© Hindawi Publishing Corp. ON A TRIVIAL ZERO PROBLEM

One trivial zero phenomenon for p-adic analytic function is considered. We then prove that the first derivative of this function is essentially the Kummer class associated with p.


Introduction.
In this paper we always fix an odd prime p > 2. For n ≥ 1, fix a p n th primitive root of unity ζ p n such that For β ∈ ᐁ, we will define a 1-admissible distribution Section 3).Consider the integral then we have ψ k (β) = (1 − p k−1 ) • Zp x k µ β , so it will have a trivial zero at k = 1.Since 1 − p k−1 is not an analytic function of k, hence we cannot take the derivative directly.But ψ k is an analytic function of k, so the derivative exists.This phenomenon in which the zero is forced by Euler factor is called trivial zero problem.Ferrero and Greenberg [4] considered the trivial zero problem for the first time in 1978 and found that the derivative has deep arithmetic meaning.The behavior of the derivative of some Kubota-Leopoldt p-adic L-function with trivial zero has a deep relation with some arithmetic Iwasawa module (see [6]).The second such trivial zero phenomenon was found by Mazur et al. in [8], and then they conjectured that the derivative has a relation with Linvariant.This conjecture was proved by Greenberg and Stevens in 1993 (see [7]).The function ψ k is very close to Coates-Wiles kth derivative (see Section 7); actually, it only differs by the factor (1−p k−1 ), and was called Coates-Wiles homomorphism in de Shalit [3].The question to find the derivative at k = 1 of ψ k was proposed by Glenn Stevens in 1997.Simultaneously, we also tried to understand how the Bloch-Kato exponential map exp Qp (1) can miss the Kummer class γ p .Glenn Stevens predicted that the derivative of ψ k at 1 will give the Kummer class γ p .We will prove this in this paper.Let C p denote the completion of Qp .For a field K ⊂ C p , let O K denote the ring of integers.Choose Iwasawa's log : C × p → C p such that log(p) = 0.In Section 2, we will review Fontaine's rings briefly and describe Bloch-Kato exponential map.In Section 3, we will define distributions and explain cohomology groups as Iwasawa module.In Section 4, we will introduce algebraic Fourier transformation and use Coleman power series to give some special distributions.In Section 5, we will review Perrin-Riou and Colmez theorems.In Section 6, we will show that Iwasawa's explicit reciprocity law is actually a special case of Perrin-Riou's theorem.In Section 7, we use the theory we developed so far to prove our theorem.
2. Fontaine's rings and Bloch-Kato exponential map.Let Ō = O Cp /pO Cp .Let denote the projective limit of the diagram where the transition maps are given by x → x p .The ring is a perfect ring with characteristic p > 0 (see [5]).For x ∈ , x = (x n ) n∈N satisfies x n ∈ Ō, and x p n+1 = x n .For each n, choose xn ∈ O Cp to be a representative of x n .Then one can show that for each m, xp n n+m exists and the limit x (m) does not depend on the choices of the representatives.Hence, x gives rise to a sequence (x (m) ) m∈N in O Cp such that (x (m+1) ) p = x (m) .On the other hand, if we have a sequence (x (m) ) m∈N in O Cp such that (x (m+1) ) p = x (m) , then ( x(m) ) m∈N is an element in .Hence, is in one-to-one correspondence with the set x (m)  m∈N | ∀m ∈ N, x (m) ∈ O Cp , x (m+1) p = x (m) . (2.2) Define a function v : → Q ∪{∞} by v x (m)  m∈N := v x (0) , ( where v is the valuation of C p such that v(p) = 1.The ring is complete with respect to v .Let W () denote the Witt vector ring of .Recall that the underlying set of W () is the set N = {(x 0 ,x 1 ,...) | x i ∈ }.The ring structure is given in terms of Witt polynomials (see [10]).Since Ō is an Fp -algebra, W () is a W ( Fp )-algebra.For x ∈ , let denote the Teichmüller representative of x.For (x 0 ,x 1 ,...,x n ,...) ∈ W (), we have the identity be defined by Then it is easy to see that θ is a Z p -homomorphism and it is surjective.The Frobenius on induces a continuous Frobenius map on W () with respect to the product topology, we denote it by ϕ, which sends (x 0 ,x 1 ,...,x n ,...) to (x p 0 ,x p 1 ,...,x p n ,...).The map ϕ is an isomorphism, semilinear over W ( Fp ).The ring W () can also be endowed with padic topology and I-adic . The kernel of θ is a principal ideal of W (), which is generated by u [5].
We will use B + dR , B dR , A crys ,B + crys , B crys , A max , and B max from Colmez [2].
Lemma 2.1.The following sequences are exact: where ϕ is the Frobenius of B dR which is induced by the one from .
For a continuous G Qp -representation V , finite-dimensional Q p -vector space, define vector space, with a Frobenius action (acts on V trivially) [5].The operator D dR has a filtration given by Fil The dimensions have the following relation: (2.10) , then V is called a crystalline representation.Note that a crystalline representation must be a de Rham representation.In the following, all representations are assumed to be de Rham representations.Similarly, we can also define For a de Rham representation V , taking tensor product with the exact sequence (2.8), we have the following exact sequence: taking the Galois cohomology, we have a map (2.12) Then the Bloch-Kato exponential map is defined as the composition The kernel of this map is Fil 0 D dR (V ) + D crys (V ) ϕ=1 , and the image is is an isomorphism.In some sense, γ p and γ 1+p should have the same positions in H 1 (Q p , Q p (1)).Note that for k = 1, the left-hand side has dimension 1 and the righthand side has dimension 2, so the image is a one-dimensional vector space, and γ p is not in the image.In this paper, we will show that the "derivative of Bloch-Kato map" is essentially γ p .To be a little bit more precise, we need the following definitions.
Let ᐄ = Hom cont (Z × p , C × p ) which is identical to B(µ p−1 , 1) and there is an obvious inclusion Z ⊂ ᐄ.Definition 2.3.Given ⊂ ᐄ, a rigid analytic subspace over Q p , an analytic family of Galois representations over is a pair (V , ρ), where (1) V is a de Rham representation of G Qp , (2) ρ : × G Qp → Gl Qp (V ) is continuous in σ and is analytic in k.Definition 2.4.Let (V , ρ) over be a family of Galois representations of G Qp and let V k denote the Galois representation of G Qp such that the underlying space is V and the action is given by (2.20) ) is said to be an analytic family if there is a cocycle representation σ → ξ k (σ ) such that for all σ ∈ G Qp , ξ k (σ ) is an analytic function of k.Now, we can go back to answer the question on γ p .In Section 7, we will show that In other words, γ p appears in the first coefficient of the "Taylor expansion" of Bloch-Kato exponential map.Let LA = {locally analytic compactly supported functions in is locally analytic and compact supported such that thereexists N ∈ N, x N f ∈ LA}.LA and LA have Morita topology.
We let A n (X) denote the Q p -affinoid algebra of B[X, p −n ].In particular, A n (X) is a Banach algebra under the Gauss norm.For a p-adic Banach space A, let Ᏸ cont (Q p ,A) := {µ : LA → A|µ is linear and continuous with respect to Morita topology}.Note that µ is continuous if and only if it is continuous when restricted on each to be the norm of the continuous linear function µ : A) denote all tempered distributions with values in A. From the above remark we see that µ is r -bounded if and only if is r -bounded.A distribution with order r is also called an r -admissible distribution.
If V is a crystalline representation of G Qp , we have a twist map which sends µ to (−tx)µ.

Proof. Obviously, we have T w(δ
For the surjectivity, given If A is a Dieudonne module, then ϕ can act on it, hence both ϕ and ϕ Ᏸ can act on Ᏸ I alg (Q p ,A).Then we define Φ = ϕ Ᏸ ⊗ ϕ.
Lemma 3.5.The twist map T w induces a map The statement about cokernel follows immediately.
Define Ᏸtemp Φ=1 , where the transition maps are given by the above twist map.Lemma 3.6.For µ ∈ Ᏸ cont (Z × p ,A), µ has order r if and only if xµ has order r .
Proof.Assume that µ has order r with r ∈ R, then there is a constant C > 0 such that for all j ≥ 0, If µ has order r , by using the expansion a+p n Zp (x−a) r (µ/x) = a+p n Zp (x −a) r (1/(a+ , this proves the lemma. For µ ∈ Ᏸ cont (Z p , C p ), define the Amice transformation ] is said to be of order r if p [nr ] a n is r -bounded.Lemma 3.8.A distribution µ ∈ Ᏸ cont (Z p , C p ) has order r if and only if Ꮽ µ (T ) has order r .Proof.See [1].

Fourier transformation and Coleman power series. Recall that we fixed ζ p n
which is a p n th root of unity.Let Obviously, this is well defined, and we get an element where From property (ii) of ε(x), if y is outside of p −m Z p , then this sum is zero, hence Ᏺ alg (f ) is well defined and compactly supported.On the other hand, since f is compactly supported, we can assume that f is supported on p −m Z p for some m.Since ε(p −m y) is locally constant, this implies that Ᏺ alg (f ) is locally constant.Extend the above definition to test function Proposition 4.1.The Fourier transformation Ᏺ alg enjoys the following properties: Proof.The properties follow easily from the definitions.For h ∈ Z, define the twist for Ᏺ alg as where f ∈ LP [1−h,+∞) , then we have Now, we define the algebraic Fourier transformation on distributions as follows.For ] be a Frobenius corresponding to π , so f π (x) ≡ πx(mod deg 2) and f π (x) ≡ x p (mod p).Let F be the one-dimensional Lubin-Tate formal group over Z p corresponding to f π and let [+] denote the formal addition.
, and K ∞ = ∪ n≥1 K n .Hence, K ∞ /Q p is a totally ramified extension with Galois group Z × p .We call this tower the Lubin-Tate tower corresponding to the formal group , where the map is with respect to the norm map.Assume that β ∈ ᐁ, then Coleman's theorem tells us that there is a unique (Coleman) power series The property (ii) of the Coleman power series implies that logg β (T ) has integral coefficients.Define an algebraic distribution Φ=1 and has the following Galois property: Proof.It is easy to see that By property (ii), we see that taking logarithm, and using the definition for µ β , we have Hence, To prove the second property, since by comparing the values at From this property, we see that To show that µ β is 1-admissible, by definition and Lemma 3.3, we only need to show that 0))/g β (0)), hence, bounded.

Perrin-Riou and Colmez theorems. Let
For a crystalline representation V , that is, a finite-dimensional Q p -vector space such that G Qp has a continuous action on it and V is crystalline, let D(V ) := D crys (V ) denote the Dieudonne module of V .Then from Colmez [2], T n can be extended to B G K∞ dR ⊗ D(V ).Then it is known that D(V ) has a Frobenius endomorphism and a filtration which we denote by Fil i D(V ).This filtration is decreasing, separated, and exhausted.That is, For I ⊂ Z, we have the algebraic distribution Ᏸ I alg (Q p , D(V )) from Section 3.For h ∈ Z, we defined the algebraic Fourier transformation then Perrin-Riou and Colmez proved that the image is fixed by G Qp , and the Perrin-Riou exponential map Exp h,V is defined as the composition of the following maps: where the last map is the connecting map of the following exact sequence: where the projective limit map is given by µ → (−tx)µ.Then Perrin-Riou [9] first proved the following theorem.
From Section 4, we know that for µ β ∈ Ᏸtemp (Q p ,D(Q p ( 1))) Φ=1 , we could have that Ᏺ alg (µ β ) is not tempered, so the miracle of this theorem is that Exp h,V sends tempered distribution to tempered distribution (not only algebraic distribution).Then Perrin-Riou gets the following theorem. ( The significance of this theorem is that for k ∈ Z p , the left-hand side (hence the righthand side) gives an analytic family of cohomology classes in the sense of Section 3.
The ring Ᏸ 0 (Z × p , Q p ) has an action on both the distribution side and the cohomology side , then the action * (which is essentially induced by the map

8) commutes with the Galois action, hence it is well defined on
The map Exp h,V is sesquilinear with respect to these actions, that is, (5.9 where √ is induced by x → x −1 and defined to be (5.10) Proof.These follow from the definitions.
For the "negative" power, Colmez proved the following theorem.
The significance of these two theorems is that for k 1, (exp * V (−k) ) −1 gives rise to an analytic family of cohomology.Theorems 5.4 and 5.6 are called explicit reciprocity law.
To get the symmetric form of the explicit reciprocity law, one defines the following pairing: (5.16) The pairing in the cohomology side is defined as (5.17) From Theorems 5.4 and 5.6, we have the following theorem.
where δ −1 is defined by

.19)
Perrin-Riou proved this theorem for V = Q p ( 1) and Colmez proved it for general crystalline representation.
Moreover, as Iwasawa modules, those pairings have the following properties.
where the last pairing is defined in Section 3.
)), the integral where the cup product is given by is sesquilinear for the first variable and linear for the second variable, that is, (5.23) (iv) (•, •) V is linear for the first variable and sesquilinear for the second variable, that is, (5.24) Proof.(i) and (ii) are just from definitions, which can also be found in Colmez [2].For (iii), we have and (iv) is similar to (iii).

Iwasawa's explicit reciprocity law. Recall that
] denote the Coleman power series.
let g β denote the Coleman power series corresponding to β, and define Then where ω n = ζ p n − 1.
In the following, we will show that Perrin-Riou-Colmez explicit reciprocity law, Theorem 5.2, implies Iwasawa's explicit reciprocity law.
Recall that we have the Bloch-Kato exponential map exp Kn,V : 1) and let U n denote the principal units of O Kn .To an element of lim ← U n , we will associate an element in Ᏸ1 (Q p , D(V )) Φ=1 .To an element in lim ← K × n , we will associate an element in and extend it to Q p by defining hence Φµ β = µ β .By Proposition 4.2, µ β is 1-admissible.Note that Coleman power series has the property On the other hand, for 1)) defined by Kummer map.By using Colmez's theorem in Section 5, we get an element β(τ which has the property We can also state this map by using integral, namely, for Especially, for β ∈ lim ← U n , we have The element (p, For example, we can take a = 1 and b = 1 + p. Then the element (.. .,(ζ a p n − 1)/(ζ b p n − 1),...) ∈ lim ← U n , hence gives a distribution, which we denote by µ ab .Recall that δ a ∈ Ᏸ 0 (Z × p , Z p ) is defined to be the Dirac measure The following lemma describes the relationship between µ β and β, µ ab and p.

Lemma 6.2. (i)
There is a homomorphism µ : lim  Hence, δ ab * µ cd = δ cd * µ ab .We denote this pseudomeasure µ ab /δ ab by µ p .For a crystalline representation W such that there is a Galois inclusion Q p (1) ⊂ W , we have the following theorem.
Theorem 6.3.For W above, the map Exp 1,V (µ) can be extended to the set including µ p by using the inclusion (6.15) Proof.We only need to show that for µ = µ p , this will follow from the definition, the sesquilinear property of the exponential map, and the pairings (This can also be seen from that if δ ab * µ = 0, then y k δ ab * µ = 0, hence y k µ = 0, hence µ = 0.) Now, we use Perrin-Riou and Colmez explicit reciprocity law to prove Iwasawa's explicit reciprocity law.Assume that α n ,β n ∈ O Kn \ {0}, β n sits in a norm coherence sequence.Then β n = u n • β n • ω j n for u n a (p − 1)th root of unity, β n ≡ 1(mod ω n ), j ≥ 0. We know that the u n will give (α n ,u n ) n = 1 and σ un = 1.So we will consider the case β n ≡ 1(mod ω n ) and the case β n = ω n separately.For β with β n ≡ 1(mod ω n ), we have crys (1) is a lifting of (log(α n )/t) ⊗ e, by the definition of exponential map and the Kummer class, we see (i).From [2] we know that since µ ab corresponds to the power series .20)and this completes the proof of this lemma.Now, we come to the proof of Iwasawa's explicit reciprocity law in two cases.
Case 1. Assume that β n ≡ 1(mod ω n ) sits in the norm coherence sequence β.Take V = Q p (1), h = 1, k = 1, and µ = µ β ∈ Ᏸtemp (Q p , D(V )) Φ=1 , and using Theorem 5.6, we have Remark 6.5.Lemma 6.4(iv) can be interpreted as a completion of the theory of Coleman power series.Namely, µ p is the distribution whose Amice transformation is log(T ) in the sense that xµ p corresponds to D log(T ).

Zp
x k µ β . (7. 2) The Euler factor 1−p k−1 forces ψ 1 = 0 at k = 1.Since 1−p k−1 is not an analytic function of k, hence we cannot take the derivative directly.But ψ k is an analytic function of k, then dψ k /dk| k=1 must exist.Glenn Stevens predicted that the derivative of ψ k at 1 will give the Kummer class γ p .Based on the previous sections, now we can prove that this is true.From the following commutative diagram: This completes the proof of the theorem.

3 .
Distributions and Iwasawa module.Let I ⊂ Z be a subset and let LP I = {x k • 1 a+p n Zp |k ∈ I, a ∈ Q p }.An algebraic I-distribution with values in M is a finitely additive function µ : LP I → M. Let Ᏸ I alg (Q p ,M) denote all the algebraic I-distributions with values in M. For X ⊂ Q p , a compact open subset, let LP I (X) = {x k • 1 (a+p n Zp )∩X }, then Ᏸ I alg (X, M) is defined with respect to these test functions.Especially, we have Ᏸ I alg (Z × p ,M), Ᏸ I alg (Z p ,M).Let Ᏸ + alg (Q p ,M) (resp., Ᏸ − alg (Q p ,M)) denote the case I = N (resp.,I = −N).Note that when we say N we always mean N = {0, 1, 2,...}.