東京理科大学 理工学部 加塩朋和

東京理科大学 理工学部 加塩朋和

E-mail: kashio [email protected]

2018 ^年 2 ^月 26 ^日 ( ^月 ) ∼ 3 ^月 1 ^日 ( ^木 ) 坂内研究室 プロジェクト研究集会 会場：箱根太陽山荘 ( ^{箱根・強羅温泉} )

∗2010年2月1日〜3月31日 坂内研究室 所属

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 1 / 39

Definition (Euler’s Γ-function) Γ(z) :=

∫

0

t

e

dt ( ℜ (z) > 0).

z Γ(z) =

∫

0

(t

)

e

dt = [

t

e

]

0

−

∫

0

t

(e

)

dt = Γ(z + 1).

Γ(1) =

∫

0

e

dt = [

− e

]

0

= 1 ⇝ Γ(n) = (n − 1)!.

Γ(1/2) = 1.772453850 . . . .

Γ(1/3) = 2.678938534 . . . , Γ(2/3) = 1.354117939 . . . . Γ(1/4) = 3.625609908 . . . , Γ(3/4) = 1.225416702 . . . . Γ(1/2)

= 3.141592653 · · · = π.

Γ(1/4)Γ(3/4)/π = 1.414213562 · · · = √ 2.

Γ(1/3)Γ(2/3)/π = 1.154700538 · · · = 2 √

3/3.

Theorem (Euler’s reflection formula) Γ(z)Γ(1 − z) = π

sin πz (

= 2πi

e

− e

)

.

Theorem (Multiplication formula)

Γ(z)Γ(z +

) = 2

√

2πΓ(2z).

d

∏

Γ(z +

) = d

√

2π

Γ(dz).

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 3 / 39

Theorem (Euler’s reflection formula) Γ(z) √

2π

Γ(1 − z)

√ 2π = 1

2 sin πz = i e

− e

z∈Q

≒ √

cyclotomic unit.

Theorem (Multiplication formula)

Γ(z) √ 2π

Γ(z +

)

√ 2π = 2

Γ(2z)

√ 2π .

d

∏

Γ(z +

)

√ 2π = d

Γ(dz)

√ 2π .

⇒ Γ

(z) := Γ(z)

√ 2π seems to be more important.

In fact, Γ

(z)

= exp (

ds

∑

k=0

(z + k)

|

) (z > 0).

Problem

“How many” monomial relations of Γ

(

) := Γ(

)/ √

2π are there?

e.g.,

∏

(a,N)=1

Γ

(

) = √

N (“cyclotomic unit”) = 1 (N ̸ = p

).

Γ

(

)

Γ

(

)

Γ

(

)

Γ

(

)

Γ

(

)

Γ

(

)

Γ

(

)

Γ

(

)

= 1.

⇝ seems to be irregular.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 5 / 39

Problem

“How many” relations of the form ∏

a=1

Γ

(

)

∈ Q

are there?

In other words,

Consider the group (under the multiplication) G

:= ⟨ Γ

(

) | 1 ≤ a ≤ N ⟩

= {

∏

a=1

Γ

(

)

| r

∈ Q }

⊂ C

/ Q

.

Then

N − 1 − (# of relations) = dim

G

= ? = ϕ(N )/2.

Theorem

G

:= ⟨ Γ

(

) | 1 ≤ a ≤ N ⟩

⊂ C

/ Q

. Then dim

G

≤ ϕ(N )/2.

Example (N = 6)

Write α ∼ β if α ≡ β mod Q

.

duplication: Γ

(a/6)Γ

(a/6 + 1/2) ∼ Γ

(2a/6) (a = 1, 2, 3)

triplication: Γ

(a/6)Γ

(a/6 + 1/3)Γ

(a/6 + 2/3) ∼ Γ

(3a/6) (a = 1, 2)



 

 



1 − 1 0 1 0 0 1 0 − 1 1

0 0 1 0 0

1 0 0 0 1

0 1 0 1 0



 

 



| {z }

rank=4



 

 



Γ

(1/6) Γ

(2/6) Γ

(3/6) Γ

(4/6) Γ

(5/6)



 

 

 =



 

 



Γ

(1/6)Γ

(2/6)

Γ

(4/6) Γ

(2/6)Γ

(4/6)

Γ

(5/6)

Γ

(3/6) Γ

(1/6)Γ

(5/6) Γ

(2/6)Γ

(4/6)



 

 

 ∼



 

 

 1 1 1 1 1



 

 



⇒ dim G

≤ 5 − 4 = 1.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 7 / 39

⇝ Shimura’s period symbol p

(σ, τ ) ∈ C

/ Q

is characterized by p

(σ, Ξ) := ∏

τ∈Ξ

p

(σ, τ ) ≡

{ π

∫

γ

ω

(σ ∈ Ξ)

∫

γ

ω

(σ / ∈ Ξ) mod Q

for abelian variety A of CM-type (K, Ξ), “K -eigen” differential form ω

:

A/ Q : abelian variety with K = End(A) ⊗

Q . Ξ = Ξ

is a half of Hom(K, C ) defined by

K ↷ H

(A, C ) ∼ = C

via ⊕

σ∈Hom(K,C)

σ

∪ ∪

K ↷ H

(A, Ω

) ∼ = C

via ⊕

σ∈Ξ

σ.

K ↷ C · ω

⊂ H

(A, C ) via σ ∈ Hom(K, C ): k

(ω

) = σ(k)ω

. γ: arbitrary closed path with ∫

γ

ω

̸= 0.

p

(σ, Ξ) ≡

{ π

∫

γ

ω

(σ ∈ Ξ)

∫

γ

ω

(σ / ∈ Ξ) mod Q

Example

-2 -1 0 1 2 3

-2 -1 0 1 2

γ

K = Q ( √

− 1), Hom(K, C ) = { id, ρ } , E : y

= x

− x, End(E) = Z[“ √

−1”], “ √

−1” : (x, y) 7→ (−x, iy).

⇝ ω

=

, ω

=

, Ξ = { id } :

“ √

− 1”

=

= i

, “ √

− 1”

=

= − i

. p

(id, id) = π

∫

γ

dx

y = 2π

∫

−1

√ dx x

− x

x=−√

=

π

∫

0

dt

t

(1 − t)

= π

B (

,

) = π

Γ(

) Γ(

) .

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 9 / 39

Shimura’s period symbol p

(σ, τ ) ( ≒ CM-periods) provides the transcendental pars of

critical values of L-functions of algebraic Hecke characters.

special values of Hilbert modular forms at CM-points.

the exponentials of the derivative values of partial zeta functions of totally real fields at s = 0

(Yoshida’s conjecture on “absolute CM-periods”).

Enjoys some “nice” properties:

For ι: K

∼ = K, p

(σ, τ ) = p

(σ ◦ ι, τ ◦ ι).

For K ⊂ L, σ ˜ ∈ Hom(L, C ), τ ∈ Hom(K, C ), p

(˜ σ|

, τ ) = p

(˜ σ, Inf(τ )) := ∏

˜

τ∈Hom(L,C),˜τ|K=τ

p

(˜ σ, τ ˜ ).

For the complex conjugation ρ, p

(σ, τ )p

(σ, ρ ◦ τ ) = 1.

Theorem (Rohrlich, Bannai-Otsubo, Yoshida) Γ

(

) ∼ π

∏

(b,N)=1

p

(id, σ

)

,

where [σ

: ζ

7→ ζ

] ∈ Gal( Q (ζ

)/ Q ), ⟨ α ⟩ ∈ (0, 1] denotes the fraction part of α ∈ Q .

sketch of proof.

Explicit computation of ∫

γ

x

y

dx on the N th Fermat curve F

: x

+ y

= 1:

∫

γ

x

y

dx ≒

∫

0

t

(1 − t)

dt = B (

,

) = Γ(

)Γ(

) Γ(

) .

& linear algebra using “nice” properties.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 11 / 39

Γ

(

) ∼ π

∏

(b,N)=1

p

(id, σ

)

.

proof of dim

G

≤ ϕ(N )/2.

G

:= ⟨ Γ

(

) | 1 ≤ a ≤ N − 1 ⟩

= ⟨π

p

(id, σ

) | (b, N ) = 1⟩ (δ

:= 1, −1, 0 if b ≡ 1, −1,otherwise)

pK(σ,τ)pK(σ,ρ◦τ)=1

= ⟨ π

p

(id, σ

) | (b, N ) = 1, 1 ≤ b ≤ N/2 ⟩ .

Γ

(

) ∼ π

∏

(b,N)=1

p

(id, σ

)

.

“Multiplication formula”

d

∏

Γ

(

+

) ∼ Γ

(

).

also follows: may assume that d | N (due to p

(˜ σ|

, τ ) = p

(˜ σ, Inf(τ ))).

d−1

∏

k=0

Γ

(

+

) ∼ π

p

(id, σ

)

∑

(b,N)=1

∑d−1 k=0 1

2−⟨^ab_N+^kb_d⟩

= π

∏

(b,N)=1

p

(id, σ

)

∼ Γ

(

) by multiplication formula ∑

k=0

B

(x +

) = B

(dx) for B

(x) = x −

.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 13 / 39

Γ

(

) ∼ π

∏

(b,N)=1

p

(id, σ

)

.

“Reflection formula”

Γ

(

)Γ

(1 −

) ∈ Q

follows from p

(σ, τ )p

(σ, ρ ◦ τ ) = 1.

Moreover the “reciprocity law”

˜ σ

( Γ

(⟨

⟩)Γ

(1 − ⟨

⟩) )

≡ Γ

(⟨

⟩)Γ

(1 − ⟨

⟩) mod µ

for any lift σ ˜

∈ Gal( Q / Q ) of σ

: ζ

7→ ζ

, follows from Coleman’s

formula on the absolute Frobenius action on F

: x

+ y

= 1.

For simplicity, assume that p ∤ 2N. Let B

⊂ B

be Fontaine’s period rings, Φ

↷ B

the absolute Frobenius action. Similarly to

∫

: H

(F

( C )) × H

(F

, Q ) → C ⇝ p

(σ, τ ) ∈ C

/ Q

, we can define the p-adic period symbol

∫

p

: H

(F

(C)) × H

(F

, Q) → B

⇝ p

(σ, τ ) ∈ (B

− { 0 } )

/( Q ∩ Q

)

.

(Note that F

has good reduction at p ∤ N .) By Complex Multiplication, the ratio [ ∫

γ

ω

: ∫

p,γ

ω

] depends only on Ξ, σ, i.e.,

∫

γωσ

∫γ′ω_σ^′

=

∫

p,γωσ

∫p,γ′ω_σ^′

∈ Q .

⇝ [p

(σ, τ ) : p

(σ, τ )] ∈ (C

× B

)/(µ

× µ

)(Q ∩ Q

)

.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 15 / 39

Theorem (Coleman, p ∤ 2N for simplicity)

G(

) :=

Γ

(

) · (2πi)

1 2−⟨N^a⟩ p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

∈ B

/µ

.

⇝ Γ

( ⟨

⟩ ) ≡ p

G( ⟨

⟩ )

Φ

(G( ⟨

⟩ )) mod µ

with Morita’s Γ

: Z

→ Z

, Γ

(n) := (−1)

∏

1≤k≤n−1 p∤k

k.

sketch of proof.

Explicit computation of Φ

on H

(F

) by the following lemma

& linear algebra using “nice” properties.

Lemma

Let C/ Q

be a projective smooth connected algebraic curve having good (or arboreal) reduction,

ω =

∑

a

t

, η =

∑

b

t

s.t. Φ

(ω) = αη.

Then

α = lim

k→∞

pσ

(a

) b

when lim

k→∞

n

a

= 0.

Note that

x

y

dx = x

(1 − x

)

dx x =

∑

( − 1)

(

N

n )

x

dx x .

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 17 / 39

G(

) := Γ

(

) · (2πi)

1 2−⟨N^a⟩

p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

, Γ

( ⟨

⟩ ) ≡ p

G( ⟨

⟩ )

Φ

(G(⟨

⟩)) mod µ

implies the “reciprocity law on cyclotomic units”

˜ σ

( Γ

( ⟨

⟩ )Γ

(1 − ⟨

⟩ ) )

≡ Γ

( ⟨

⟩ )Γ

(1 − ⟨

⟩ ) mod µ

since

G(⟨

⟩)G(1 − ⟨

⟩) ≡ Γ

(⟨

⟩)Γ

(1 − ⟨

⟩) mod µ

, Γ

(z)Γ

(1 − z) ∈ µ

,

Φ

= ˜ σ

on Q

∩ Q .

Coleman’s formula

G(

) := Γ

(

) · (2πi)

1 2−⟨_N^a⟩

p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

,

Γ

(⟨

⟩) ≡ p

G(⟨

⟩)

Φ

(G( ⟨

⟩ )) mod µ

also implies the “Anderson-Gross-Koblitz formula”

d−1

∏

k=0

Γ

(⟨

⟩) ≡ “Gauss sum” := ∏

(b,N)=1

π

1

h(⟨^ab_N⟩−¹₂)·σ_b⁻¹

P

mod µ

for d := order of p ∈ ( Z /N Z )

, the maximal subfield H ⊂ Q (ζ

) where p splits completely, the prime ideal P corresponding to id : H , → Q

,

P

= (π

) with h := h

.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 19 / 39

G(

) := Γ

(

) · (2πi)

1 2−⟨N^a⟩ p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

, Γ

( ⟨

⟩ ) ≡ p

G( ⟨

⟩ )

Φ

(G(⟨

⟩)) mod µ

sketch of proof of ∏

k=0

Γ

( ⟨

⟩ ) ≡ ∏

(b,N)=1

π

1

h(⟨^ab_N⟩−¹₂)·σ⁻¹_b

P

.

Note that d = deg P

. Hence, for an algebraic Hecke character χ of Q (ζ

) whose infinity type is the “reflex of the CM-type”,

d−1

∏

k=0

Γ

( ⟨

⟩ ) ≒ “Φ

↷ H

(M (χ))” ≒ χ(P

).

Morita’s Γ

: Z

→ Z

is the unique continuous function satisfying Γ

(0) = 1, Γ

(z + 1) = z

Γ

(z), z

:=

{ −z (z ∈ Z

),

− 1 (z ∈ p Z

).

Proposition (for the proof, see Koblitz, Robert, Schikhof, etc.) Γ

(z) mod µ

satisfies “multiplication formula”:

d

∏

Γ

(z +

) ≡ d

Γ

(dz) mod µ

(p ∤ d ∈ N ), where z

∈ Z

is defined by z = z

+ pz

with z

∈ { 1, 2, . . . , p } . Note that z +

∈ / Z

when z ∈ Z

.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 21 / 39

Definition

G(

) := Γ

(

) · (2πi)

1 2−⟨_N^a⟩ p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

, Γ

( ⟨

⟩ ) := p

G( ⟨

⟩ )

Φ

(G( ⟨

⟩ )) on Z

∩ (0, 1).

We can derive “multiplication formula”:

d

∏

Γ

(z +

) ≡ d

Γ

(dz) mod µ

(p ∤ d ∈ N ),

from properties of classical Γ-function or period symbols, independently of

Coleman’s formula Γ

(z) ≡ Γ

(z).

G(

) := Γ

(

) · (2πi)

1 2−⟨_N^a⟩ p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

, Γ

( ⟨

⟩ ) := p

G( ⟨

⟩ )

Φ

(G( ⟨

⟩ )) on Z

∩ (0, 1).

Proof of “multiplication formula” for period symbols.

z := ⟨ px ⟩ (x ∈ Z

∩ (0, 1)) ⇒ z = px − n (0 ≤ n ≤ p − 1)

⇒ Z

∋ x =

⇒ n = p − z

(z = z

+ pz

, 1 ≤ z

≤ p) ⇒ x = z

+ 1 i.e., Γ

(z) = p

cris(G(z1+1))

(z ∈ Z

∩ (0, 1)) z ∈ (0,

) ⇒

=

∏d−1

k=0Γ1(z+^k_d)

Γ1(dz)

·

k=0Γ1((z+^k_d)1+1)

·

“p-power” · “period symbols” ≡ d

· d

≡ d

.

Note { (z +

)

+ 1 | k = 0, · · · , d − 1 } = {

+

| k = 0, · · · , d − 1 } since both sets are contained in d

Z ∩ [

,

+ 1 −

].

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 23 / 39

Definition

G(

) := Γ

(

) · (2πi)

1 2−⟨_N^a⟩

p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

, Γ

(z) := p

G(z)

Φ

(G(z

+ 1)) on z ∈ Z

∩ (0, 1).

Remark

p-adic continuity of Γ

( ⟨

⟩ ) follows from the facts that We can take γ so that ∫

γ

x

y

dx = B(

,

) (Bannai-Otsubo).

Coefficients of x

y

dx = ∑

n=0

(−1)

(

) x

are

“continuous” on

,

∈ Z

.

Then we can define Γ

(z) on Z

.

Problem

Is Γ

(z) mod µ

characterized by “multiplication formula”

d−1

∏

k=0

Γ

(z +

) ≡ d

Γ

(dz) mod µ

(p ∤ d ∈ N )?

⇝ ? ⇝ an alternative proof for Coleman’s formula without explicit calculation.

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 25 / 39

Problem

d

∏

f (z +

) ≡ f (dz) (p ∤ d) ⇒ ? ⇒ f (z) ≡ 1

∏

k=0

f (z +

) ≡ f (dz)

⇒ ∏

k=1

f (z +

) ≡ f (dz + 1) (replace z with z +

)

⇒

≡

⇒ g(z) :=

≡ g(dz) (p ∤ d ∈ N )

⇒ g(z) ≡ c

( ∃ c

depends only on n := ord

z)

Problem

d−1

∏

k=0

f (z +

) ≡ f (dz) (p ∤ d)

⇒ g(z) :=

≡ c

(n := ord

z)

⇒ ? ⇒ f (z) ≡ 1

Write z − 1 = x

+ x

p + · · · + x

p

+ · · · ∈ Z

(0 ≤ x

≤ p − 1)

⇒ f (z) ≡ lim

f (1)g(1)g(2) · · · g(x

+ x

p + · · · + x

p

)

≡ lim

f(1)α

α

· · · α

(α

:= c

c

· · · c

c

)

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 27 / 39

Problem

d−1

∏

k=0

f (z +

) ≡ f (dz) (p ∤ d)

⇒ f (1 +

∑

x

p

) ≡ f (1)

∏

α

(c

:=

, α

:= c

· · · c

c

→ 1)

⇒ ? ⇒ f (z) ≡ 1

Consider the case d = 2, z =

: f (

)f (1) ≡ f (1)

⇒ f (

) = f (1 +

) = f(1 + ∑

k=0 p−1

2

p

) ≡ f (1) ∏

k=0

α

p−1 2

k

≡ 1

⇒ f (1) ≡ ∏

k=0

α

p−1 2

k

⇒ f (1 + ∑

k=0

x

p

) ≡ ∏

k=0

α

p−1 2

k

Proposition

A continuous function f (z) on Z

satisfies

d

∏

f(z +

) ≡ f (dz) (p ∤ d)

⇔ ∃ α

satisfying f(1 +

∑

x

p

) ≡

∏

α

p−1 2

k

(0 ≤ x

≤ p − 1).

In particular, there exist infinitely many parameters.

sketch of proof.

z+z^′= 1⇒xk+x^′_k=p−1⇒f(z)f(z^′)≡∏^∞

k=0

α⁰_k≡1⇒the casez= 0:

d∏−1

k=1

f(^k_d)≡1.

Then “mathematical induction” by

d−1∏

k=0

f(z+ 1 +^k_d)≡

d−1∏

k=0

f(z+^k_d)g(z+^k_d), f(dz+d)≡f(dz)g(dz)· · ·g(dz+d−1)andg(dz+k)≡g(z+^k_d)(k= 0, . . . , d−1).

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 29 / 39

Proposition

Assume a continuous function f (z) on Z

satisfies

d−1

∏

k=0

f (z +

) ≡ f (dz) (p ∤ d) and put c

:=

. Then

1

c

≡ c

≡ · · · ⇔ f (z) ≡ c

1 2

0

.

2

c

≡ c

≡ · · · ⇔ f (z) ≡ c

1 2

0

(c

/c

)

(z = z

+ pz

, 1 ≤ z

≤ p).

Proof.

1

α

= c

· · · c

c

= c

⇒ f (1 + ∑

k=0

x

p

) ≡ ∏

k=0

α

p−1 2

k

= c

∑_∞

k=0xkp^k−^p−₂¹p^k

0

= c

1 2

0

.

2

α

= c

, α

= c

(c

/c

)

(k ≥ 1)

⇒ f (1 + ∑

k=0

x

p

) ≡ c

∑_∞

k=0xkp^k−^p−₂¹p^k

0

(c

/c

)

.

Compare

n+1

of Γ

(z), Γ

(z).

Idea:

Γperiod(pz)Γperiod(z+1)

Γ_period(pz+1)Γ_period(z)

= ? ≡

= {

1 (p | z) z (p ∤ z)

⇑

= ?

⇑ Γ

(z) := p

cris(G(z1+1))

(z ∈ Z

∩ (0, 1), z = z

+ pz

) G(

) := Γ

(

) · (2πi)

1 2−⟨N^a⟩ p

∏

(b,N)=1

p

(id, σ

)

(2πi)

∏

(b,N)=1

p

(id, σ

)

. Seems to be natural to put Γ

(z) := p

cris(G(???))

even for z ∈ p

Z

∩ (0, 1).

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 31 / 39

When p | N , J (F

) has potentially good reduction

⇝ [p

(σ, τ ) : p

(σ, τ )] ∈ ( C

× (B

Q

)

)/(µ

× µ

) Q

. for the composite ring B

Q

of B

and Q

in B

.

The action of the Weil group:

W

:= { τ ∈ Gal( Q

/ Q

) | τ |

= Frob

with deg τ ∈ Z} , τ ∈ W

⇝

{

Φ

:= Φ

⊗ τ ↷ B

Q

= B

⊗

Q

, τ ↷ Q ∩ (0, 1) by τ (ζ

) = ζ

⇒ τ (

) :=

. Define p-adic gamma function on z ∈ ( Q − Z

) ∩ (0, ∞ ) by

Γ

(z) := exp

(

ds

p-adic interpolation of ∑

k=0

(z + k)

|

) .

[p

(σ, τ ) : p

(σ, τ )] ∈ ( C

× (B

Q

)

)/(µ

× µ

) Q

. τ ∈ W

⇝ Φ

↷ (B

Q

)

/µ

, τ ↷ Q ∩ (0, 1).

Γ

(z) := exp

(

ds

p-adic interpolation of ∑

k=0

(z + k)

|

) . Theorem (Coleman, p ̸ = 2,

∈ ( Q − Z

) ∩ (0, 1))

G(

) :=

a N)(2πi)

1 2−⟨a

N⟩

p ∏

p_Q(ζN),p(id,σb)^{12−⟨abN⟩} (2πi)^{12−⟨Na⟩}∏

p_Q(ζN)(id,σ_b)^{12−⟨abN⟩}

⇒ Φ

(

)

≡

Γp(τ(_N^a))p^{e( 12−τ(Na))}

(e := ord

N ). As a result, Γ

(z) := p

G(z)

Φ

(τ

(z)) (z ∈ ( Q − Z

) ∩ (0, 1)) is “continuous” at (z, τ

(z)).

We need only the continuity!

東京理科大学 理工学部 加塩朋和 On special values ofΓ-function 2月27日15:20–16:20 33 / 39