Workshop “Special values of automorphic L-functions, periods of automorphic forms and related topics,”

multiple gamma functions

Tomokazu Kashio

13:20 – 14:20, Sep. 19 (Tue), 2017

Workshop “Special values of automorphic L-functions, periods of automorphic forms and related topics,”

Celebrating the 60th Birthday of Professor Masaaki Furusawa

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 1 / 29

This talk is based on the following two papers.

[K1]

K., Fermat curves and a refinement of the reciprocity law on cyclotomic units, J. Reine Angew. Math.

[K2]

K., On a common refinement of Stark units and Gross-Stark units (preprint, arXiv:1706.03198).

Besides I used many of the ideas written in

[Y]

H. Yoshida, Absolute CM-Periods, Math. Surveys Monogr. 106,

Amer. Math. Soc., 2003.

First recall “CM periods” and “Stark units” shortly.

Let K be a CM field (that is, an imaginary quadratic extension of a totally real field). For two complex embeddings σ, τ ∈

,

),

Shimura’s period symbol (or, CM period symbol) p

(σ, τ ) ∈

/Q

,

is defined in terms of periods of abelian varieties with CM by K . Note that their values are well-defined only up to multiplication by an algebraic number.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 3 / 29

Example

Let K be an imaginary quadratic field. We consider an elliptic curve E : y

= x

+ ax + b with a, b ∈

, K ∼ =

) ⊗

. Let ρ be the complex conjugation. Then there are 4 CM periods, which are defined as the integral of suitable differential forms on E :

p

(id,

p

(ρ, ρ) := π

∫

γ

dx

y mod

, p

(ρ,

p

(id, ρ) :=

∫

γ

xdx

y mod

,

where γ ⊂ E (

) is an arbitrary non-trivial closed path. Note that H

(E ) = ⟨

,

⟩ .

When [K :

] > 2, we consider an abelian variety A/

with CM by K , take a suitable ω ∈ H

(A,

), and need to “decompose”

ω.

CM periods are closely related to the theme of this workshop:

By using the symbol p

, we can express the transcendental parts of critical values of L-functions associated with algebraic Hecke characters of K .

special values of Hilbert modular forms at CM-points ∈ K . Example

Let f be an elliptic modular form of weight k whose Fourier coefficients

∈

. For any imaginary quadratic field K =

(τ ) (Im(τ ) > 0), we have f (τ )/p

(id,

∈

.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 5 / 29

Let H/F be an abelian extension of number fields. We consider the partial zeta function:

ζ (s , σ) :=

a⊂OF,(_H/F

a

)

=σ

Na

(σ ∈

)).

We assume that F is totally real and that H has a real place ι: H , →

. (In particular, a real place of F splits completely in H/F .) Then Stark’s conjecture states that

exp(2ζ

(0, σ)) ∈ H

(in fact, ∈ ι(H

)), τ (exp(2ζ

(0, σ))) = exp(2ζ

(0, τ σ)) (σ, τ ∈

)),

“exp(2ζ

(0, σ)) is a unit in many cases”, “H(exp(ζ

(0, σ)))/F is abelian”.

exp(2ζ

(0, σ)) is called a Stark unit.

Let n ≥ 3, ζ

:= e

. We consider a CM field

(ζ

) and its maximal real subfield

(ζ

+ ζ

). These are abelian over

. Write

[σ

: ζ

→ ζ

] ∈

(ζ

)/

) =

(ζ

),

) (a ∈ (

/n

)

).

By Rohrlich’s formula (reformulated by Yoshida) and Lerch’s formula, we can write them explicitly:

K :=

(ζ

) p

(σ

, σ

) ≡ π

δab 2

∏

c∈(Z/nZ)^×

Γ(

)

∑ η∈([Z/nZ)×

− η(a⁻¹bc) L(0, η)φ(n)

|

H :=

(ζ

+ ζ

) exp(2ζ

(0, σ

|

)) =

n)Γ(n−a n ) 2π

)2

|

=

2−ζ_n^a−ζ^−an

Q

Here δ

:= 1, − 1, 0 if a = b, a = − b, otherwise, respectively.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 7 / 29

Problem

Rohrlich’s formula: p

(σ

, σ

) in terms of Γ(

)’s.

Lerch’s formula: exp(2ζ

(0, σ

|

)) in terms of Γ(

)’s.

By using these, without using Euler’s formulas, can we derive Stark’s conjecture with F =

?

Since CM periods are defined only up to

, we can only derive the algebraicity of Stark units with F =

: roughly speaking,

monomial relations on CM periods

Rohrlich’s formula

⇝

monomial relations on Γ(

)’s

Lerch’s formula

⇝

the algebraicity of exp(2ζ

(0, σ

|

)).

Let F

: x

+ y

= 1 be the nth Fermat curve. J(F

) has CM by

(ζ

).

H

(J(F

)) ∼ = H

(F

) = ⟨ η

:= x

y

| 0 < r , s < n, r + s ̸ = n ⟩ . Rohrlich’s formula.

∫

γ

η

≡ B(

,

) := Γ(

)Γ(

)

Γ(

) mod

(ζ

)

. The cup product induces a correspondence

H

(F

) × H

(F

) → H

(F

) =

( − 1) (the Lefschetz motive).

Since the period of

, we obtain “monomial relations”

B(

,

)B(

,

) ≡

∫

γ

η

∫

γ^′

η

≡ 2πi mod

. Noting that Γ(

)

= Γ(r)

k=1

B(

,

), we obtain

Γ(

)Γ(

) ∈ 2πi ·

, that is, exp(ζ

(0, σ

|

)) = Γ(

)Γ(

) 2π ∈

.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 9 / 29

When F =

, the algebraicity exp(2ζ

(0, σ

|

)) ∈

follows from K :=

(ζ

)

γ

η

Rohrlich’s formula

≡

n )

mod

(ζ

)

|

H :=

+ ζ

) exp(2ζ

(0, σ

|

))

=

(Γ(a n)Γ(n−a

n ) 2π

)2

and monomial relations on CM periods

γ

η

γ

η

≡ 2πi mod

. Moreover, in [K1], we show that Coleman’s formula on the absolute Frobenius action on Fermat curves implies the reciprocity law

τ

exp(2ζ

(0, σ

|

))

≡ exp(2ζ

(0, τ ◦ σ

|

)) mod µ

(τ ∈

/

)).

Here µ

denotes the group of all roots of unity.

p-adic periods are defined by comparison isomorphisms of p-adic Hodge theory (instead of the de Rham isomorphism):

H

(F

) × H

(F

) → B

, (γ, η) 7→

∫

p,γ

η.

Here B

denotes Fontaine’s p-adic period ring. Since abelian varieties with CM have potentially good reduction, we have

p,γ

η ∈ B

. The Weil group W

⊂

/

) acts this subring as:

Φ

:= Φ

⊗ τ

B

⊗

= B

(τ ∈ W

).

Assume p ̸= 2, p | n, p

rs(r + s). (A similar argument works when p

n.) Then Coleman’s formula on Φ

H

(F

/

) implies that

Φ

( ∫

p,γ

η

B

(

,

)

)

≡ p

∫

p,γ

η

B

(

,

) mod µ

, where we define τ (r) by τ (ζ

) = ζ

, B

(α, β) :=

p(α+β)

.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 11 / 29

Now by using Rohrlich’s formula again, we can define the period-ring-valued beta function

B(_n^r

,

) := B(

,

)

∫

γ

η

∫

p,γ

η

B

(

,

) ∈ B

. Then Coleman’s formula implies the reciprocity law on

Φ

(B(

,

)) ≡ p

,

) mod µ

(τ ∈ W

).

Then, by using these formulas, we obtain the reciprocity law on Stark’s units exp(2ζ

(0, σ

|

)) =

(Γ(a n)Γ(n−a

n ) 2π

)2

up to µ

as follows:

B(^r_n

,

) := B(

,

)

∫

γ

η

∫

p,γ

η

B

(

,

) , Φ

(B(

,

)) ≡ p

,

).

By the correspondence H

(F

) × H

(F

) →

( − 1)

⇝B(_n^r

,

)B(

,

) = (2πi )

2πi

B(

,

)B(

,

)

B

(

,

)B

(

,

) ∈ (2πi)

·

. Φ

= Φ

⊗ τ acts on

as τ , on (2πi)

as p

⇝

τ

1

2πi

B(

,

)B(

,

) B

(

,

)B

(

,

)

)

≡ 1 2πi

B(

,

)B (

,

) B

(

,

)B

(

,

) . Γ

(

)

= Γ(r)

k=1

B

(

,

) ( ∗ =, p), Γ

(

)Γ

(

)

⇝

τ

1

2πi

Γ(

)Γ(

)

≡≡≡≡≡≡≡

Γ

(

)Γ

(

)

≡ 1 2πi

Γ(

)Γ(

)

≡≡≡≡≡≡≡≡≡≡

Γ

(

)Γ

(

) mod µ

only for τ ∈ W

⊂

/

) ⊂

/

). Then we vary p. Q.E.D.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 13 / 29

n n Γp( _n )

Morita’s p-adic gamma function:

Γ

(z ) :

→

, z 7→ lim

N∋n→z

( − 1)

1≤k≤n−1,p∤k

k.

Recall Lerch’s formula:

Γ(x) √ 2π = exp

(

d ds

[ _∞

∑

m=0

(x + m)

s=0

)

.

We need to put Γ

(

) := exp

( p^N

d

ds

∑

m=0

(

+ m)

(p

)

p-adic interpolation

]

s=0

) ¹

pN

for some N ∈

since exp

(z) :=

k=0 z^k

k!

converges only on a

neighborhood of 0. To be honest there is one more reason: the “µ

-part”

Summary. In [K1], studying classical or p-adic CM periods for CM fields abelian over

, we prove the reciprocity law on

,

) and provide an alternative proof of a part of Stark’s conjecture with F =

.

Note that not only an alternative proof, but also a refinement since exp(2ζ

(0, σ

|

)) = (

a n)Γ(n−a

n )

2π

)

is a finite product of

,

)’s.

Motivation. Generalize this to totally real fields F . We have some good Guidelines.

We have Shimura’s period symbol p

(σ, τ ), which is a generalization of periods

γ

η

of Fermat curves.

We have Shintani’s formula on partial zeta functions of totally real fields, which is a generalization of Lerch’s formula

exp(ζ

(0, σ

|

)) =

a n)Γ(n−a

n )

2π

.

We have Yoshida’s conjecture on “absolute CM periods”, which is a conjectural generalization of Rohrlich’s formula

γ

η

≡

.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 15 / 29

Γ(x) √ 2π = exp

(

d ds

[ _∞

∑

m=0

(x + m)

s=0

)

.

For a “good” subset Z ⊂

, we put Γ(Z ) := exp





d ds

[∑

z∈Z

z

s=0





Here we say Z is “good” if

z∈Z

z

converges for

) >> 0, has a meromorphic continuation, is analytic at s = 0. In particular, for x > 0,

:= (ω

, . . . , ω

) with ω

, . . . , ω

> 0, the lattice-like set

L

:= { x + m

ω

+ · · · + m

ω

| 0 ≤ m

, . . . , m

∈

is “good” and Γ(L ) is called Barnes’ multiple gamma function.

Let F be a totally real field,

an integral ideal of

C

the ideal class group modulo

D be Shintani’s fundamental domain of F

/ O

. For c ∈ C

, we take an ideal

∈ c and consider a subset:

Z

:= { z ∈ D ∩

| z

∈ c } ⊂ F

. Shintani provided an expression Z

=

i=1

L

. In particular ι(Z

) is

“good” for any ι ∈

,

). Yoshida defined a class invariant Γ(c , ι) := Γ(ι(Z

)) ×

i

ι(a

)

(c ∈ C

, ι ∈

,

)) for certain a

, b

∈ F . Although Z

, a

, b

depend on D,

∏

ι∈Hom(F,R)

Γ(c, ι) = exp(ζ

(0, c )): Shintani’s formula, which is a generalization of Lerch’s formula.

Γ(c , ι) mod ι( O

)

does not depend on the choices of D,

that is, Γ(c , ι; D,

, ι; D

,

) = ι(ϵ)

with ϵ ∈ O

, N ∈

We fix

F , →

and put Γ(c ) := Γ(c,

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 17 / 29

Example (F = Q )

F

/ O

=

/ { 1 } , D =

.

C

= { [(a)] | 1 ≤ a ≤ f , (a, f ) = 1 } ∼ = (Z /f

)

(f ∈

).

Z

:= { z ∈

∩ (a)

| (za) ∈ [(a)] } = {

| 0 ≤ k ∈

. Γ([(a)]) := Γ(Z

) × a correction term

= exp





d ds



∑

k≥0

(_a+kf

a

)₋s





s=0





× a

= Γ(

)f

(2π)

= exp(ζ

(0, [(a)])).

Note that

,

) = {

} .

Example ([F : Q ] = 2)

F

/ O

= F

/ ⟨ ϵ ⟩ , D = { a + bϵ | a, b ∈

, a ≥ 0, b > 0 } . Let c ∈ C

,

∈ c, Z

:= {z ∈ D ∩

| za ∈ c }. We defined

Γ(c ) := Γ(Z

) × correction terms. In this case, it can be expressed in terms of Barnes’ double gamma function:

Γ(x, (ω

, ω

)) := exp( d ds [

k1,k2≥0

(x + k

ω

+ k

ω

)

]

).

∃ a finite set R ⊂ D, an element α ∈ F , a generator π

of (af)

s.t.

Γ(c , ι) =

x∈R

Γ(ι(x), (1, ι(ϵ))) × ι(ϵ)

× ι(π

)

ζ(0,c) h+

F

, exp(ζ

(0, c )) =

ι∈Hom(F,R)

Γ(c , ι).

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 19 / 29

Yoshida formulated a conjecture in [Y] which expresses Shimura’s period symbol p

as a finite product of rational powers of Γ(c)’s. Here we introduce its slight generalization: The original conjecture in [Y] is equivalent to

Conjecture

Assume that the narrow ray class field H

modulo

contains a CM field.

Let K be the maximal CM subfield of H

. Then we have for

)

∏

c∈Art⁻¹(σ)

Γ(c) ≡

c∈Art⁻¹(σ)

π

c^′∈Cf

p

(c , c

)

ζ(0,c′)

[Hf:K]

mod

.

Strictly speaking, c, c

in p

(c, c

) are the images of c, c

under the Artin map

: C

→

/F ).

When F =

, this conjecture holds true by Rohrlich’s formula.

Let

be the prime ideal corresponding to the

. Let D be Shintani’s fundamental domain of F

/ O

. For c ∈ C

with

we take an ideal

∈ c and put

Z

:= { z ∈ D ∩

| z

∈ c } ⊂ F , →

. For c ∈ C

, we define

Γ

(c) := Γ

(Z

) ×

i

exp

(b

log

a

) (a

, b

∈ F ),

Γ

(Z

) := exp





d ds



∑

z∈Zc

z

analytic continuation

p-adic interpolation





s=0





for some a

, b

∈ F satisfying that

Γ

(c ) mod ( O

)

does not depend on

D.

Moreover the “ratio” [Γ(c ) : Γ

(c)] mod µ

does not depend on

D, i.e., Γ(c; D,

; D

,

) ≡ Γ

(c ; D,

(c ; D

,

) mod µ

.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 21 / 29

Hecke character χ of K

:

the infinite type of χ = ℓ · (τ

− ρ ◦ τ

) with ℓ large enough.

χ(a) ∈ K for all

⊂ O

relatively prime to the conductor of χ.

and consider the associated motive M (χ)/K

with coefficients in K . By comparison isomorphisms of

H

(M(χ)) ⊗

∼ = H

(M (χ)) ⊗

, c

⊗ P

(χ) → c

⊗ 1, K ⊗

∼ =

σ∈Hom(K,BdR)

B_dR

, P

(χ) → (P

(σ, χ))

with c

a K or K ⊗

K

-basis of H

(M(χ)). Then we define

p

(σ, τ ) ≡ (2πi)

P

(σ, χ)

mod

,

where we put δ

:= 1, − 1, 0 if σ = τ, ρ ◦ τ , otherwise, respectively.

Moreover [p (σ, τ ) : p (σ, τ )] mod µ

is well-defined.

So far, I have explained the following things: For a totally real field F , c ∈ C

with

|

[Γ(c ) : Γ

(c )] ∈ (

/µ

×

/µ

)/( O

)

.

For a CM field K , σ, τ ∈

,

), we defined the ratio of periods [p

(σ, τ ) : p

(σ, τ )] ∈ (

/µ

× B

/µ

)/

. Then (a slight generalization of) Yoshida’s conjecture implies that

G(c) :=

Γ(c ) (2πi )

c^′∈Cf

p

(c, c

)

ζ(0,c′) [Hf:K]

(2πi )

∏

c^′∈Cf

p

(c , c

)

ζ(0,c′) [Hf:K]

Γ

(c )

∈ B

/µ

is well-defined. Furthermore, by CM,

G(c)

∈ (B

− {0})

/µ

where τ ∈ W

acts as Φ

= Φ

⊗ τ .

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 23 / 29

Now we can state the main result in [K2].

Conjecture

Assume that

|

c ∈ C

, τ ∈ W

⊂

/F

), we have the following reciprocity law on this period-ring-valued function:

Φ

(G(c)) ≡

c ) mod µ

where c

:=

(τ |

) ∈ C

. In [K2], we proved the following:

Theorem

Conjecture holds true when H

is abelian over

and

2.

The case F =

follows from Rohrlich’s formula and Coleman’s formula as

we have seen. We reduce the problem to the case F =

, as follows:

G(c) := ^Γ(c)

(2πi)^ζ(0,c)∏

c′∈CfpK(c,c^′)

ζ(0,c′) [Hf:K]

(2πi)^ζ(0,c)p ∏

c′∈CfpK,p(c,c^′)

ζ(0,c′) [Hf:K]

Γp(c)

. Recall that exp(ζ

(0, c )) =

ι∈Hom(F,R)

Γ(c, ι). When H

/

is abelian, ι(F ), ι(f) and ι(c) ∈ C

do not depend on ι ∈

,

). Hence, we obtain an expression like exp(ζ

(0, c)) = Γ(c )

× (correction terms) by Yoshida’s technique. Since the same holds true for exp

(ζ

(0, c)) we have

[exp(ζ

(0, c )) : exp

(ζ

(0, c ))] ≡ [Γ(c )

: Γ

(c )

] mod µ

. Put G :=

/

) and consider C

=

/F ) ⊂ G . Then we have

∑

c∈C_f

χ(c )ζ

(s , c) = L

(s , χ) =

ψ∈Gb,ψ|Cf=χ

L

(s , ψ) (χ ∈ C

, ∗ = ∅ , p).

Hence we obtain an explicit relation between [Γ(c ) : Γ

(c)]’s of H

/F and those of H

/

. The part of [p

(. . . ) : p

(. . . )] is simpler. Q.E.D.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 25 / 29

Remark

We also formulated a conjecture in the case

complicated since we do not have Γ

(c) with

Our conjecture is consistent with Stark’s conjecture w.r.t. real places and Gross’ p-adic analogue:

Slight generalization of Yoshida’s conjecture “implies” the algebraicity of Stark’s units:

Our conjectures in both casesp|f,p∤fimply the reciprocity law on Stark’s units up toµ_∞.

Our conjecture in the casep∤fimplies Gross’p-adic analogue which was proved by Dasgupta-Darmon-Pollack and Ventullo, and its refinements by K.-Yoshida under a certain assumption.

Conjecture (Slight generalization of Yoshida’s conjecture)

Assume that the narrow ray class field H

modulo

contains a CM field.

Let K be the maximal CM subfield of H

. Then we have for

)

∏

c∈Art⁻¹(σ)

Γ(c) ≡

c∈Art⁻¹(σ)

π

c^′∈C_f

p

(c , c

)

ζ(0,c′)

[Hf:K]

mod

.

This “difference” is equivalent to the algebraicity of Stark’s units:

exp(ζ

(0, σ)) ∈

for σ ∈

)

if F is a totally real field, H has a real place, H/F is abelian.

Namely, we have

Generalized version implies this algebraicity.

“Original version + this algebraicity” implies generalized version.

(K., On the algebraicity of some products of special values of Barnes’

multiple gamma function, to appear in Amer. J. Math.)

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 27 / 29

Remark

We also formulated a conjecture in the case

complicated since we do not have Γ

(c) with

Our conjecture is consistent with Stark’s conjecture w.r.t. real places and Gross’ p-adic analogue:

Slight generalization of Yoshida’s conjecture “implies” the algebraicity of Stark’s units.

Our conjectures in both casesp|f,p∤fimply the reciprocity law on Stark’s units up toµ_∞.

Our conjecture in the casep∤fimplies Gross’p-adic analogue which was proved by Dasgupta-Darmon-Pollack and Ventullo, and its refinements by K.-Yoshida under a certain assumption.

Conjecture

Assume that

c ∈ C

; D,

Γ(c ; D,

(2πi)

c^′

p

(c , c

)

ζ(0,c′) [Hf:K]

(2πi )

c^′

p

(c , c

)

ζ(0,c′) [Hf:K]

∈ (B

− { 0 } )

/µ

. Then we have for τ ∈ W

with deg

τ = 1

Φ

(G(c; D,

≡ π

ζ(0,[p]c) h+

p F G([p]c;

D,

∏

˜ c∈Cfp

˜

c7→[p]c∈Cf

Γ

(˜ c; D,

mod µ

.

The truth of Conjecture does not depend on the choices of D,

F =

follows from Rohrlich’s formula and Coleman’s (other) formula.

T. Kashio, Tokyo Univ. of Sci. CM periods, Stark units & mult. Γ-functions Sep. 19, 2017 29 / 29