Note on Coleman's formula for the absolute Frobenius on Fermat curves

TL;DR

This paper explores the p-adic properties of Coleman's formula for the Frobenius on Fermat curves, revealing that p-adic continuity underpins much of Coleman's explicit calculation through relations involving p-adic gamma functions and CM-periods.

Contribution

It demonstrates that p-adic continuity explains a significant part of Coleman's explicit Frobenius formula on Fermat curves, linking functional equations of p-adic gamma functions to CM-period relations.

Findings

p-adic continuity underlies Coleman's formula

Functional equations of p-adic gamma functions relate to Frobenius

Connections between p-adic gamma functions and CM-periods established

Abstract

Coleman calculated the absolute Frobenius on Fermat curves explicitly. In this paper we show that a kind of $p$-adic continuity implies a large part of his formula. To do this, we study a relation between functional equations of the ($p$-adic) gamma function and monomial relations on ($p$-adic) CM-periods.

Equations

$${\Gamma }_{\infty }(z):=\frac{\Gamma (z)}{\sqrt{2\pi }}=\exp ({\zeta }^{\prime }(0,z))(z>0)$$

$${\Gamma }_{\infty }(z){\Gamma }_{\infty }(1-z)=\frac{1}{2\sin \pi z},$$

$$\prod _{k=0}^{d-1}{\Gamma }_{\infty }(z+\frac{k}{d})=d^{\frac{1}{2}-dz}{\Gamma }_{\infty }(dz)(d\in \mathbb{N}).$$

$$I_K:=\underset{\sigma \in \mathrm{Hom}(K,\mathbb{C})}{⨁}\mathbb{Q}\cdot \sigma .$$

$$p_K:I_K\times I_K\to {\mathbb{C}}^{\times }/{\overline{\mathbb{Q}}}^{\times }$$

$$k^{\ast }({\omega }_{\sigma })=\sigma (k){\omega }_{\sigma }(k\in K),$$

$$\Xi =\{\sigma \in \mathrm{Hom}(K,\mathbb{C})\mid {\omega }_{\sigma }\text{ is holomorphic}\},$$

$$p_K(\sigma ,\Xi )\equiv \{\begin{array}{ll}{\pi }^{-1}{\int }_{\gamma }{\omega }_{\sigma }& (\sigma \in \Xi )\\ {\int }_{\gamma }{\omega }_{\sigma }& (\sigma \in \mathrm{Hom}(K,\mathbb{C})-\Xi )\end{array}\mathrm{mod}{\overline{\mathbb{Q}}}^{\times }$$

$$p_K(\sigma ,\tau )p_K(\rho \circ \sigma ,\tau )\equiv p_K(\sigma ,\tau )p_K(\sigma ,\rho \circ \tau )\equiv 1\mathrm{mod}{\overline{\mathbb{Q}}}^{\times }(\sigma ,\tau \in \mathrm{Hom}(K,\mathbb{C})).$$

$$p_K(\sigma ,\tau )\equiv p_{K^{\prime }}(\sigma \circ \iota ,\tau \circ \iota )\mathrm{mod}{\overline{\mathbb{Q}}}^{\times }(\sigma ,\tau \in \mathrm{Hom}(K,\mathbb{C})).$$

$$\mathrm{Res}:I_L\to I_K,\tilde{\sigma }\mapsto \tilde{\sigma }{|}_K(\tilde{\sigma }\in \mathrm{Hom}(L,\mathbb{C})),$$

$$\mathrm{Inf}:I_K\to I_L,\sigma \mapsto \sum _{\begin{array}{c}\tilde{\sigma }\in \mathrm{Hom}(L,\mathbb{C})\\ \tilde{\sigma }{|}_K=\sigma \end{array}}\tilde{\sigma }(\sigma \in \mathrm{Hom}(K,\mathbb{C})).$$

$$p_K(\mathrm{Res}(X),Y)\equiv p_L(X,\mathrm{Inf}(Y))\mathrm{mod}{\overline{\mathbb{Q}}}^{\times }(X\in I_L,Y\in I_K).$$

$${\int }_{\gamma }{\eta }_{r,s}\equiv \frac{\Gamma (\frac{r}{N})\Gamma (\frac{s}{N})}{\Gamma (\frac{r+s}{N})}\mathrm{mod}\mathbb{Q}({\zeta }_N{)}^{\times }.$$

$${\Xi }_{r,s}:=\{{\sigma }_b\mid 1\le b\le N,(b,N)=1,⟨\frac{br}{N}⟩+⟨\frac{bs}{N}⟩+⟨\frac{b(N-r-s)}{N}⟩=1\}.$$

$$p_{\mathbb{Q}({\zeta }_N)}(\mathrm{id},{\Xi }_{r,s})\equiv \{\begin{array}{ll}{\pi }^{-1}{\int }_{\gamma }{\eta }_{r,s}& (r+s<N)\\ {\int }_{\gamma }{\eta }_{r,s}& (r+s>N)\end{array}\mathrm{mod}{\overline{\mathbb{Q}}}^{\times }.$$

$${\Gamma }_{\infty }(\frac{a}{N})\equiv {\pi }^{\frac{1}{2}-⟨\frac{a}{N}⟩}{p}_{\mathbb{Q}({\zeta }_N)}\left(\mathrm{id},\sum _{(b,N)=1}\left(\frac{1}{2}-⟨\frac{ab}{N}⟩\right)\cdot {\sigma }_b\right)\mathrm{mod}{\overline{\mathbb{Q}}}^{\times }.$$

$${\Gamma }_{\infty }(\frac{a}{N}){\Gamma }_{\infty }(\frac{N-a}{N})\equiv 1\mathrm{mod}{\overline{\mathbb{Q}}}^{\times },$$

$$\prod _{k=0}^{d-1}{\Gamma }_{\infty }(\frac{a}{N}+\frac{k}{d})\equiv {\Gamma }_{\infty }(\frac{da}{N})\mathrm{mod}{\overline{\mathbb{Q}}}^{\times }.$$

$$\begin{gathered} {\pi }^{\stackrel{d-1}{\sum _{k=0}}\frac{1}{2}-⟨\frac{a}{N}+\frac{k}{d}⟩}{p}_{\mathbb{Q}({\zeta }_N)}\left(\mathrm{id},\sum _{(b,N)=1}\left(\sum _{k=0}^{d-1}\frac{1}{2}-⟨\frac{ab}{N}+\frac{kb}{d}⟩\right)\cdot {\sigma }_b\right) \\ \equiv {\pi }^{\frac{1}{2}-⟨\frac{ad}{N}⟩}{p}_{\mathbb{Q}({\zeta }_N)}\left(\mathrm{id},\sum _{(b,N)=1}(\frac{1}{2}-⟨\frac{dab}{N}⟩)\cdot {\sigma }_b\right). \end{gathered}$$

$$\sum _{k=0}^{d-1}{B}_1(x+\frac{k}{d})=B_1(dx)$$

$${\int }_p:H_1^{\mathrm{B}}(A(\mathbb{C}),\mathbb{Q})\times H_{\mathrm{d}R}^1(A,\overline{\mathbb{Q}})\to B_{\mathrm{c}ris}\overline{{\mathbb{Q}}_p},(\gamma ,\omega )\mapsto {\int }_{\gamma ,p}\omega$$

$$p_{K,p}:I_K\times I_K\to (B_{\mathrm{c}ris}\overline{{\mathbb{Q}}_p}-\{0\}{)}^{\mathbb{Q}}/{\overline{\mathbb{Q}}}^{\times }$$

$$\left[{\int }_{\gamma }{\omega }_{\sigma }:{\int }_{\gamma ,p}{\omega }_{\sigma }\right]\in ({\mathbb{C}}^{\times }\times (B_{\mathrm{c}ris}\overline{{\mathbb{Q}}_p}-\{0\}))/{\overline{\mathbb{Q}}}^{\times }$$

$$\frac{{\int }_{{\gamma }^{\prime }}{\omega }_{\sigma }^{\prime }}{{\int }_{\gamma }{\omega }_{\sigma }}=\frac{{\int }_{{\gamma }^{\prime },p}{\omega }_{\sigma }^{\prime }}{{\int }_{\gamma ,p}{\omega }_{\sigma }}\in {\overline{\mathbb{Q}}}^{\times }.$$

$$[p_K:p_{K,p}]:I_K\times I_K\to ({\mathbb{C}}^{\times }\times (B_{\mathrm{c}ris}\overline{{\mathbb{Q}}_p}-\{0\}{)}^{\mathbb{Q}})/({\mu }_{\infty }\times {\mu }_{\infty }){\overline{\mathbb{Q}}}^{\times }$$

$$\begin{gathered} [p_K:p_{K,p}](\sigma ,\Xi ) \\ \equiv \{\begin{array}{ll}[(2\pi i{)}^{-1}{\int }_{\gamma }{\omega }_{\sigma }:(2\pi i{)}_p^{-1}{\int }_{\gamma ,p}{\omega }_{\sigma }]& (\sigma \in \Xi )\\ [{\int }_{\gamma }{\omega }_{\sigma }:{\int }_{\gamma ,p}{\omega }_{\sigma }]& (\sigma \in \mathrm{Hom}(K,\mathbb{C})-\Xi )\end{array}\mathrm{mod}({\mu }_{\infty }\times {\mu }_{\infty }){\overline{\mathbb{Q}}}^{\times }. \end{gathered}$$

$$[p_K:p_{K,p}](\sigma ,\tau )\cdot [p_K:p_{K,p}](\rho \circ \sigma ,\tau )\equiv 1\mathrm{mod}({\mu }_{\infty }\times {\mu }_{\infty }){\overline{\mathbb{Q}}}^{\times },$$

$$[p_K:p_{K,p}](\sigma ,\tau )\cdot [p_K:p_{K,p}](\sigma ,\rho \circ \tau )\equiv 1\mathrm{mod}({\mu }_{\infty }\times {\mu }_{\infty }){\overline{\mathbb{Q}}}^{\times }.$$

$$[p_K:p_{K,p}](\sigma ,\tau )\equiv [p_{K^{\prime }}:p_{K^{\prime },p}](\sigma \circ \iota ,\tau \circ \iota )\mathrm{mod}({\mu }_{\infty }\times {\mu }_{\infty }){\overline{\mathbb{Q}}}^{\times }.$$

$$[p_K:p_{K,p}](\mathrm{Res}(X),Y)\equiv [p_L:p_{L,p}](X,\mathrm{Inf}(Y))\mathrm{mod}({\mu }_{\infty }\times {\mu }_{\infty }){\overline{\mathbb{Q}}}^{\times }.$$

$${\exp }_p:{\mathbb{C}}_p\to {\mathbb{C}}_p^1$$

$${\alpha }^{\beta }:={\exp }_p(\beta {\log }_p\alpha )$$

Full text

1 Introduction

We modify Euler’s gamma function $\Gamma(z)$ into

[TABLE]

and focus on its special values at rational numbers. Here we put $\zeta(s,z):=\sum_{k=0}^{\infty}(z+k)^{-s}$ to be the Hurwitz zeta function. The last equation is due to Lerch. One has a“simple proof” in [Yo, p17]. The gamma function enjoys some functional equations:

[TABLE]

For proofs, see [Ar, §3, 4]. The main topic of this paper is a relation between such functional equations and monomial relations of CM-periods, and its $p$-adic analogue. We introduce some notations.

Definition 1.1**.**

Let $K$ be a CM-field. We denote by $I_{K}$ the $\mathbb{Q}$-vector space formally generated by all complex embeddings of $K$:

[TABLE]

We identify a subset $S\subset\mathrm{Hom}(K,\mathbb{C})$ as an element $\sum_{\sigma\in S}\sigma\in I_{K}$. Shimura’s period symbol is the bilinear map

[TABLE]

characterized by the following properties $(\mathrm{P}_{1})$, $(\mathrm{P}_{2})$.

1. $(\mathrm{P}_{1})$

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$, having CM of type $(K,\Xi)$. Namely, for each $\sigma\in\mathrm{Hom}(K,\mathbb{C})$, there exists a non-zero “$K$-eigen” differential form $\omega_{\sigma}$ of the second kind satisfying

[TABLE]

where $k^{*}$ denotes the action of $k\in K$ via $K\cong\mathrm{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}$ on the de Rham cohomology $H^{1}_{\mathrm{d}R}(A,\mathbb{C})$. Then we have

[TABLE]

for an arbitrary closed path $\gamma\subset A(\mathbb{C})$ satisfying $\int_{\gamma}\omega_{\sigma}\neq 0$. 2. $(\mathrm{P}_{2})$

Let $\rho$ be the complex conjugation. Then we have

[TABLE]

Strictly speaking, Shimura’s $p_{K}$ in [Sh, §32] is a bilinear map on $\bigoplus_{\sigma\in\mathrm{Hom}(K,\mathbb{C})}\mathbb{Z}\cdot\sigma$. The period symbol also enjoys the following relations:

1. $(\mathrm{P}_{3})$

Let $\iota\colon K^{\prime}\cong K$ be an isomorphism of CM-fields. Then we have

[TABLE] 2. $(\mathrm{P}_{4})$

Let $K\subset L$ be a field extension of CM-fields. We define two linear maps defined as

[TABLE]

Then we have

[TABLE]

The following results by Gross-Rohrlich and the above relations $(\mathrm{P}_{3})$, $(\mathrm{P}_{4})$ provide an explicit formula [Yo, Theorem 2.5, Chap. III] on $p_{K}$ for $K=\mathbb{Q}(\zeta_{N})$ ($\zeta_{N}=e^{\frac{2\pi i}{N}}$, $N\geq 3$). We can rewrite it in the form (5) by the arguments in [Ka2, §6]. Let $\sigma_{b}\in\mathrm{Gal}(\mathbb{Q}(\zeta_{N})/\mathbb{Q})$ ($(b,N)=1$) be defined by $\sigma_{b}(\zeta_{N}):=\zeta_{N}^{b}$, $\langle\alpha\rangle\in(0,1)$ denote the fraction part of $\alpha\in\mathbb{Q}-\mathbb{Z}$.

Theorem 1.2** ([Gr, Theorem in Appendix]).**

Let $F_{N}:x^{N}+y^{N}=1$ be the $N$th Fermat curve, $\eta_{r,s}:=x^{r-1}y^{s-N}dx$ its differential forms of the second kind $(0<r,s<N$, $r+s\neq N)$. Then we have for any closed path $\gamma$ on $F_{N}(\mathbb{C})$ with $\int_{\gamma}\eta_{r,s}\neq 0$

[TABLE]

Theorem 1.3** ([Gr, §2], [Yo, §2, Chap. III]).**

The CM-type corresponding to $\eta_{r,s}$ is

[TABLE]

That is, we have

[TABLE]

Corollary 1.4** ([Ka2, Theorem 3]).**

We have for any $\frac{a}{N}\in\mathbb{Q}-\mathbb{Z}$

[TABLE]

Here the sum runs over all $b$ satisfying $1\leq b\leq N$, $(b,N)=1$.

Note that (5) holds true even if $(a,N)>1$, essentially due to $(\mathrm{P}_{4})$. Although the following is just a toy problem, we provide its proof by using the period symbol, in order to explain the theme of this paper: we may say that some functional equations of the gamma function “correspond” to some monomial relations of CM-periods.

Proposition 1.5** (A toy problem).**

The explicit formula (5) implies the following “functional equations $\bmod\overline{\mathbb{Q}}^{\times}$” on $\Gamma(\frac{a}{N})$:

[TABLE]

Proof.

“Reflection formula” follows from $(\mathrm{P}_{2})$ immediately. Concerning “Multiplication formula”, we may assume that $d\mid N$. Under the expression (5), “Multiplication formula” is equivalent to

[TABLE]

This follows from the multiplication formula

[TABLE]

for the $1$st Bernoulli polynomial $B_{1}(x)=x-\frac{1}{2}$. ∎

The aim of this paper is to study a $p$-adic analogue of such “correspondence”. More precisely, we shall characterize the $p$-adic gamma function by its functional equations and some special values. Then we show that the period symbol and its $p$-adic analogue satisfy the corresponding properties to such functional equations. As an application, we provide an alternative proof of a large part of Coleman’s formula (Theorem 2.4-(i)): originally, Coleman’s formula was proved by calculating the absolute Frobenius on all Fermat curves. We shall see that it suffices to calculate it on only one curve (Remark 3.7).

Remark 1.6**.**

Yoshida and the author formulated conjectures in [KY1, KY2, Ka2] which are generalizations of Coleman’s formula, from cyclotomic fields to arbitrary CM-fields: Coleman’s formula implies “the reciprocity law on cyclotomic units” [Ka1] and “the Gross-Koblitz formula on Gauss sums” [GK, Co1] simultaneously. The author conjectured a generalization [Ka2, Conjecture 4] of Coleman’s formula which implies a part of Stark’s conjecture and a generalization of (the rank $1$ abelian) Gross-Stark conjecture simultaneously. The results in this paper (in particular Remark 3.7) are very important toward this generalization, since we know only a finite number of algebraic curves (e.g., [BS]) whose Jacobian varieties have CM by CM-fields which are not abelian over $\mathbb{Q}$.

The outline of this paper is as follows. First we introduce Coleman’s formula [Co2] for the absolute Frobenius on Fermat curves in §2. The author rewrote it in the form of Theorem 2.4: roughly speaking, we write Morita’s $p$-adic gamma function $\Gamma_{p}$ in terms of Shimura’s period symbol $p_{K}$, its $p$-adic analogue $p_{K,p}$, and modified Euler’s gamma function $\Gamma_{\infty}$. In §3, we show that some functional equations almost characterize $\Gamma_{p}$ (Corollary 3.3), and the corresponding properties ((13), Theorem 3.5) hold for $p_{K},p_{K,p},\Gamma_{\infty}$. Then we see that a large part (Corollary 3.6) of Coleman’s formula follows automatically, without explicit computation, under assuming certain $p$-adic continuity properties. Unfortunately, our results have a root of unity ambiguity although the original formula is a complete equation, since some definitions are well-defined only up to roots of unity. In §4, we confirm that we can show (at least, a part of) needed $p$-adic continuity properties relatively easily.

2 Coleman’s formula in terms of period symbols

Coleman explicitly calculated the absolute Frobenius on Fermat curves [Co2]. The author rewrote his formula in [Ka1, Ka2] as follows.

2.1 $p$-adic period symbol

Let $p$ be a rational prime, $\mathbb{C}_{p}$ the $p$-adic completion of the algebraic closure $\overline{\mathbb{Q}_{p}}$ of $\mathbb{Q}_{p}$, and $\mu_{\infty}$ the group of all roots of unity. For simplicity, we fix embeddings $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C},\mathbb{C}_{p}$ and consider any number field as a subfield of each of them. Let $B_{\mathrm{c}ris}\subset B_{\mathrm{d}R}$ be Fontaine’s $p$-adic period rings. We consider the composite ring $B_{\mathrm{c}ris}\overline{\mathbb{Q}_{p}}\subset B_{\mathrm{d}R}$. Let $A$ be an abelian variety with CM defined over $\overline{\mathbb{Q}}$, $\gamma$ a closed path on $\subset A(\mathbb{C})$, and $\omega$ a differential form of the second kind of $A$. Then the $p$-adic period integral

[TABLE]

is defined by the comparison isomorphisms of $p$-adic Hodge theory, instead of the de Rham isomorphism (e.g., [Ka1, §6], [Ka2, §5.1]). Here $H^{B}$ denotes the singular (Betti) homology. Then, in a similar manner to $p_{K}$, we can define the $p$-adic period symbol

[TABLE]

satisfying $p$-adic analogues of $(\mathrm{P}_{1})$, $(\mathrm{P}_{2})$, $(\mathrm{P}_{3})$, $(\mathrm{P}_{4})$. Here we put $(B_{\mathrm{c}ris}\overline{\mathbb{Q}_{p}}-\{0\})^{\mathbb{Q}}:=\{x\in B_{\mathrm{d}R}\mid\exists n\in\mathbb{N}\text{ s.t.\ }x^{n}\in B_{\mathrm{c}ris}\overline{\mathbb{Q}_{p}}-\{0\}\}$. Moreover the “ratio”

[TABLE]

depends only on $\sigma\in\mathrm{Hom}(K,\mathbb{C})$ and the CM-type $\Xi$. That is, if we replace $A,\omega_{\sigma},\gamma$ with $A^{\prime},\omega_{\sigma}^{\prime},\gamma^{\prime}$ for the same $\Xi,\sigma$, then we have

[TABLE]

Therefore we may consider the following ratio of the symbols $[p_{K}:p_{K,p}]$, which is well-defined up to $\mu_{\infty}$.

Proposition 2.1** ([Ka2, Proposition 4]).**

There exists a bilinear map

[TABLE]

satisfying the following.

1. (i)

Let $A,\Xi,\sigma,\omega_{\sigma},\gamma$ be as in $(\mathrm{P}_{1})$. Then

[TABLE]

Here $(2\pi i)_{p}\in B_{\mathrm{c}ris}$ is the $p$-adic counterpart of $2\pi i$ defined in, e.g., [Ka2, §5.1]. 2. (ii)

We have for $\sigma,\tau\in\mathrm{Hom}(K,\mathbb{C})$ and for the complex conjugation $\rho$

[TABLE] 3. (iii)

Let $\iota\colon K^{\prime}\cong K$ be an isomorphism of CM-fields. Then we have for $\sigma,\tau\in\mathrm{Hom}(K,\mathbb{C})$

[TABLE] 4. (iv)

Let $K\subset L$ be a field extension of CM-fields. Then we have for $X\in I_{L}$, $Y\in I_{K}$

[TABLE]

2.2 Coleman’s formula

Theorem 2.4 below is essentially due to Coleman [Co2, Theorems 1.7, 3.13]. Note that the original formula does not have a root of unity ambiguity. First we prepare some notations. We assume that $p$ is an odd prime.

Definition 2.2**.**

1. (i)

Let $\mathbb{C}_{p}^{1}:=\{z\in\mathbb{C}_{p}^{\times}\mid|z|_{p}=1\}$. We fix a group homomorphism

[TABLE]

which coincides with the usual power series $\exp_{p}(z):=\sum_{k=0}^{\infty}\frac{z^{k}}{k!}$ on the convergence region. For $\alpha\in\mathbb{C}_{p}^{\times}$, $\beta\in\mathbb{C}_{p}$, we put

[TABLE]

with $\log_{p}$ Iwasawa’s $p$-adic $\log$ function. 2. (ii)

For $z\in\mathbb{C}_{p}^{\times}$, we put

[TABLE]

Here we define $\mathrm{ord}_{p}\hskip 1.0ptz\in\mathbb{Q}$ by $|z|_{p}=|p|_{p}^{\mathrm{ord}_{p}\hskip 1.0ptz}$. Note that $z\equiv z^{\flat}\bmod\mu_{\infty}$ $(z\in\mathbb{C}_{p}^{\times})$. 3. (iii)

We define the $p$-adic gamma function on $\mathbb{Q}_{p}$ as follows.

1. (a)

On $\mathbb{Z}_{p}$, $\Gamma_{p}(z)$ denotes Morita’s $p$-adic gamma function which is the unique continuous function $\Gamma_{p}\colon\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}^{\times}$ satisfying

[TABLE] 2. (b)

On $\mathbb{Q}_{p}-\mathbb{Z}_{p}$, we use $\Gamma_{p}\colon\mathbb{Q}_{p}-\mathbb{Z}_{p}\rightarrow\mathcal{O}_{\overline{\mathbb{Q}_{p}}}^{\times}$ defined in [Ka1, Lemma 4.2], which is a continuous function satisfying

[TABLE]

Such a continuous function on $\mathbb{Q}_{p}-\mathbb{Z}_{p}$ is unique up to multiplication by $\mu_{\infty}$. 4. (iv)

For $z\in\mathbb{Z}_{p}$, we define $z_{0}\in\{1,2,\dots,p\}$, $z_{1}\in\mathbb{Z}_{p}$ by

[TABLE]

Note that when $p\mid z$, we put $z_{0}=p$, instead of [math]. 5. (v)

Let $W_{p}$ be the Weil group defined as

[TABLE]

Here $\mathbb{Q}_{p}^{ur}$ denotes the maximal unramified extension of $\mathbb{Q}_{p}$, $\sigma_{p}$ the Frobenius automorphism on $\mathbb{Q}_{p}^{ur}$. 6. (vi)

We define the action of $W_{p}$ on $\mathbb{Q}\cap[0,1)$ by identifying $\mathbb{Q}\cap[0,1)=\mu_{\infty}$. Namely

[TABLE] 7. (vii)

Let $\Phi_{\mathrm{c}ris}$ be the absolute Frobenius automorphism on $B_{\mathrm{c}ris}$. We consider the following action of $W_{p}$ on $B_{\mathrm{c}ris}\overline{\mathbb{Q}_{p}}\cong B_{\mathrm{c}ris}\otimes_{\mathbb{Q}^{ur}}\overline{\mathbb{Q}_{p}}$:

[TABLE] 8. (viii)

For $\frac{a}{N}\in\mathbb{Q}\cap(0,1)$ we put

[TABLE]

This definition makes sense since

[TABLE]

by (5) and the ratio $[p_{K}:p_{K,p}]$ is well-defined up to $\mu_{\infty}$ by Proposition 2.1.

Remark 2.3**.**

1. (i)

*Let $\mu_{p-1}$ be the group of all *$(p-1)$st roots of unity, $p^{\mathbb{Z}}:=\{p^{n}\mid n\in\mathbb{Z}\}$, $1+p\mathbb{Z}_{p}:=\{1+pz\mid z\in\mathbb{Z}_{p}\}$. Then we have the canonical decomposition

[TABLE]

where $\omega$ denotes the Teichmüller character. The maps $z\mapsto z^{*},z^{\flat}$ provide a similar (but non-canonical) decomposition of $\mathbb{C}_{p}^{\times}$. Moreover, we note that the maps $z\mapsto\exp_{p}(z),z^{*},z^{\flat}$ are continuous homomorphisms. 2. (ii)

We easily see that

[TABLE]

Theorem 2.4** ([Ka2, Theorem 3]).**

Let $p$ be an odd prime.

1. (i)

Assume that $z\in\mathbb{Z}_{(p)}\cap(0,1)$. Then we have

[TABLE] 2. (ii)

Assume that $z\in(\mathbb{Q}-\mathbb{Z}_{(p)})\cap(0,1)$. Then we have

[TABLE]

Remark 2.5**.**

As a result, we see that the right-hand sides of Theorem 2.4-(i), (ii) are $p$-adic continuous on $z$, $(z,\tau(z))$ respectively, since the left-hand sides are so. We use only the $p$-adic continuity in the next section, in order to recover Theorem 2.4-(i).

3 Main results

Morita’s $p$-adic gamma function $\Gamma_{p}\colon\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}^{\times}$ is the unique continuous function satisfying

[TABLE]

In this section, we study other functional equations characterizing $\Gamma_{p}$ and provide an alternative proof of Coleman’s formula in the case $z\in\mathbb{Z}_{(p)}$. Strictly speaking, we only “assume” that the right-hand sides of Theorem 2.4-(i), (ii) are continuous on $z$, $(z,\tau(z))$ respectively (of course, this is correct). Then we can recover a “large part” (Corollary 3.6) of Theorem 2.4-(i). We assume that $p$ is an odd prime.

3.1 A characterization of Morita’s $p$-adic gamma function

$\Gamma_{p}(z)$ satisfies the following $p$-adic analogues of multiplication formulas, which we consider only up to roots of unity in this paper. For the detailed formulation and its proof, see [Ko, “Basic properties of $\Gamma_{p}$” in §2 of Chap. IV].

Proposition 3.1**.**

Let $d\in\mathbb{N}$ with $p\nmid d$. Then we have for $z\in\mathbb{Z}_{p}$

[TABLE]

Note that if $p\mid d$, then $z+\frac{k}{d}$ is not in the domain of definition of Morita’s $\Gamma_{p}$. In the rest of this subsection, we show that multiplication formulas (8) and some conditions characterize Morita’s $p$-adic gamma function (at least up to $\mu_{\infty}$).

Proposition 3.2**.**

Assume a continuous function $f(z)\colon\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}^{\times}$ satisfies

[TABLE]

Then the following holds.

1. (i)

$\frac{f(z+1)}{f(z)}\bmod\mu_{\infty}$* depends only on $\mathrm{ord}_{p}\hskip 1.0ptz$.* 2. (ii)

The values

[TABLE]

characterize the function $f(z)$ up to $\mu_{\infty}$. More precisely, for $z\in\mathbb{Z}_{p}$, we write the $p$-adic expansion of $z-1$ as

[TABLE]

Then we have

[TABLE]

Conversely, assume that

[TABLE]

for constants $\alpha_{k}\in\mathbb{C}_{p}^{\times}$ satisfying $\alpha_{k}\rightarrow 1$ $(k\rightarrow\infty)$. Then $f(z)$ satisfies the functional equations (9).

Proof.

We suppress $\bmod\mu_{\infty}$. Assume (9). Replacing $z$ with $z+\frac{1}{d}$, we obtain $\prod_{k=1}^{d}f(z+\frac{k}{d})\equiv f(dz+1)$. It follows that $\frac{f(z+1)}{f(z)}\equiv\frac{f(dz+1)}{f(dz)}$. That is,

[TABLE]

Then the assertion (i) is clear. Let $c_{k}:=(g(p^{k}))^{\flat}$, $a_{n}:=x_{0}+x_{1}p+\dots+x_{n}p^{n}$ ($0\leq x_{i}\leq p-1$). We easily see that

[TABLE]

Then we can write

[TABLE]

with $\alpha_{k}=c_{k}\prod_{i=0}^{k-1}c_{i}^{p^{k-1-i}(p-1)}$. Since $\lim_{n\rightarrow\infty}f(a_{n}+1)$ converges, so do $\lim_{n\rightarrow\infty}f(a_{n}+1)^{\flat}$ and $\prod_{k=0}^{\infty}\alpha_{k}^{x_{k}}$. Moreover we can write

[TABLE]

Consider the case of $d=2$, $z=\frac{1}{2}$ of (9): $f(\frac{1}{2})f(1)\equiv f(1)$. Therefore, noting that $-\frac{1}{2}=\sum_{k=0}^{\infty}\frac{p-1}{2}p^{k}$, we obtain

[TABLE]

Then the assertion (ii) is also clear.

Next, assume (10). When $\mathrm{ord}_{p}\hskip 1.0ptz=k$, we see that $\frac{f(z+1)}{f(z)}\equiv\frac{\alpha_{k}}{\alpha_{k-1}^{p-1}}$ (resp. $\alpha_{0}$) if $k>0$ (resp. $k=0$). In particular, $g(z):=\frac{f(z+1)}{f(z)}\bmod\mu_{\infty}$ depends only on $\mathrm{ord}_{p}\hskip 1.0ptz$. When $z+z^{\prime}=1$, the $p$-adic expansions $z-1=\sum_{k=0}^{\infty}x_{k}p^{k}$, $z^{\prime}-1=\sum_{k=0}^{\infty}x_{k}^{\prime}p^{k}$ satisfy $x_{k}+x_{k}^{\prime}=p-1$ for any $k$. Then we have

[TABLE]

Therefore the case $z=0$ of (9) holds true since $\left(\prod_{k=1}^{d-1}f(\frac{k}{d})\right)^{2}=\prod_{k=1}^{d-1}f(\frac{k}{d})f(1-\frac{k}{d})\equiv 1$. Then (9) for $z\in\mathbb{N}$ follows by mathematical induction on $z$ noting that

[TABLE]

Since $\mathbb{N}$ is dense in $\mathbb{Z}_{p}$, we see that (9) holds for any $z\in\mathbb{Z}_{p}$. ∎

The following corollary provides a nice characterization of $\Gamma_{p}(z)\bmod\mu_{\infty}$ in terms of functional equations and one or two special values.

Corollary 3.3**.**

Assume a continuous function $f(z)\colon\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}^{\times}$ satisfies

[TABLE]

and put

[TABLE]

Then the following equivalences hold:

1. (i)

$c_{0}=c_{1}=\cdots$* $\Leftrightarrow$ $f(z)\equiv c_{0}^{z-\frac{1}{2}}\bmod\mu_{\infty}$.* 2. (ii)

$c_{1}=c_{2}=\cdots$* $\Leftrightarrow$ $f(z)\equiv c_{0}^{z-\frac{1}{2}}(c_{1}/c_{0})^{z_{1}+\frac{1}{2}}\bmod\mu_{\infty}$.*

Proof.

We suppress $\bmod\mu_{\infty}$. For (i), assume that $c_{0}=c_{1}=\cdots$. Then

[TABLE]

Hence we have by Proposition 3.2

[TABLE]

The opposite direction is trivial by definition $c_{n}:=(\frac{f(p^{n}+1)}{f(p^{n})})^{\flat}$. For (ii), the assumption $c_{1}=c_{2}=\cdots$ implies $\alpha_{0}=c_{0}$, $\alpha_{k}=c_{0}^{p^{k}}(c_{1}/c_{0})^{p^{k-1}}$ ($k\geq 1$). In this case we have

[TABLE]

since $\sum_{k=1}^{\infty}x_{k}p^{k-1}=\frac{z-1-x_{0}}{p}=z_{1}$. ∎

3.2 Alternative proof of a part of Coleman’s formula

We fix $\tau\in W_{p}$ with $\deg\tau=1$ and put

[TABLE]

Here we added $(\ )^{\flat}$ to the right-hand sides of Coleman’s formulas (Theorem 2.4), in order to resolve a root of unity ambiguity, only superficially. Note that $G_{2}$ corresponds to Theorem 2.4-(ii) replaced $z$ with $\tau^{-1}(z)$.

By Theorem 2.4-(i), we see that $G_{1}$ is continuous for the $p$-adic topology. $G_{2}$ is not $p$-adically continuous in the usual sense, on the whole of $(\mathbb{Q}-\mathbb{Z}_{(p)})\cap(0,1)$ (for details, see Remark 3.8). Theorem 2.4-(i) only implies the following “continuity”:

[TABLE]

In Corollary 3.6, oppositely, we show that the $p$-adic continuity of $G_{1},G_{2}$ implies a “large part”

[TABLE]

of Theorem 2.4-(i):

[TABLE]

Besides we shall show the continuity of $G_{1}(z)$ in §4, independently of Theorem 2.4.

Hereinafter in this section, we forget Theorem 2.4. We assume the following Assumption instead.

Assumption 3.4**.**

$G_{1}(z)$* is $p$-adically continuous and $G_{2}(z)$ is continuous in the sense of (12). In particular, we regard $G_{1}$ as a $p$-adic continuous function:*

[TABLE]

First we derive “multiplication formula”:

[TABLE]

independently of Theorem 2.4.

Proof of (13).

We suppress $\bmod\mu_{\infty}$. Let $z\in\mathbb{Z}_{(p)}\cap(0,\frac{1}{d})$. By Definition 2.2-(viii) and (11) we can write

[TABLE]

where the “products of classical or $p$-adic periods” become trivial by (6), as we saw in the proof of Proposition 1.5. Besides we see that

[TABLE]

To see this, it suffices to show that $\{\tau^{-1}(\zeta_{N}^{a}\zeta_{d}^{k})\mid k=0,\dots,d-1\}$ and $\{\tau^{-1}(\zeta_{N}^{da})^{\frac{1}{d}}\zeta_{d}^{k}\mid k=0,\dots,d-1\}$ coincide with each other. We easily see that both of them are the inverse image of $\tau^{-1}(\zeta_{N}^{da})$ under the $d$th power map $\mu_{\infty}\rightarrow\mu_{\infty}$, $x\mapsto x^{d}$. Hence we obtain

[TABLE]

by (2), (6). For the last “$\equiv$”, we note that $\Phi_{\tau}$ acts on $\overline{\mathbb{Q}_{p}}\ni d^{\tau^{-1}(dz)-\frac{1}{2}}$ as $\tau$. By Remark 2.3-(ii), we have $\tau^{-1}(dz)=(dz)_{1}+1$. Then the assertion is clear. ∎

Furthermore we can show that $c_{n}=\left(\frac{f(p^{n}+1)}{f(p^{n})}\right)^{\flat}$ for $f(z):=\frac{G_{1}(z)}{\Gamma_{p}(z)}$ is constant, at least for $n\geq 1$.

Theorem 3.5**.**

We assume Assumption 3.4 and put $f(z):=\frac{G_{1}(z)}{\Gamma_{p}(z)}$.

1. (i)

The following functional equations hold.

[TABLE] 2. (ii)

We have $c_{1}=c_{2}=\cdots$ for $c_{n}:=\left(\tfrac{f(p^{n}+1)}{f(p^{n})}\right)^{\flat}$.

Proof.

We suppress $\bmod\mu_{\infty}$. (i) follows from (8), (13). For (ii), we need for $z\in p\mathbb{Z}_{p}$

[TABLE]

Since the right-hand side is equal to $\begin{cases}1&(p\mid z)\\ z&(p\nmid z)\end{cases}$ by (7), it suffices to show that

[TABLE]

Note that we can not use the definition (11) directly since $z,z+1,pz,pz+1$ are not contained in $(0,1)$ simultaneously. Therefore a little complicated argument is needed as follows. Let $z\in\mathbb{Z}_{(p)}\cap(0,\frac{1}{p})$. By Remark 2.3-(ii), we have

[TABLE]

We can write

[TABLE]

Here we note that $\mathrm{ord}_{p}(z+\frac{k}{p})=-1$ for $k=1,\dots,p-1$. We have

[TABLE]

since both of $\{\tau^{-1}(\zeta_{N}^{a}\zeta_{p}^{k})\mid k=0,\dots,p-1\}$, $\{\zeta_{pN}^{a+Nk}\mid k=0,\dots,p-1\}$ are the set of the $p$th roots of $\zeta_{N}^{a}$ when $z=\frac{a}{N}$. Therefore the $p$-power parts of $H_{1}$ become

[TABLE]

Moreover the “period parts” of $H_{1}$ become trivial by (6), (14). Namely we can write

[TABLE]

By using the original Multiplication formula (2) for $\Gamma_{\infty}$, we obtain

[TABLE]

Next, let $z=\frac{a}{N}\in\mathbb{Z}_{(p)}\cap(-\frac{1}{p},0)$. Then we have

• •

$\tau(z+1)=pz+1$. Hence $\tau^{-1}(pz+1)=z+1$.

• •

$\{\tau^{-1}(\zeta_{N}^{a}\zeta_{p}^{k})\mid k=1,\dots,p\}=\{\zeta\mid\zeta^{p}=\zeta_{N}^{a}\}=\{\zeta_{pN}^{a+Nk}\mid k=1,\dots,p\}$. Hence $\{\tau^{-1}(z+\frac{k}{p})\mid k=1,\dots,p\}=\{\tfrac{z+k}{p}\mid k=1,\dots,p\}$.

Then we can prove similarly that

[TABLE]

Here $H_{i}(z)\equiv 1\bmod\mu_{\infty}$ implies $H_{i}(z)=1$ ($i=1,2$) since we have $x^{\flat}=\exp_{p}(\log_{p}x)=\exp_{p}(0)=1$ for $x\in\mu_{\infty}$. ($G_{1}(z),G_{2}(z)$ are in the image under $(\ )^{\flat}$ by definition, so are $H_{i}(z)$.) In particular, we have

[TABLE]

Let $z\in p\mathbb{Z}_{(p)}$. Then there exist $z_{n}^{+}\in p\mathbb{Z}_{(p)}\cap(0,\frac{1}{p})$, $z_{n}^{-}\in p\mathbb{Z}_{(p)}\cap(-\frac{1}{p},0)$ which converge to $z$ when $n\rightarrow\infty$ respectively. Then we can write

[TABLE]

Recall that $G_{2}(z)$ is continuous in the sense of (12). Clearly we have for $k=1,\dots,p-1$

[TABLE]

Additionally we see that

[TABLE]

by noting that $\tau^{-1}(z+z^{\prime})\equiv\tau^{-1}(z)+\tau^{-1}(z^{\prime})\bmod\mathbb{Z}$ ($\forall z,z^{\prime}$), $\tau^{-1}(z)\equiv\tfrac{z}{p}\bmod\mathbb{Z}$ if $p\mid z$, $\tfrac{z_{n}^{\pm}}{p}\in(-\frac{1}{p},\frac{1}{p})$, $\tau^{-1}(\tfrac{k}{p})\in[\frac{1}{p},\frac{p-1}{p}]$. It follows that

[TABLE]

Then the assertion is clear. ∎

By Corollary 3.3, we obtain the following.

Corollary 3.6**.**

Assume Assumption 3.4. Then there exist constants $a,b$ satisfying

[TABLE]

Remark 3.7**.**

In addition to the above results, by computing the absolute Frobenius on only one Fermat curve, we obtain Coleman’s formula $G_{1}(z)\equiv\Gamma_{p}(z)\bmod\mu_{\infty}$. For example, when $p=3$, we obtain it for $z=\frac{1}{5},\frac{2}{5}$ by the computation on $F_{5}$. It follows that $a^{\frac{-3}{10}}b^{\frac{-1}{10}}\equiv a^{\frac{-1}{10}}b^{\frac{3}{10}}\equiv 1$, hence $a\equiv b\equiv 1$.

Remark 3.8**.**

We used the assumption $p\mid z$ only in the last paragraph of the proof for Theorem 3.5 because $G_{2}$ is not $p$-adically continuous on the whole of $(\mathbb{Q}-\mathbb{Z}_{(p)})\cap(0,1)$. For example, we put

[TABLE]

and take $\tau\in W_{p}$ with $\deg\tau=1$ so that

[TABLE]

In particular we see that

$z_{n}\rightarrow\frac{1}{p^{2}}$* for the $p$-adic topology.*

On the other hand we see that

[TABLE]

Hence we have

[TABLE]

Then, by Theorem 2.4-(ii), we see that $G_{2}(z_{n})=(\Gamma_{p}(z_{n})/\Gamma_{p}(\tau^{-1}(z_{n})))^{\flat}$ does not converge $p$-adically although $z_{n}$ does.

4 On the $p$-adic continuity

In the previous section, we showed that the $p$-adic continuity of the right-hand sides of Theorem 2.4-(i), (ii) implies a large part of Theorem 2.4-(i) itself. In this section, we see that it is relatively easy to show such $p$-adic continuity properties, without explicit computation. For simplicity, we consider only the case $z\in\mathbb{Z}_{p}$. Assume that $p\nmid N$.

Lemma 4.1** ([Co1, §VI]).**

Let $1\leq r,s<N$ with $r+s\neq N$. We consider the formal expansion of the differential form $\eta_{r,s}=x^{r}y^{s-N}\frac{dx}{x}$ on $F_{N}\colon x^{N}+y^{N}=1$ at $(x,y)=(0,1)$:

[TABLE]

Let $\Phi$ be the absolute Frobenius on $H_{\mathrm{d}R}^{1}(F_{N},\mathbb{Q}_{p})$. Then there exists $\alpha_{r^{\prime},s^{\prime}}\in\mathbb{Q}_{p}$ satisfying

[TABLE]

Then we have

[TABLE]

We note that $\alpha_{r^{\prime},s^{\prime}}$ depends only on $(\frac{r^{\prime}}{N},\frac{s^{\prime}}{N})$. That is $\alpha_{r^{\prime},s^{\prime}}$ with $N=N_{1}$ is equal to $\alpha_{tr^{\prime},ts^{\prime}}$ with $N=tN_{1}$.

Proposition 4.2**.**

$\alpha_{r^{\prime},s^{\prime}}$* is $p$-adically continuous on $(\frac{r^{\prime}}{N},\frac{s^{\prime}}{N})\in(\mathbb{Z}_{(p)}\cap(0,1))^{2}$.*

Proof.

It suffices to show that $\alpha_{r^{\prime}_{1},s^{\prime}_{1}}$ with $N=N_{1}$ is close to $\alpha_{r^{\prime}_{2},s^{\prime}_{2}}$ with $N=N_{2}$ when $\frac{r^{\prime}_{1}}{N_{1}}$ is close to $\frac{r^{\prime}_{2}}{N_{2}}$ and $\frac{s^{\prime}_{1}}{N_{1}}$ is close to $\frac{s^{\prime}_{2}}{N_{2}}$. We may assume $N:=N_{1}=N_{2}$ by considering $N=N_{1}N_{2}$. First we fix $r^{\prime}:=r^{\prime}_{1}=r^{\prime}_{2}$ and assume that $s_{1}^{\prime}$ is close to $s_{2}^{\prime}$. Then we can take the same $k$ for the limit expressions (15) of $\alpha_{r^{\prime},s^{\prime}_{1}}$, $\alpha_{r^{\prime},s^{\prime}_{2}}$. We easily see that if $p^{l}\mid(s_{1}^{\prime}-s_{2}^{\prime})$, then $p^{l-1}\mid(s_{1}-s_{2})$. In fact, we can write $s_{i}^{\prime}=ps_{i}-l_{i}N$ with $l_{i}=0,1,\dots,p-1$ since $0<s_{i},s_{i}^{\prime}<N$ for $i=1,2$. If $p\mid(s_{1}^{\prime}-s_{2}^{\prime})$, then we have $p\mid(l_{1}-l_{2})$, so $l_{1}=l_{2}$. Therefore we obtain $s_{1}-s_{2}=\frac{s_{1}^{\prime}-s_{2}^{\prime}}{p}$. It follows that $s_{1}$ also is close to $s_{2}$. Hence the continuity on $\frac{s^{\prime}}{N}$ is clear since the numerator (resp. the denominator) of the expression (15) is a polynomial on $\frac{s}{N}$ (resp. $\frac{s^{\prime}}{N}$).

For the variable $\frac{r^{\prime}}{N}$, we replace $x$ with $y$. In other words, replace the point $(x,y)=(0,1)$ for the expansion with $(1,0)$. Then the continuity on $\frac{r^{\prime}}{N}$ also follows from the same argument. ∎

Corollary 4.3**.**

$G_{1}(z)$* defined in (11) is $p$-adically continuous on $z\in\mathbb{Z}_{(p)}\cap(0,1)$. In particular, we may regard $G_{1}(z)$ as a continuous function on $\mathbb{Z}_{p}$.*

Proof.

CM-types $\Xi_{r,s}$ of (4), corresponding to $\eta_{r,s}$, generate the $\mathbb{Q}$-vector space $\{\sum_{\sigma}c_{\sigma}\cdot\sigma\mid c_{\sigma}+c_{\rho\circ\sigma}$ is a constant$\}$. More explicitly, we claim that

[TABLE]

where $s$ runs over $1\leq s<N$ with $a+s\neq N$ in the first sum of the right-hand side. By the definition (4), $\sigma_{b}\in\Xi_{a,s}$ if and only if $\langle\frac{ab}{N}\rangle+\langle\frac{sb}{N}\rangle<1$. Namely $\langle\frac{sb}{N}\rangle=\frac{1}{N},\frac{2}{N},\dots,1-\frac{1}{N}-\langle\frac{ab}{N}\rangle$. The number of such $b$ is congruent to $-1-ab\bmod N$. Hence we have

[TABLE]

Here we note that $ab\not\equiv 0\bmod N$ since $(b,N)=1$, $a\not\equiv 0\bmod N$. Then the above claim follows. By substituting this into Definition 2.2-(viii), we can write

[TABLE]

since the part $\sum_{(b,N)=1}\sigma_{b}$ becomes trivial by Proposition 2.1-(ii). We can strengthen the congruence relation $\equiv$ of the formula (3) into an equality $=$, by selecting a specific closed path $\gamma_{0}$ (e.g., $\gamma_{0}=N\gamma_{N}$ with $\gamma_{N}$ in [Ot, Proposition 4.9]). Then we have

[TABLE]

where we put

[TABLE]

Since (2) implies that

[TABLE]

we obtain

[TABLE]

For the last equality we used (1) and the difference equation $\Gamma(z+1)=z\Gamma(z)$. Take $\tau\in W_{p}$ with $\deg\tau=1$. Then we have

[TABLE]

by noting that $\Phi_{\tau}((2\pi i)_{p})=p(2\pi i)_{p}$ and $\Phi_{\tau}(\sin(\tfrac{a}{N}\pi))=\tau(\sin(\tfrac{a}{N}\pi))=\pm\sin(\tfrac{a^{\prime}}{N}\pi)$. Here $a^{\prime},s^{\prime}$ denote integers satisfying $1\leq a^{\prime},s^{\prime}<N$, $pa\equiv a^{\prime}\bmod N$, $ps\equiv s^{\prime}\bmod N$ as above. By Proposition 4.2, $\alpha_{a^{\prime},s^{\prime}}$ are continuous for $a^{\prime}$. When $a$ is in a small open ball, as we saw in the proof of Proposition 4.2, we may write $a^{\prime}=pa-M$ for a fixed $M$ ($M$ is $lN$ in the proof of Proposition 4.2). Then the remaining part becomes

[TABLE]

which is also continuous as desired. ∎