =
1 if τ = 1,
−1 if τ =ρ, 0 otherwise.
For a proof, see [Y5], Chapter III, Theorem 2.6. In view of this theorem, it is natural to conjecture the following in the non-abelian case.
Conjecture A (Colmez [C], Yoshida [Y3]). Let K be a CM-field normal over Q. Put G= Gal(K/Q). Let cbe a conjugacy class of G. Then we have
Y
τ∈c
pK(id, τ)∼π−µ(c)/2 Y
χ∈Gˆ−
exp(χ(c)|c|
|G|
L′(0, χ)
L(0, χ)), τ ∈G.
Here Gb− denotes the set of characters of all irreducible odd representations of G,
µ(c) =
1 if c={1},
−1 if c={ρ}, 0 otherwise.
Remark. In Conjecture A,ρdenotes the complex conjugation;ρbelongs to the center of G. A representation ω of G is called odd if ω(ρ) = −id.
1It is known that the jacobian variety of the Fermat curve containsJ as its factor.
Now let F be a totally real algebraic number field of degree n. Put JF = {σ1, . . . , σn}. We assume that a CM-field K is abelian over F. Put G= Gal(K/F). From Conjecture A, we can derive the relation
(4)
Yn i=1
pK(σi, τ σi)∼π−nµ(τ)/2 Y
χ∈Gˆ−
exp(χ(τ)
|G|
L′(0, χ)
L(0, χ)), τ ∈G.
Here Gb− denotes the set of all characters χ of G such that χ(ρ) = −1;
µ(τ) = 1, −1, 0 if τ = 1, ρ, otherwise. We use the same letter σi for its extension to an element of JK;pK(σi, τ σi) modQ× does not depend on the choice of this extension. By a property of the period symbol, we have
pK(σi, τ σi)∼pKσi(id, σi−1τ σi).
Hence (4) can be written as (5)
Yn i=1
pKσi(id, σi−1τ σi)∼π−nµ(τ)/2 Y
χ∈Gˆ−
exp(χ(τ)
|G|
L′(0, χ)
L(0, χ)), τ ∈G.
§4. Factorization
The left hand side of (5) is factored as the product of conjugate terms over F. In §6, we will show that the right hand side can also be factored naturally using Shintani’s formula. In this way, we can deepen Conjecture A. In this section, we will recall two such examples with which the author has some familiarity.
The first example concerns Hilbert modular forms. For an elliptic mod- ular cusp form f of weight k, which is a primitive Hecke eigenform, it is a well known theorem of Shimura that
L(m, f, ω)
(2πi)mc±(f) ∈Q, 1≦m≦k−1, m∈Z.
Hereωis a Dirichlet character andL(s, f, ω) =P∞
n=1a(n)ω(n)n−sforf(z) = P∞
n=1a(n)e2πinz; c±(f) are the periods of f and ± is chosen so that ±1 =
(−1)mω(−1). Let g be another primitive Hecke eigenform of weight l. We assume that k > l. Then we have
L(m, f ⊗g)
(2πi)2m−l−1c+(f)c−(f) ∈Q l≦m < k, m ∈Z.
Now let f be a Hilbert modular cusp form with respect to a congruence subgroup of SL(2,OF). Here F is a totally real algebraic number field of degree n. The weight of f is given by (k(τ))τ∈JF ∈ ZJF. We assume that f is a primitive Hecke eigenform. For a Hecke character ω of finite order of FA×/F×, we obtain a period invariant U(f, ǫ) which describes the critical values L(m, f, ω); ǫ is a mapping from JF to {±1} which depends only on m and the parity of ω at the infinite places. 2 Let g be another primitive Hecke eigenform of weight (l(τ))τ∈JF such that k(τ) > l(τ) for τ ∈ δ and k(τ) < l(τ) for τ ∈ δ′ with subsets δ and δ′ of JF satisfying JF = δ ⊔δ′. We have period invariants Q(f, δ) and Q(g, δ′) which describes the critical values L(m, f ⊗g). These facts were proved by Shimura. For f, there are 2n+1 period invariants U(f, ǫ) and Q(f, δ).
Now a conjecture of Shimura states that these period invariants can be factorized naturally; there are 2n period invariants c±τ(f) such that
U(f, ǫ)∼ Y
τ∈JF
cǫ(τ)τ (f), Q(f, δ)∼Y
τ∈δ
c+τ(f)c−τ(f).
For an interpretation of this conjecture by the theory of motives, see [Y1].
In [Y2], the author proved this conjecture when the weight of f is not too small and satisfies the parity condition, based on results of M. Harris and H.
Hida.
The next example is the Stark conjecture. LetK be an algebraic number field. From the analytic class number formula of Dirichlet–Dedekind and the functional equation, it follows that
ζK(s)∼ −hKRK
wK
sr1+r2−1, s →0.
2Strictly speaking, to obtainU(f, ǫ), we have to assume thatk(τ) mod 2 is independent ofτ.
HerewK is the number of roots of unity contained inK;r1 andr2 denote the number of real and imaginary infinite places ofK respectively. Now suppose that K is normal over an algebraic number field F. Put G = Gal(K/F) and let Gb denote the set of all irreducible characters of G. The quotient of the Dedekind zeta function can be factorized as the product of the Artin L-functions:
ζK(s)
ζF(s) = Y
16=χ∈Gˆ
L(s, χ, K/F)dim(χ).
For χ∈G, Stark defined a generalized regulatorb R(χ) so that RK
RF ≡ Y
16=χ∈Gˆ
R(χ)dim(χ) mod Q×
holds 3 and conjectured that
L(s, χ, K/F)∼R(χ)sr(χ), s→0.
Here r(χ) is a nonnegative integer which can be easily determined fromχ.
§5. Shintani’s formula
T. Shintani discovered some essential aspects of Stark’s conjecture inde- pendently. He derived a formula for the derivative ats = 0 of the partial zeta function. This formula is of pivotal importance for the factorization process.
Forω = (ω1, . . . , ωr),ωi >0 for 1≦ i≦ r and x >0, we define ther-ple zeta function by
ζr(s, ω, x) =
X∞ m1,...,mr=0
(x+m1ω1+· · ·+mrωr)−s.
This series converges when Re(s)> rand can be continued meromorphically to the whole s-plane. It is holomorphic at s= 0. We define
∂
∂sζr(s, ω, x)
s=0 = logΓr(x, ω) ρr(ω) ,
3R. Brauer had a similar idea in 1950’s.
−logρr(ω) = lim
x→+0
∂
∂sζr(s, ω, x)
s=0
+ logx
.
Γr(x, ω) is the r-ple gamma function introduced by Barnes at the beginning of the twentieth century. If r = 1, we have
Γ1(x,1) = Γ(x), ρ1(x) =√ 2π.
LetF be a totally real algebraic number field of degree n. For r linearly independent vectors v1, . . ., vr in Rn, we put
C(v1, . . . , vr) ={x1v1+· · ·+xrvr |xi >0,1≦i≦r}
and call it an r-dimensional open simplicial cone. Shintani showed the exis- tence of a cone decomposition:
Rn+ =⊔ǫ∈EF+ǫ(⊔j∈JCj).
Here R+ is the set of all positive real numbers and Cj =C(vj1, . . . , vjr(j)) is an r(j)-dimensional open simplicial cone with vji ∈ OF; J is a finite set of indices. We put vj = (vj1, . . . , vjr(j)).
Leta1, . . ., ah0 be integral ideals which represent narrow ideal classes of F. Let f be an integral ideal ofF. By Cf, we denote the ideal class group of F modulo f∞1· · · ∞n. Let c ∈Cf. We take aµ so that c and aµf belong to the same narrow ideal class. We put
R(Cj, c) ={z ∈(aµf)−1 |z = Xr(j)
i=1
xi(z)vji,0< xi(z)≦1,(z)aµf≡c in Cf}. This is a finite set. Let ζF(s, c) be the partial zeta function of the class c, i.e.,
ζF(s, c) =X
a∈c
N(a)−s, Re(s)>1.
Then Shintani’s formula has the following form:
ζF′ (0, c) =X
j∈J
X
z∈R(Cj,c)
X
σ∈JF
logΓr(j)(zσ, vjσ)
ρr(j)(vjσ) + correction terms.
We put
G(c) =X
j∈J
X
z∈R(Cj,c)
logΓr(j)(z, vj) ρr(j)(vj) . V(c) is the term of the form
V(c) =X
i
ailogǫi, ai ∈F, ǫi ∈EF+.
We omit the details of the definition. 4 We put W(c) =−1
nlogN(aµf)ζF(0, c), X(c) =G(c) +V(c) +W(c).
SinceX(c) depends on the choice of{Cj}andaµ, we write it asX(c;{Cj},{aµ}) when we have to be precise. Then Shintani’s formula can be written as
Theorem. ζF′ (0, c) = P
σ∈JF X(cσ;{Cjσ},{aσµ}).
§6. Absolute CM-period symbol
Let F be a totally real algebraic number field and K be an abelian ex- tension of F. We assume thatK is a CM-field. We put G= Gal(K/F). By Gb−, we denote the set of all odd characters of G. For χ ∈ G,b f(χ) denotes the finite part of the conductor of χ. For τ ∈G, we define
(6) gK(id, τ) =π−µ(τ)/2exp( 1
|G| X
χ∈Gˆ−
χ(τ) L(0, χ)
X
c∈Cf(χ)
χ(c)X(c)).
We call it the absolute CM-period symbol. It depends on the choice of {Cj}, {aµ} and also on F. When we have to be precise, we write it as gK/F(id;τ,{Cj},{aµ}).
4The definition ofV(c) is a technical core in the construction of the absolute CM-period symbol. We refer the reader to [Y4], (4.4) or [Y5], Chapter III, (3.29), (3.30).
Using the theorem in§5, which gives the factorization ofζF′ (0, c), we have
(7)
Yn i=1
gKσi(id, σi−1τ σi;{Cjσi},{aσµi})
=π−nµ(τ)/2exp( 1
|G| X
χ∈Gˆ−
χ(τ) L(0, χ)
X
c∈Cf(χ)
χ(c)ζF′ (0, c)).
Therefore the formula (5), which is derived from Conjecture A, can be written as
(8)
Yn i=1
gKσi(id, σi−1τ σi;{Cjσi},{aσµi})∼ Yn i=1
pKσi(id, σi−1τ σi).
We conjecture a stronger form of (8):
Conjecture B. pK(id, τ)∼gK(id, τ).
There are overwhelming numerical evidences which support Conjecture B. The reader is referred to [Y4], [Y5].
§7. Remarks
In this section, we will give several remarks to clarify the meaning of our conjectures.
(1) LetL be a CM-field and χ be an algebraic Hecke character of L×A. A theorem of Shimura states that critical values L(m, χ) can be descibed using the period symbol pL. We can refine Conjecture B so that we are able to describe the behavior under Aut(C) of the quotient of a product of such crit- ical values by an absolute CM-period. In this form, the conjecture becomes amenable to numerical tests. We can show that this refined conjecture is consistent with the theory of motives.
(2) Our absolute CM-period symbol gK/F(id, τ;{Cj},{aµ}) depends on the choice of {Cj}, {aµ} and F. We can prove that the validity of Conjec- ture B does not depend on the choices of a cone decomposition {Cj} and representatives of the narrow ideal classes {aµ}. The dependence on F is more subtle. For partial independence results, see [Y5], Chapter III, §3.8.
(3) Conjecture B can predict the values of pK by the multiple gamma function. To see this, letK be a CM-field andM be the normal closure ofK over Q. Letσ,τ ∈JK. We take ˜σ ∈JM so thatσ = ResM/K(˜σ). A property of the period symbol gives
pK(σ, τ)∼pM(˜σ,InfM/K(τ)).
We may identify JM with Gal(M/Q). Now pM(α, β)∼ pM(id, α−1β) for α, β ∈JM; hence it suffices to considerpM(id, γ),γ ∈Gal(M/Q). LetF be the fixed field of the group generated by γ and ρ. Then F is totally real and M is abelian over F. Therefore pM(id, γ) is predicted by Conjecture B.
(4) One may ask that Conjecture B is truly stronger than Conjecture A.
LetK be a CM-field and Φ be a CM-type ofK. Conjectures of Shimura and Deligne state that pK(σ,id), σ ∈ Φ are algebraically independent (cf. [Y5], Appendix III). Now assume that K is normal over Q. Then pK(σ,id) ∼ pK(id, σ−1), pK(id, σρ) ∼ pK(id, σ)−1, σ ∈ JK. Admitting conjectures of Shimura and Deligne, we have [K :Q]/2 transcendental invariants. On the otherhand, Conjecture A can produce transcendental invariants at most the number of conjugacy classes of Gal(K/Q). In general, the former number is considerably bigger than the latter. For example, there exists a CM-field K such that Gal(K/Q)∼=Sn×Z/2Z. Then we have
[K :Q]/2 =n!, logn!∼nlogn.
On the otherhand, the number of the conjugacy classes of Gal(K/Q) is 2p(n), where p(n) denotes the number of partitions of n. A famous formula gives
logp(n)∼ r2
3πn1/2.
(5) Let F be a totally real algebraic number field of degree n ≧ 2. Let M be a class field over F of conductor f∞2· · · ∞n. Here one has to notice the subtle difference of the infinite part of the conductor compared to our previous situation. Put G = Gal(M/F). For σ ∈ G, let ζF(s, σ) be the partial zeta function of σ. A conjecture of Stark–Shintani tells that there exists a unit u∈EM such that
ζF′ (0, σ) = loguσ, ∀σ ∈G.
By our invariants X(c), periods and units can be comprehended in a unified manner.
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