arXiv:1502.04397v2 [math.NT] 10 Mar 2015 ojcue1 Conjecture a eoopi otnaint h hl complex whole the to continuation meromorphic a has Here ucin emtcre,CM-periods, , Fermat function, ne h ri ybl( symbol Artin the under efis ealtern bla tr ojcue Let conjecture. Stark abelian 1 rank the recall first We Introduction 1 l nnt lcsadalrmfigpae.Te h ata eafunc zeta partial the Then places. ramifying all σ S, and places infinite all fnme ed with fields number of emtcre n h eirct a ncyclotomic on law reciprocity the and curves Fermat ∗ 2010 e od n phrases. and words Key oy nvriyo Science, of University Tokyo ∈ a mle htteepnnilo h eiaie at derivatives the Rohrlic of applicati of exponential following results the the have some that also on implies We based curves. is Fermat proof cerning The values. special od,tercpoiylwgvni hsppri refinement a is paper this units. in cyclotomic ring-valued alte given on period an law the provide reciprocity on we the law paper, words, reciprocity this the In using by field. proof number cyclotomic rational of under the properties law is and reciprocity formulas a Euler’s from satisfy follows which numbers algebraic are (Asmp) G usoe l nerliel of ideals integral all over runs ahmtc ujc classification(s). subject Mathematics edfiea“eidrn-audbt ucin n ieareci a give and function” beta ring-valued “period a define We sdfie by defined is Tern bla tr conjecture) Stark abelian 1 rank (The S otisaplace a contains G tr’ ojcue yltmcuis ( units, cyclotomic conjecture, Stark’s =Gal( := K/F ζ S ∗ kashio ( ,σ s, seulto equal is ) p := ) K/F ai periods. -adic ooauKashio Tomokazu [email protected] pi 6 2018 26, April v a ⊂O hc piscmltl in completely splits which .Let ). F Abstract units 10,1M5 12,1R2 18,14H45 11S80, 11R42, 11R27, 11M35, 11M06, F eaieypiet l rmsin primes all to prime relatively , ( σ a 1 ,S hssre ovre hnRe( when converges series This . X )=1 S eafiiesto lcsof places of set finite a be , ( K/F a ) s . = ∗ σ s suethat Assume fprilzt functions zeta partial of 0 = -plane. p N ai)gmafnto,( function, gamma -adic) K/F a − nt hntebs field base the when units s eafnto.I other In function. beta . n tr’ conjecture Stark’s on. eti odtos It conditions. certain ftercpoiylaw reciprocity the of ntv adpartial) (and rnative n oea con- Coleman and h K/F ea bla extension abelian an be rct a nits on law procity and inascae to associated tion S | S F hs image whose ≥ | s containing p ) ai)beta -adic) > 2 . and 1 We choose a place w of K lying above the splitting place v and put eK to be the number of roots of unity contained in K. Then there exists an element ε, which is called a Stark unit, satisfying the following conditions:

(Alg) ε K×. ∈

(Unit) If S > 2, then ε is a v -unit. Otherwise, we can write S =: v, v′ and ε is an | | { } { } S-unit satisfying that ε ′ is a constant for all places w′ of K lying above v′. | |w σ (Rec) log ε = e ζ′ (0, σ) for all σ G. | |w − K S ∈ 1 (Abel) K ε eK /k is an abelian extension.   We note that (Asmp) implies ζS(0, σ) = 0 for all σ G. We consider the case when v is a real place in this paper. Automatically we may assume∈ that F is totally real since it is known that Conjecture 1 holds true if S contains two places splitting completely in K/F . Namely, we assume that

F is a totally real field and v,w are real places.

Then we can write the Stark unit explicitly as ε = exp( 2ζ′ (0, σ)) (assuming it exists). − S Here eK = 2 since K is not totally complex and we regard K as a subfield of R by the place w. From now on, we focus on two conditions (Alg),(Rec) of Stark’s conjecture when v is a real place. These are equivalent to the following (Alg′),(Rec′).

Conjecture 2 (A part of Stark’s conjecture). Let F be totally real, K its finite abelian extension, and G := Gal(K/F ). Assume that K = Q and that 6 .there exists an embedding K ֒ R → We regard K as a subfield of R by this embedding. Then we have

(Alg′) u (σ) := exp( 2ζ′(0, σ)) Q× for σ G. F − ∈ ∈

(Rec′) τ(u (σ)) = u (τσ) for σ G, τ G := Gal(Q/F ). F F ∈ ∈ F Here taking the minimal S := infinite places ramifying places , we drop the 0 { }∪{ } symbol S0 from partial zeta functions ζ(s, σ) := ζS0 (s, σ). In the case of F = Q, Conjecture 2 (and Conjecture 1 also) follow from Euler’s formulas and properties of cyclotomic units. We provide a sketch of the proof: When F = Q, the + 1 2π√ 1 case of K = Q(ζm) := Q(ζm + ζm− ) with ζm := exp( m− ), 3 m N is essential. Any + ≤ ∈1 a a element in G = Gal(Q(ζ ) /Q) can be expressed as [σ a : ζ + ζ− ζ + ζ− ] with m m m m m m m ± 7→ 0

2π 2 a uQ(σ )= a m a . (1) ± m Γ( )Γ( − )  m m 

2 We can derive the expression (1) from the following formulas (2): Let ζ(s,v,z) := s n∞=0(z + vn)− be the Hurwitz zeta function (z,v > 0). Then we have

P ζ(s, σ a )= ζ(s, m, a)+ ζ(s, m, m a), ± m − 1 a ζ(0, m, a)= , (2) 2 − m 1 a 2 ζ(0,m,a) exp (ζ′(0, m, a)) = Γ( m )(2π)− m− . Furthermore, we see that

a 2 a a uQ(σ a )= 2 sin( π) =2 ζm + ζm− (3) ± m m − by using Euler’s formulas:   π Γ(z)Γ(1 z)= , (4) − sin(πz) exp(z√ 1) exp( z√ 1) sin(z)= − − − − . (5) 2√ 1 − + Then Conjecture 2 in the case of F = Q, K = Q(ζm) follows immediately from (3). For any totally real field F , Shintani expressed the special values ζ(0, σ) and the derivative values ζ′(0, σ) in terms of Bernoulli polynomials and Barnes’ multiple gamma functions, with some correction terms. Shintani’s formulas are generalizations of formulas (2). (For detail, see [Shin] or [Yo, Theorem 3.3, Chapter II].) On the other hand, we do not have any generalization of Euler’s formulas (4),(5) nor cyclotomic units for general totally real fields. Therefore the author believes that it is worthwhile to provide an alternative proof for Conjecture 2 even in the known case F = Q. In fact, we obtain a weaker result (Corollaries 1,2 in 7) § u (σ) Q (σ Gal(Q(ζ )+/Q)), Q ∈ ∈ m + (6) τ(uQ(σ)) uQ(τσ) mod µ (σ Gal(Q(ζm) /Q), τ Gal(Q/Q)) ≡ ∞ ∈ ∈ without Euler’s formulas (4),(5) nor cyclotomic units. Here we denote by µ the group ∞ of roots of unity. The outline of this paper is structured as follows. We find that we can write a Stark unit over Q in terms of a product of special values of the beta function in 2. Therefore § we can reduce the problem to an algebraicity property and a reciprocity law on such products. In 3, we introduce a result of Rohrlich. It relates special values of the beta § function to periods of Fermat curves. Besides, we need some p-adic argument: We define a modified p-adic gamma function on Qp in 4. Then we introduce a result of Coleman which expresses the absolute Frobenius action§ on the Fermat curves in 5. In particular, § we rewrite it in terms of our p-adic gamma function. We note that when the Fermat has a bad reduction, the expression in Theorem 1-(ii) becomes simpler than Coleman’s original formula [Co, Theorem 3.13], although the root of unity ambiguity occurs. In 6, we define p-adic periods of Fermat curves and study their basic properties. We will state§ the main results in 7: We define a “period ring-valued beta function” and provide § a reciprocity law on its special values (Theorem 3). Moreover we derive (6) from this reciprocity law.

3 Remark 1. We will provide not only an alternative proof but also a refinement. By (3), Stark’s conjecture in the case of F = Q, v = implies the following reciprocity law on special values of the sine function: ∞ τ sin( a π) = sin τ( a )π (τ Gal(Q/Q)). (7) m ± m ∈ Here we define the action of Gal(Q/Q) on Q (0, 1] by identifying the set of roots of a a ∩ unity µ and Q (0, 1], ζm . Our main result (Theorem 3) is a refinement of this ∞ m reciprocity law (7),∩ from the↔ sine function to “the period ring-valued beta function”. Remark 2. The proof of our main results is based on some formulas by Rohrlich and Coleman. Hiroyuki Yoshida formulated a conjecture [Yo, Conjecture 3.9, Chapter III] which is a generalization of Rohrlich’s formula, from the rational number field to gen- eral totally real fields. Yoshida and the author also conjectured its p-adic analogue in [KY1],[KY2], which is a generalization of Coleman’s formula. These conjectures are one of the motivation of providing an alternative proof of Stark’s conjecture in the case of F = Q by using Rohrlich’s formula and Coleman’s formula.

2 The gamma function in terms of the beta function

The beta function B(α, β) is defined by

1 α 1 β 1 B(α, β) := t − (1 t) − dt − Z0 for 0 < α,β R and can be written in terms of the gamma function: ∈ Γ(α)Γ(β) B(α, β)= . Γ(α + β) Conversely, we can write special values Γ(α) at rational numbers α Q in terms of those ∈ of the beta function by virtue of the functional equation Γ(z +1)= zΓ(z). For example, 1 to obtain Γ( 3 ), we compute the product Γ( 1 )2Γ( 1 )2 Γ( 2 )Γ( 2 ) B( 1 , 1 )2B( 2 , 2 )= 3 3 3 3 = 3Γ( 1 )3. 3 3 3 3 2 2 4 3 Γ( 3 ) Γ( 3 ) For later use, we give an explicit formula of the product Γ(α)Γ(1 α) for α Q (0, 1) in terms of the beta function. − ∈ ∩ Lemma 1. We write α Q (0, 1) as α = a with a, m N, (a, m)=1, 0

t 2k−2 1 2t 1 2t−1 1 1 2k−1 1 1 1 2k 1 2k 1 (Γ( 2t )Γ( 2−t )) = B( 2 , 2 ) 2k−−1 B( 2k , 2k )B( 2−k , 2−k ) . Yk=2   Proof. We put γ(α) := Γ(α)Γ(1 α), β(α) := (1 2α)B(α,α)B(1 α, 1 α). Then 2 − − − − we have β(α) = γ(α) if α = 1 . First we assume that 2 ∤ m. We take f N so that γ(2α) 6 2 m ∈ 2fm 1 mod m. Then we obtain ≡ fm 1 a fm 2a fm−1 2fm−1a 2 2 2 f − l fm−1−l γ( ) γ( ) γ( ) 2 m 1 β( 2 a )2 = m m ... m = γ a − , m γ( 2a )2fm−1 γ( 4a )2fm−2 2fm a ± m l=0 m m γ( m ) Y  a since γ( m ) mod 1 depends only on a mod m by Γ(z +1) = zΓ(z). Next, assume that {± }t 2 ∤ a and write m =2 m0 with (2, m0) = 1. Then we have

t a 2 a 4 a 2t k−1 γ( ) γ( ) γ( t ) t a a 2 a 2m0 4m0 2 m0 a 2 γ( ) β( k ) = γ( ) ... = γ( ) m0 2 m0 m0 a a 2 a 2t−1 m γ( ) γ( ) γ( t−1 ) m0 2m0 2 m0 Yk=1 if a = 1. Combining these, we obtain the first formula. The case of a = 1 follows from m0 6 m0 a similar but simpler argument.

3 Periods of Fermat curves

m m Let Fm be the mth Fermat curve defined by the affine equation x +y =1(3 m N). i 1 j m ≤ ∈ We consider differentials of the second kind η i , j := x − y − dx (0 < i, j < m, i + j = m m 6 m). The following fact is well-known ([Gr, Theorem in Appendix by Rohrlich]). For all γ H (F (C), Q), we have ∈ 1 m η i j γ m , m i j Q(ζm). (8) BR ( m , m ) ∈

Moreover we can take γ0 H1(Fm(C), Q)(γ0 = mγm with γm in [Ot, Proposition 4.9]) so that ∈ i j η i j = B( , ). (9) m , m m m Zγ0 4 p-adic gamma functions

We prepare p-adic analogues of the gamma function. Morita [Mo] constructed the p-

adic gamma function Γp : Zp Zp×, which is continuous and characterized by Γp(n) = n n 1 → ( 1) k=1− , (p,k)=1 k for n N. We note that the p-adic counterpart of the formula (4) is “degenerate”:− ∈ Q Γ (z)Γ (1 z)= 1. (10) p p − ± For the proof, see [GK, Lemma 2.3].

5 We define Γ on Q Z as follows. For simplicity, assume that p is odd. We denote p p − p Iwasawa’s p-adic log function by logp. We define the p-adic exponential function expp(z) zn ∞ on pZp by the usual power series n=0 n! . In order to define expp(z) for z Qp, we 1 ∈ choose values expp( e ) × = z Qp z p =1 for 0 e Z so that p ∈ OQp { P∈ || | } ≤ ∈ p 1 1 expp( pe+1 ) = expp( pe ). (11)   n e+1 For any z Qp, we can write z = pe + z0 with n, e Z, 0 e, 0 n

exp (z1 + z2) = exp (z1) exp (z2) (z1, z2 Qp), p p p ∈ (12) exp (log (z)) = z∗ (z Q×). p p ∈ p

ordp z 1 ordp z Here z∗ = zω(zp− )− p− is defined by the usual decomposition:

Z Qp× = µp 1 p (1 + pZp), − ordpz × ordp z × z = ω(zp− ) p z∗ × ×

with µp 1 the set of (p 1)st roots of unity, ω the Teichm¨uller character. We shall give − − ni a proof of (12): Write zi = pei + zi,0 (i =1, 2) in a manner similar to the above. We may assume that e e . Then we have 1 ≤ 2 e −e n1p 2 1 +n2 1 n1 n2 expp( pe2 ) expp(z1,0 + z2,0) ( pe1 + pe2

Lemma 2. There exists a continuous function Γp : Qp Zp × satisfying − → OQp

2z 1 1 Γ (z +1) = z∗Γ (z), Γ (2z)=2 − 2 Γ (z)Γ (z + ) (z Q Z ). (13) p p p p p 2 ∈ p − p 2z 1 1 Here we put 2 − 2 := exp ((2z ) log 2). Moreover, such a function Γ is unique up to p − 2 p p multiplication by a root of unity. Strictly speaking, if continuous functions Γp, Γp′ satisfy (13), then for all z Qp Zp we have Γp(z) Γp′ (z) mod µ . ∈ − ≡ ∞

6 Proof. In [Ka1], the p-adic logarithmic gamma function LΓ (a, (a )) (a Q , a Q×) p,1 1 ∈ p 1 ∈ p is defined as the derivative value at s = 0ofthe p-adic Hurwitz zeta function ζp,1(s, (a1), a) ([Ka1, (5.10)]). We omit the precise definition but we write LΓp,1(a, (a1)) in the form of [Ka1, (5.12)]. We introduce some notations and definitions: For a function f(X): Z p → Qp, we define pl 1 1 − JX (f(X)) := lim f(n) l pl →∞ n=0 X if it converges in Qp. We easily see that

n n b J ((aX + b) )= a B ( ) (0 n Z, a Q×, b Q ), X n a ≤ ∈ ∈ p ∈ p

where Bn(x) it the nth Bernoulli polynomial ([Ka1, Lemma 5.1]). For a Zp×, 0 = a1 pZ , we define the function f (X): Z Q by ∈ 6 ∈ p a,a1 p → p

(a1X + a) fa,a1 (X) := − (1 logp(a1X + a)) a1 − k ∞ k 1 ( 1) k! l l+1 k l = − (a1X + a)+ − ω(a)− (a1X + a) ( 1) − . a1 k l!(k l)! − !! Xk=1 Xl=0 − We put

LΓp,1(a, (a1)) := JX (fa,a1 (X)) k ∞ k a ( 1) k! l l a k l (14) = B1( a ) − ω(a)− a1Bl+1( a )( 1) − . − 1 − k l!(k l)! 1 − ! Xk=1 Xl=0 −

We see that for c Z× ∈ p LΓ (ca, (ca )) = LΓ (a, (a )) + B ( a ) log c (15) p,1 1 p,1 1 1 a1 p

since f (X) = (a1X+a) log c + f (X). The equation [Ka1, (5.12)] (in the case of ca,ca1 a1 p a,a1 r = 1) is stated only when a 1 p < 1, but it is also valid for all a Zp× since we have | 1 − | 1 ∈ LΓp,1(a, (a1)) = LΓp,1(ω(a)− a, (ω(a)− a1)) by (15). So the definition (14) of LΓp,1 in this paper is consistent with that in [Ka1]. The function LΓp,1(a, (a1)) is continuous on a Z×, a pZ by [Ka1, Lemma 5.4-1]. ∈ p 1 ∈ p Now we define the p-adic Γ function on Q Z . For z Z×, e N, we put p − p ∈ p ∈ z e Γp( pe ) := expp(LΓp,1(z, (p ))). (16)

It is continuous since so are expp, LΓp,1. One can derive the desired properties from the expression (14) and some properties of Bernoulli polynomials. Indeed, the well-known formulas

m 1 − n 1 n 1 k B (x +1)= B (x)+ nx − , B (mx)= m − B (x + ) (m N) n n n n m ∈ Xk=0 7 imply that for a Z×, a pZ , we have ∈ p 1 ∈ p LΓ (a + a , (a )) LΓ (a, (a )) p,1 1 1 − p,1 1 k ∞ k ( 1) k! 1 l k l = 1 − (l + 1)(ω(a)− a) ( 1) − − − k l!(k l)! − ! Xk=1 Xl=0 − ∞ k ∞ ( 1) 1 k 1 1 k 1 (17) = 1 − (ω(a)− a 1) ( ω(a)− a) (1 ω(a)− a) − = log (a), − − k − − − − p k=1 k=1 X m 1 X − LΓ (ma, (a )) = LΓ (ma + ka , (ma )) (m N, (p, m)=1). p,1 1 p,1 1 1 ∈ Xk=0

By (15) and (17), we have for z Z×, e, m N with (p, m)=1 ∈ p ∈ e e e LΓp,1(z + p , (p )) = logp z + LΓp,1(z, (p )), (18) m 1 − e LΓ (mz, (pe)) = ( z + k 1 ) log m + LΓ (z + kp , (pe)) . (19) p,1 pe m − 2 p p,1 m Xk=0   Then the assertions (13) follow from (12),(16),(18) and (19) in case of m = 2. In order to

prove the uniqueness of Γp up to µ , it suffices to show that logp of Γp is unique, since the ∞ kernel of logp is generated by µ and rational powers of p. (Note that we assumed that Γp ∞ takes values in × .) To do this, let f1, f2 be continuous functions on Qp Zp satisfying Qp O 1 − 2z 2 1 fi(z+1) = logp z+fi(z), fi(2z) = logp(2 − )+fi(z)+fi(z+ 2 ). Then h(z) := f1(z) f2(z) 1 − satisfies h(z +1) = h(z), h(2z)= h(z)+ h(z + 2 ). Since h is continuous, h(z +1) = h(z) 1 means that h(z + a) = h(z) for all a Zp. Hence h(2z) = h(z)+ h(z + 2 )=2h(z). z0 ∈ Write z = pe with z0 Zp×, e N. Since (2,p) = 1, there exists f, m N satisfying f e ∈ ∈f f e ∈ 2 = 1+ mp . Then we get 2 h(z) = h(2 z) = h(z + mp z) = h(z + mz0) = h(z). Therefore we obtain f (z) f (z)= h(z) = 0 as desired. 1 − 2 Definition 1. We fix Γ on Q Z satisfying (13) and define the p-adic gamma function p p − p Γ on Q by gathering our Γ on Q Z and Morita’s Γ on Z . We define two kinds of p p p p − p p p p-adic beta functions B( , ), B , : Q Q × by ∗ ∗ h∗ ∗i × → OQp Γ (α)Γ (β) Γ ( α )Γ ( β ) B (α, β) := p p , B α, β := p h i p h i . p Γ (α + β) ph i Γ ( α + β ) p p h i Here we denote the fractional part (0, 1] of α Q by α . ∈ ∈ h i Remark 4. The functional equations (13) are p-adic analogues of classical formulas

2z 1 2 2 Γ(z +1)= zΓ(z), Γ(2z)= − Γ(z)Γ(z + 1 ). (20) √2π 2 We note that Artin [Ar1, Ar2] showed that the classical gamma function is also charac- terized by functional equations (20) and some conditions.

8 Remark 5. By (19), we also obtain the multiplication formula for m N with (p, m)=1. Namely, for z Q Z we have ∈ ∈ p − p m 1 − k 1 z+ m 2 k Γp(mz)= m − Γp(z + m ) kY=0 k 1 z+ m 2 k 1 with m − := expp((z + m 2 ) logp m). We need some adjustments for m with p m. In particular, one can show that− for z Q | ∈ p p 1 − Γ (pz)= Γ (z + k ) (pz / Z ), p p p ∈ p k=0 Y p 1 − k Γp(pz) Γp(z + ) mod µ (pz Zp). ≡ p ∞ ∈ k=0, pz+k Z× Y ∈ p

Note that the latter formula gives a relation between Morita’s Γp on Zp and our Γp on Qp Zp. We provide a brief sketch of the proof: When pz / Zp, we can write z0− e∈ 1 k z = pe with z0 Zp×, e 2. Then Γp(pz) = expp(LΓp,1(z0, (p − ))), Γp(z + p ) = ∈e 1 e ≥ e 1 expp(LΓp,1(z0 + kp − , (p ))). Therefore we only need the formula LΓp,1(z0, (p − )) = p 1 e 1 e k−=0 LΓp,1(z0 + kp − , (p )). It follows form (14) similarly to the proof of (19). When pz Zp, it follows from [Ka1, Lemma 5.5] since LΓp,1(a, (1)) (a Zp) in [Ka1] was P ∈ p 1 ∈ defined as − × LΓ (a + k, (p)). k=0, pz+k Z p,1 ∈ p P Coleman [Co] also generalized Morita’s Γp to a continuous function on Qp. We denote it by Γcol in this paper. It is characterized by the following conditions ([Co, the sentence containing the equation (2.2)]):

Γ (z +1)= z∗Γ (z) (z Q Z ), col col ∈ p − p Γ (z)=1 (z Z[ 1 ] [0, 1)). col ∈ p ∩ For later use, we compare Coleman’s Γ and our Γ . For z Q Z , we write z = z +z col p ∈ p − p p 0 with z Z[ 1 ] (0, 1), z Z . Then we have for each ♯ = p, col p ∈ p ∩ 0 ∈ p n 1 − Γ♯(z)=Γ♯(zp + z0) = lim Γ♯(zp + n)=Γ♯(zp) lim (zp + l)∗. N n z0 N n z0 ∋ → ∋ → Yl=0

Hence by Γcol(zp) = 1, we have

Γ (z)=Γ (z)Γ (z ) (z Q Z ). (21) p col p p ∈ p − p 5 The Frobenius matrices of Fermat curves

In this section, we recall Coleman’s results in [Co]. In [Co], the semi-linear action ρcris of the (crystalline) Weil group on the de Rham cohomology of a curve with arboreal reduction

9 is introduced. We express this action on Fermat curves in terms of the comparison of ur some cohomologies for later use. Let Qp be the maximal unramified extension of Qp and W Gal(Q /Q ) the Weil group. Then for all τ W , there exists n Z such that p ⊂ p p ∈ p ∈ τ ur is the nth power of the Frobenius automorphism. We put deg τ := n. |Qp Let Jm be the of the mth Fermat curve Fm. Even if Fm dose not have good reduction at p (i.e., the case of p m), J has potentially good reduction since | m it has CM. Hence there exist a finite extension field K of Qp and a smooth model m over with the canonical isomorphism J OK 1 1 HdR(Fm, Q) Q K = Hcris( m K FK, WK) WK K. (22) ⊗ ∼ J ×O ⊗ Here we denote by , F and W the ring of integers, its residue field and the Witt ring OK K K over FK respectively. We may assume K is normal over Qp. We denote by Φcris the action 1 of the absolute Frobenius on Hcris( m K FK, WK). Then for τ Wp we may consider deg τ J ×O ∈ the action Φcris τ on the right hand side of (22). We denote by Φτ the corresponding 1 ⊗ action on HdR(Fm, Q) Q K. Our Φτ is equal to ρcris(τ) in [Co]. Therefore it is also equal 1⊗ to Φ∗(τ) in [Co] on HdR(Fm, Q). We need a few more notations in [Co]: Let i, j, m N with 0 < i, j < m, i + j = m. We put ε( i , j ) := i + j i + j . That is ∈ 6 m m h m i h m i−h m m i i + j (i + j < m) 0 (i + j < m) i + j = m m , ε( i , j )= . h m m i i + j 1 (i + j > m) m m 1 (i + j > m) ( m m − ( The element v H1 (F , Q) in [Co] is the class of m,r,s ∈ dR m

i j mη i j (i + j < m) i j ε( , ) m , m m + m m η i j = (23) m m m , m h i ((i + j m)η i , j (i + j > m) − m m i j for i, j with r = m , s = m . Coleman calculated Φ∗(τ)(vm,r,s) explicitly in [Co] as follows. We recall that we define the action of Gal(Qp/Qp) on Q (0, 1] by identifying Q (0, 1] = a a ∩ ∩ µ , ζm. ∞ m ↔ Theorem 1 ([Co, Theorems 1.7, 3.13]). By abuse of notation, we write η i j for its class m , m in H1 (F , Q) (0 < i, j < m, i + j = m). Then for τ W we can write dR m 6 ∈ p i j Φτ (η i j )= γ(τ, , )η i j m , m m m τ( m ),τ( m ) with γ(τ, i , j ) Q . Moreover the values γ(τ, i , j ) satisfy the following conditions. m m ∈ p m m (i) Assume that (p, m)=1. Then we have for τ W with deg τ =1 ∈ p 1 ε( i , j ) ε(τ( i ),τ( j )) i j ε(τ( i ),τ( j )) p m m ( 1) m m τ( )+ τ( ) m m γ(τ, i , j )= − − h m m i . m m i j ε( i , j ) i j + m m Bp τ( ), τ( ) h m m i h m m i

(ii) Assume that p > 2, p m, (p, ij(i + j)) = 1. Then we have for τ Wp with deg τ =1 | ∈ i j i j 1 Bp( m , m ) γ(τ, , ) p 2 mod µ . m m i j ∞ ≡ Bp(τ( m ), τ( m ))

10 j i j ε(τ( i ),τ( )) j τ( )+τ( ) m m j Proof. Our γ(τ, i , ) is Coleman’s h m m i β (τ( i ), τ( )) by (23), [Co, m m j i j τ m m i + ε( m , m ) h m m i Proposition 1.4]. Therefore the former assertion follows from [Co, Theorem 1.7] immedi- ately. In the latter case, [Co, Proposition 3.12, Theorem 3.13] state

1 i 1 j γ(τ, τ − ( m ), τ − ( m )) i j ε( i , j ) i j i j + m m Dτ ( )Dτ ( )Γτ ( )Γτ ( ) (24) h m m i m m m m mod µ . i j ε(τ −1( i ),τ −1( j )) i j i j ≡ τ 1( )+ τ 1( ) m m Dτ ( + )Γτ ( + ) ∞ h − m − m i h m m i h m m i

The notations are as follows: Coleman defined functions Dτ , Aτ , Γτ satisfying

f 2k−1 k f Dτ (z) (Aτ ((z/2 )p) 2 −1 mod µ , ≡ ∞ kY=1 ( 1)(2z)p −1 Γ (z + − ) τ ((2z)p)τ+(2z)p col p 2 Aτ (z)=(2− )∗ −1 , ( 1)τ ((2z)p) 1 − Γcol(τ − (zp)+ 2 ) 1 τ−p τ−1 −1 Γcol(τ − (z)) 1 2(p−1) 2 τ (z)τ z Γτ (z)= pΓp( 2 )( p) (2z) κ(z) − − − Γcol(z)

for z Q× (0, 1) with ordp z < 0 ([Co, p179, p182]). Here f is the order of 2 in ∈1 ∩ z 1 (Z /z− Z )×, κ(z) := and z is the unique element in Z[ ] (0, 1] satisfying z p p z∗ p p ∩ ≡ r 1 r zp mod Zp. We define x m with x Qp,r Z[Wp], m N as (x m ) by taking an mth root 1 ∈ z ∈ ∈ x m Q of x. In order to define ( 1) p , we regard z as its image under the composite ∈ p − p map Z[ 1 ] Z Z /2Z = 0, 1 . We put p ⊂ 2 → 2 2 { } f 2k−1 k f D(z) := A((z/2 )p) 2 −1 mod µ , ∞ k=1 Y (2z)p (2z)p ( 1) A(z) := 2 Γcol(zp + − ) mod µ . 2 ∞ Then by (21) we have

( 1)(2z)p Γp(zp + − ) A(z) 2(2z)p 2 mod µ . ≡ Γp(zp) ∞

1 2(zp) 2 1 Since Γp(2(zp))=2 − Γp(zp)Γp(zp + 2 ) by (13), we get

( 1)(2z)p 1 Γ (z + − )Γ (2(z )) +(2z)p 2(zp) p p 2 p p A(z) 2 2 − mod µ . 2 1 ∞ ≡ Γp(zp) Γp(zp + 2 )

If 2(z )=(2z) then ( 1)(2z)p = 1 so we have p p −

1 Γp((2z)p) 2 A(z) 2 2 mod µ . (25) ≡ Γp(zp) ∞

11 If 2(z )=(2z) +1 then ( 1)(2z)p = 1 so we get p p − − 1 1 1 Γp(zp 2 )Γp((2z)p + 1) A(z) 2 2 − − mod µ . 2 1 ∞ ≡ Γp(zp) Γp(zp + 2 )

Hence the formula (25) is also valid in this case since Γp((2z)p +1) = ((2z)p)∗Γp((2z)p)= 1 1 1 f (2(z ) 1)∗Γ ((2z) ), Γ (z + )=(z )∗Γ (z ). By (21),(25) and (z/2 ) = z , p − p p p p 2 p − 2 p p − 2 p p we see that

f−1 1 2 2 f f f 1 f 1 Γp(zp) 2 −1 1 Γp((z/2)p) 2 −1 1 Γp((z/2 − )p) 2 −1 2 2 2 D(z) 2 2 2 2 2 ... 2 f 2 ≡ Γp((z/2)p) Γp((z/2 )p) Γp((z/2 )p)  1 1      2 2 2 2 Γ (z) col mod µ . ≡ Γp(zp) ≡ Γp(z) ∞

D(z) τ−p −1 τ−1 1 2(p−1) 2 2 We easily see that Dτ (z) D(τ −1(z)) , Γp( 2 ) 1 (by (10)), ( p) p , (2z) 1, −1 ≡ ≡ − ≡ ≡ τ −1(z)τ z pτ (z)ordp z κ(z) − zordp z mod µ . Therefore we can write ≡ p ∞ −1 τ (z)ordp z 1 1 p Γp(τ − (z)) 2 Dτ (z)Γτ (z) p zord z mod µ . ≡ p p Γp(z) ∞ By substituting this into (24), we get the desired result

1 i 1 j i j 1 i 1 j 1 Γp(τ − ( m )) Γp(τ − ( m )) Γp( m + m ) γ(τ, τ − ( ), τ − ( )) p 2 mod µ m m i j 1 i 1 j ∞ ≡ Γp( m ) Γp( m ) Γp(τ − ( m )+ τ − ( m ))

j j j j i j j i i i i ε( m , m ) i since ordp m = ordp m = ordp m + m , Γp( m + m )=( m + m ∗) Γp( m + m ) j i j ε( i , ) h i h i h i ≡ + m m j h m m i i i j i j Γp( m + m ). pε( m , m )ordp h m + m i h i 6 p-adic periods of Fermat curves

We rewrite Theorem 1 by using p-adic periods of Fermat curves. Let Bcris, BdR be the p-adic period rings introduced by Fontaine. We may consider Qp, Bcris are subrings of B . Then for any subfield K of Q , the composite ring B K B is well-defined. dR p cris ⊂ dR We denote by Φcris the action of the absolute Frobenius on Bcris and define actions Φτ (τ W ) on the ring B K by ∈ p cris Φ := Φdeg τ τ. (26) τ cris ⊗ Let K, be as in (22): K is a finite normal extension of Q , is a smooth model Jm p Jm ,over K of the Jacobian variety Jm of Fm. For simplicity, we fix embeddings Q ֒ C → .Q ֒ OQ . The following results are well-known → p There exists a canonical isomorphism between the singular cohomology group and • the p-adic ´etale cohomology group:

H1 (F (C), Q) Q = H1 (F Q, Q ). (27) B m ⊗Q p ∼ p,´et m ×Q p

12 There exists a canonical isomorphism between the p-adic ´etale cohomology group • and the de Rham cohomology group: H1 (F Q, Q ) B K = H1 (F ,K) B K, (28) p,´et m ×Q p ⊗Qp cris ∼ dR m ⊗K cris which is compatible with the action of the Weil group Wp. The element τ Wp deg τ ∈ acts on the left hand side by 1 Φτ , on the right hand side by Φcris Φτ . It follows from the canonical isomorphism⊗ ⊗ 1 1 Hp,´et( m K Q, Qp) Qp Bcris = Hcris( m K FK, WK) WK Bcris J ×O ⊗ ∼ J ×O ⊗ 1 Φ Φ Φ ⊗ cris ↔ cris ⊗ cris 1 1 1 by identifying HdR(Fm,K) = Hcris( m K FK, WK) WK K, Hp,´et( m K Q, Qp)= J ×O ⊗ J ×O H1 (F Q, Q ). We note that the isomorphism (28) is a restriction of the p,´et m ×Q p following canonical isomorphism H1 (F Q, Q ) B = H1 (F , Q) B . (29) p,´et m ×Q p ⊗Qp dR ∼ dR m ⊗Q dR The singular cohomology group is the dual of singular homology group. Namely, we • have the non-degenerate pairing: H (F (C), Q) H1 (F (C), Q) Q. (30) 1 m × B m → Combining (27), (28) and (30), we get a period ring valued pairing H (F (C), Q) H1 (F , Q) B K. (31) 1 m × dR m → cris We denote by p,γ η the image of (γ, η) under the map (31). This is a p-adic counterpart of the usual period η, H (F (C), Q) H1 (F , Q) C. R γ 1 m × dR m → Theorem 2. Let p be a prime, i, j, m integers satisfying 0 < i, j < m, i + j = m. R 6 (i) Assume that (p, m)=1. Then we have

k k k−1 k−1 deg τ ε( p i , p j ) 1 ε( p i , p j ) ( 1) h m i h m i p − h m i h m i j Φτ η i = − k k m , m p i p j p,γ Bp , ! Z  kY=1 h m m i j i j i ε(τ( m ),τ( m )) τ( m )+ τ( m ) j h i i j η i i j ε( , ) τ( m ),τ( m ) × + m m p,γ h m m i Z for γ H (F (C), Q), τ W with deg τ 0. ∈ 1 m ∈ p ≥ (ii) Assume that p> 2, p m, (p, ij(i + j))=1. Then we have |

η i j deg τ η i j p,γ m , m p,γ τ( m ),τ( m ) Φτ p 2 mod µ i j i j ∞ BR p( m , m )! ≡ BRp(τ( m ), τ( m )) for γ H (F (C), Q), τ W . ∈ 1 m ∈ p Proof. Since the isomorphism (28) is compatible with the action of Wp and Wp acts 1 1 trivially on HB(Fm(C), Q) Q Qp ∼= Hp,´et(Fm QQ, Qp), the assertion follows from Theorem ⊗ i pdeg τ i× i 1 immediately. (Note that τ( m ) = m in the former case, and that τ(Γp( m )) i i pN h i ≡ Γp( ) mod µ since Γp( ) Qp for large enough N in the latter case.) m ∞ m ∈

13 7 A reciprocity law on a BcrisQp-valued beta function We will formulate a “reciprocity law on a period ring-valued beta function” which is a refinement of (7). We first define BcrisQp-valued beta function by

i j B( m , m ) i j ε( i , j ) + m m η i j ((p, m)=1), m m p,γ m , m  γ η i , j h i B( i , j ) := m m m m  i j j R B( , ) p,γ η i ,  R m m m m  i j (p 2, p m, (p, ij(i + j)) = 1). η i j Bp( , ) ≥ | γ m , m R m m   R for i, j N withi, j < m, i + j = m and γ H1(Fm(C), Q) with γ η i , j = 0. ∈ 6 ∈ m m 6 Lemma 3. The value B( i , j ) B Q does not depend on the choiceR of m, γ whenever m m ∈ cris p η i , j =0. γ m m 6 Proof.R Put G := Z/mZ Z/mZ. Then H (F (C), Q) is the rank one Q[G ]-module in m ⊕ 1 m m the sense of [Ot, Proposition 4.9, (i)]. Let γm H1(Fm(C), Q) be the generator. Then we can write any element in H (F (C), Q) as ργ∈ with ρ Q[G ]. Hence the dependences 1 m m ∈ m of η i j , η i j on γ are canceled out since η i j is a simultaneous eigenvector γ m , m p,γ m , m m , m for any g Gm. In order to see the dependence on m, we consider the canonical map R ∈ R n n f : Fnm Fm, (x, y) (x ,y ) (2 n N). Then we have f ∗η i , j = nη ni , nj , so the → 7→ ≤ ∈ m m nm nm assertion follows.

i j i j By (9), we can take γ0 so that B( , ) = η i j . Then we have B( , ) = m m γ0 m , m m m R η j j i j p,γ0 i , i ε( m , m ) m m + η i j (resp. i j ) if (p, m)R = 1 (resp. p m). Therefore Theorem m m p,γ0 , B ( , ) h i m m p m m | 2 implies the followingR reciprocity law on our beta function. Theorem 3. Let p be a prime, i, j, m integers satisfying 0 < i, j < m, i + j = m, and τ W . 6 ∈ p (i) Assume that (p, m)=1, deg τ 0. Then we have ≥ k k k−1 k−1 deg τ ε( p i , p j ) 1 ε( p i , p j ) ( 1) m m p m m Φ (B( i , j )) = h i h i − h i h i B(τ( i ), τ( j )). τ m m − pki pkj m m Bp , ! Yk=1 h m m i (ii) Assume that p> 2, p m, (p, ij(i + j))=1. Then we have | i j deg τ i j Φτ (B( , )) p 2 B(τ( ), τ( )) mod µ . m m ≡ m m ∞ i j m i m j We prove the key formula (32) which states that the product B( m , m )B( m− , m− ) of i j m i m j our beta function is essentially equal to the product B( m , m )B( m− , m− ) of the classical beta function. To do this, we prepare some Lemmas. Lemma 4. (i) Assume that (p, m)=1. Then we have

i j m i m j B , B − , − = 1. ph m m i ph m m i ±

14 (ii) Assume p> 2, p m, (p, ij(i + j))=1. Then we have | i j m i m j m Bp( m , m )Bp( m− , m− ) ( m i j )∗ mod µ . ≡ − − ∞ Proof. The first assertion follows from (10). In the case of i , j Q Z , we have m m ∈ p − p i j m i m j i j m i m j Γp( m )Γp( m )Γp( m− )Γp( m− ) m Bp( m , m )Bp( m− , m− )= i+j m i j ( m i j )∗. Γp( m )Γp( −m− ) − − by (13). Therefore it suffices to show that Γp(α)Γp(1 α) 1 mod µ for α Qp Zp, − ≡ ∞ ∈ − which is a generalization of (10). It reduces to the formula

LΓ (a, (a )) + LΓ (a a, (a ))=0 p 1 p 1 − 1 for a Zp×, a1 pZp by (16). This formula follows from a property of Bernoulli polyno- mials:∈B (1 x∈)=( 1)nB (x) (0 n Z). In fact, by (14) we can write n − − n ≤ ∈ LΓ (a, (a )) + LΓ (a a, (a )) p,1 1 p 1 − 1 a a = B1( )+ B1(1 ) − a1 − a1 k ∞ k  ( 1) k! l l a l a k l − a ω(a)− Bl+1( )+(ω(a1 a))− Bl+1(1 ) ( 1) − . − k l!(k l)! 1 a1 − − a1 − Xk=1 Xl=0 −   l a l a Since ω(a a)= ω(a), we have ω(a)− B ( )+(ω(a a))− B (1 ) = 0. Then 1 l+1 a1 1 l+1 a1 the assertion− is clear.− − −

Rp,γ η i j Rp,γ η m−i m−j 1 m , m 2 m , m Lemma 5. The ratio does not depend on γ1,γ2, i, j, m whenever Rγ η i j Rγ η m−i m−j 1 m , m 2 m , m γ η i , j , γ η m−i , m−j =0. To be precise, there exists a constant (2πi)p BdR× , which is a 1 m m 2 m m 6 ∈ p-adic counterpart of 2πi, satisfying R R

γ η i , j γ η m−i , m−j p,γ η i , j p,γ η m−i , m−j 1 m m 2 m m = 1 m m 2 m m Q R 2Rπi R (2Rπi)p ∈

for all i, j, m with 0 < i, j < m, i + j = m and for all γ1,γ2 H1(Fm(C), Q). Here we 6 1 ∈ define (2πi)p as follows. Taking the P (= F1 : x + y = 1) and the basis c H (P1, Q), c H2 (P1, Q), we put b ∈ 2 dr ∈ dR

(2πi) cdr p,cb (2πi)p := , cdr cbR

where : H (P1, Q) H2 (P1, Q) C, : HR (P1, Q) H2 (P1, Q) B are given 2 × dR → p 2 × dR → cris similarly to (31). In particular, we see that (2πi)p Bcris and that Φcris((2πi)p)= p(2πi)p R2 1 1 R ∈ since H ´et(P Q, Q ) = H ´et(G Q, Q ) = the Lefschetz motive Q ( 1). p, ×Q p ∼ p, m ×Q p ∼ p −

15 Proof. Let γ1,γ2 H1(Fm(C), Q) satisfy γ η i , j , γ η m−i , m−j = 0. First we note that ∈ 1 m m 2 m m 6 we can write via the de Rham isomorphism R R H1 (F (C), Q(ζ )) C = H1 (F , Q) C B m m ⊗Q(ζm) ∼ dR m ⊗Q γ1∗ γ η i , j η i , j 1 ⊗ 1 m m ↔ m m ⊗ 1 with an element γ∗ H (F (CR), Q(ζ )). (By the , we can take a 1 ∈ B m m tentative element γ1∗ so that γ1∗ c η i j 1 with c C×. Then we normalize it as m , m ∗ ⊗ ↔ ⊗ ∈ γ1 ∗ by using the paring (γ1,γ1∗) of (30).) We take γ2∗ in the same manner. Since the de (γ1,γ1 ) Rham isomorphism is a ring isomorphism under cup products, we have H2 (F (C), Q(ζ )) C = H2 (F , Q) C B m m ⊗Q(ζm) ∼ dR m ⊗Q (γ1∗ γ2∗) ( γ η i , j γ η m−i , m−j ) (η i , j η m−i , m−j ) 1. ∪ ⊗ 1 m m 2 m m ↔ m m ∪ m m ⊗ Similarly we have R R H2 (F (C), Q(ζ )) B = H2 (F , Q) B B m m ⊗Q(ζm) dR ∼ dR m ⊗Q dR (γ1∗ γ2∗) ( η i , j η m−i , m−j ) (η i , j η m−i , m−j ) 1, ∪ ⊗ p,γ1 m m p,γ2 m m ↔ m m ∪ m m ⊗ 2 since (27), (29) are alsoR ring isomorphisms.R We note that dimQ HdR(Fm, Q) = 1 and that m m 2 1 the canonical map ιm : Fm F1, (x, y) (x ,y ) induces isomorphisms H (P , Q) = 2 → 7→ ∗ ∼ H (Fm, Q)( = B, dR). Hence it suffices to show that the cup product η i , j η m−i , m−j = ∗ ∗ m m ∪ m m 6 ˇ 0. In the following of this proof, we compute η i , j η m−i , m−j directly by using Cech m m ∪ m m (m) (m) m cohomology. We take the covering U1 , U2 of the projective Fermat curve Fm : X + m m (m) Y = Z so that they corresponds to Z = 0, Y = 0 respectively. Namely, U1 (resp. (m) m m6 6 m U2 ) is the affine curve defined by x + y = 1 with x = X/Z, y = Y/Z (resp. x′ +1= m z′ with x′ = X/Y , z′ = Z/Y ). Then we can write

1 (m) 0 (m) (m) 1 (ω1,ω2, f) ωi Ω (Ui ), f Ω (U1 U2 ),ω1 ω2 = df HdR(Fm, Q)= { | ∈ ∈ ∩ − }, (df , df , f f ) f Ω0(U (m)) { 1 2 1 − 2 | i ∈ i } 1 (m) (m) 2 Ω (U1 U2 ) HdR(Fm, Q)= ∩ , ω ω df ω Ω1(U (m)), f Ω0(U (m) U (m)) { 1 − 2 − | i ∈ i ∈ 1 ∩ 2 } (ω ,ω , f) (ω′ ,ω′ , f ′)= f ′ω + fω′ , 1 2 ∪ 1 2 − 1 2 i 1 j m i m m i j 1 (x − y − dx, x′ − z′ − − − dz′, 0) if i + j < m, η i j = m , m i 1 j m j m i m 2m i j 1 1 i j m ((x − y − dx, i+−j m x′ − z′ − − − dz′, i+j m x y − ) if i + j > m. − − Therefore we get for i, j with i + j < m xmy−m dx 1 dx η i , j η m−i , m−j = i+j m x = ιm∗ ( m(i+j m) 1 x ). m m ∪ m m − − − 1 dx We easily see that m(i+j m) 1 x is non-zero in − − 1 (1) (1) 2 Ω (U1 U2 ) HdR(F1, Q)= ∩ 1 (1) 0 (1) (1) ω1 ω2 df ωi Ω (Ui ), f Ω (U1 U2 ) { − − | ∈1 ∈ ∩ } Q[x, x 1 ]dx − = 1 dx 1 Q[x]dx + Q[ x 1 ] (x 1)2 + d(Q[x, x 1 ]) − − − 16 dx since Resx=1 1 x = 0. Hence the assertion is clear. − 6

Rγ η i j Rγ η m−i m−j By Lemma 1 and (8), the monomial relations 1 m , m 2 m , m Q in Lemma 5 2πi ∈ imply the algebraicity of Stark units over the rational number field. Namely, we have the following Corollary.

Corollary 1. (Alg′) for F = Q holds true. That is, we have

u (σ) Q (σ Gal(Q(ζ )+/Q)). Q ∈ ∈ m j i j m j m−i m−j j i ε( m , m ) m i ε( m , m ) i By Lemma 4-(ii),Lemma 5 and m + m m− + m− = ( m + m 1), we can write h i h i ± −

i j m i m j i j m i m j B( , )B( − , − ) B( , )B( − , − ) m m m m i j ∗ m m m m 1 m m mod µ (32) (2πi)p ≡ − − 2πi ∞  in both cases: (p, m)=1or p> 2, p m, (p, ij(i + j)) = 1. Therefore Lemma 5 and (8) i j m i m j | 1 i j m i m j 1 imply not only B( , )B( − , − )π− Q, but also B( , )B( − , − )(2πi)− Q. m m m m ∈ m m m m p ∈ Finally in this section, we derive (Rec′) modµ in the case of F = Q from Theorem i j m i m j ∞ 3. By Lemma 4-(i) and ε( m , m )+ ε( m− , m− ) = 1, we see that Theorem 3 implies

i j m i m j deg τ i j m i m j Φτ (B( , )B( − , − )) p B(τ( ), τ( ))B(τ( − ), τ( − )) mod µ (τ Wp) m m m m ≡ m m m m ∞ ∈ in both cases. Besides, we have (32), Φcris((2πi)p) = p(2πi)p, and Φτ is τ-semi linear, so we get

i j m i m j B( , )B( − , − ) i j ∗ m m m m τ 1 m m − − π !  i j m i m j i j B(τ( m ), τ( m ))B(τ( m− ), τ( m− )) 1 τ( ) τ( ) ∗ mod µ (τ Wp). ≡ − m − m π ∞ ∈  Hence we obtain the desired formula τ(uQ(σ)) uQ(τσ) mod µ for τ Wp by (1) and ∞ Lemma 1. Since we can vary p and the embedding≡ Q ֒ Q , the same∈ formula is valid → p for any τ Gal(Q/Q). Namely, ∈ Corollary 2. (Rec′) for F = Q holds true up to multiplication by a root of unity. That is, we have

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