The field $\C_p$

Algebraic Properties

This is an algebraically closed field after Theorem 3.10. This field plays, for a lot of questions, the role of $\C$ in $p$ adic. It is abstractly isomorphic to $\C$ by the reasons of cardinality, but Tate demonstrated that, it does not contain a reasonable analogue of $2\ci\pi.$
  If $r\in \R$ and $a\in\C_p$, we denote $D(a,r)$ as a closed ball $\{x\in\C_p,v_p(x-a)\geq r\}$ and $D(a,r^+)$, the open ball $\{x\in\C_p, v_p(x-a)>r\}$. In particular, $D(0,0)=\oo_{\C_p}$ and $D(0,0^+)=\id{m}_{\C_p}$.
Teichmuller Representatives The residue field of $\C_p$ is an algebraic closure $\bar{\F}_p$ of $\F_p$, after the corollary 3.14. We now exhibit a system of privileged Representatives of $\bar{F}_p$ in $\oo_{\C_p}$.

Proposition 4.1 If $x\in\tio{\bar{\F}_p}$, there exists in $\oo_{\C_p}$, a unique root of unity $[x]$ of order prime to $P$ whose image in $\bar{\F}_p$ is $x$.
Proof There exists $n$ such that $x$ belongs to $\F_{p^n}$. The elements of $\tio{\F_{p^n}}$ are the solutions of the equations $P(X)=X^{p^n-1}-1=0$. Or the polynomial $X^{p^n-1}$ has a discriminant prime to $p$ (its discriminant upto a sign is $\prod_{\eta^{p^n-1}}P'(\eta)=\pm(p^n-1)^{p^n-1}$), this signifies that all the images are distinct $\mod\id{m}_{\C_p}$; we have thus an injection of a set with $p^n-1$ elements into a set of $p^n-1$ elements and thus reduction $\mod\id{m}_{\C_p}$ is a bijection permitting the conclusion.

Remark 4.2 The uniqueness of $[x]$ implies that $[xy]=[x]\cdot[y]$ and denoting $[0]=0$ we can construct a mulitplicative system of representatives, called teichmuller representatives of $\bar{\F}_p$ in $\oo_{\C_p}$.

Remark Let $\zeta$ be the root of unity
(i) If $\zeta$ is primitive of order $p^n,n\geq 1$, then $v_p(\zeta-1)=1/(p-1)p^{n-1}$.
(ii) If $\zeta$ does not have an order a power of $p$, then $v_p(\zeta-1)=0$.

4.1.3 The mulitplicative group $\tio{\C_p}-$ We have $v_p(\tio{\C_p})=v_p(\tio{\bar{\Q}_p})=\Q$. Choose a morphism of groups of $\Q$ in $\tio{\C_p}$ sending $1$ to $p$. We denote $p^r$, the image of $r\in\Q$ by the morphism. If $x\in\tio{\oo_{\C_p}}$ we denote $\omega(x)$ as the unique root of unity of order prime to $p$ such that $v_p(x-\omega(x))>0$. If $\bar{x}$ denotes the image of $x$ in $\bar{\F}_p$, we have $\omega(x)=[\bar{x}]$.

Proposition 4.3 The maps $$ \begin{align*} x&\ra p^{v_p(x)}, \\ x&\ra \tilde{\omega}(x)=\omega(xp^{-v_p(x)})\\ x&\ra \lr{x}=xp^{-v_p(x)}\tilde{\omega}(x)^{-1} \end{align*} $$ are the isomorphisms of the groups of $\tio{\C_p}$ in, respectively, $p^{\Q}$, the groups of roots of unity of order prime to $p$ and $D(1,0^+)$, and we have $x=p^{v_p(x)}\tilde{\omega}(x)\lr{x}$.

Rudimentary $p$ adic analysis

4.2.1 The logarithmic function
Lemma 4.4 If $v_p(x)>0$, the series $\log(1+x)=-\sum_{n\geq 1}\dfrac{(-x)^n}{n}$ converge in $\C_p$. Moreover, if $v_p(x)>0$ and $v_p(y)>0$, then $\log((1+x)(1+y))=\log(1+x)+\log(1+y)$.

Proof We have $$v_p\left(\dfrac{(-x)^n}{n}\right)=nv_p(x)-v_p(n)\geq nv_p(x)-\dfrac{\log n}{\log p}$$ Thus if $v_p(x)>0$, $v_p\left(\dfrac{(-x)^n}{n}\right)\ra \infty$ as $n\ra\infty$. This demonstrates the convergence of $\log(1+x)$ if $v_p(x)>0.$

Now, we have $\log((1+X)(1+Y))=\log(1+X)+\log(1+Y)$ being formal series in $X,Y$ (just derive). A brutal expansion of $\log(1+(X+Y+XY))$ shows that the two series above are also equal to $$ \sum_{i_1+i_2+i_3\geq 1}\dfrac{(-1)^{i_1+i_2+i_3}(i_1+i_2+i_3)!}{(i_1+i_2+i_3)i_1!i_2!i_3!} X^{i_1+i_3}Y^{i_2+i_3}. $$ Now, the triple series $$ \sum_{i_1+i_2+i_3\geq 1}\dfrac{(-1)^{i_1+i_2+i_3}(i_1+i_2+i_3)!}{(i_1+i_2+i_3)i_1!i_2!i_3!} x^{i_1+i_3}y^{i_2+i_3}. $$ converges because the general term tends to zero, when $i_1+i_2+i_3\ra+\infty$ and we can re order the terms as we want, permitting the conclusion.

Remark 4.5 If $\zeta$ is a root of unity of order a power of $p$, then $v_p(\zeta-1)>0$. We can thus calculate $\log\zeta$ by the above formula. Moreover, we have $\log(1+x)^n=n\log(1+x)$ if $n\in\N$ and $x\in D(0,0^+)$. Taking for $n$ a power of $p$ sufficiently large we can conclude that $\log\zeta=0$ if $\zeta$ is a root of unity or order a power of $p$, this is a manifestation of non existence of $2\pi\ci$ in $\C_p$.

Lemma 4.6 If $x\in D(1,0^+)$ verifies $\log x=0$, then $x$ is a root of unity of order a power of $p$.
Proof We have $$\log x=0\text{ if and only if }x=1 $$ if $v_p(x-1)>1/(p-1)$ because then the only term of maximal valuation in $\sum{n\geq 1}(1-x)^n/n$ is the first, and $v_p(\log x)=v_p(x-1)$.

Now, we have $x^p-1=(x-1)^p+p(x-1)\sum_{i=1}^{p-1}p^{-1}\binom{p}{i}(x-1)^{i-1}$. We deduce $$v_p(x^p-1)\geq\inf(1+v_p(x-1),pv_p(x-1)), $$ this permits to prove that $v_p(x^{p^n}-1)>1/(p-1)$ if $n$ is large and $x\in D(1,0^+)$. As $\log x=0$ if and only if $\log x^{p^n}=0$, this permits the conclusion.

Proposition 4.7 If $\scr{L}\in\C_p$, the function $\log$ is a unique embedding denoted by $\log_{\scr{L}}$ in $\tio{\C_p}$, verifying the following three conditions.
(i) $\log_{\scr{L}}=\log_{\scr{L}}(x)+=\log_{\scr{L}}(y)$
(ii)$\log_{\scr{L}}(x)=-\sum_{n\geq 1}(1-x)^n/n$ if $x\in D(1,0^+)$;
(iii)${\log_\scr{L}}p=\scr{L}$

Proof We see that if we fix a morphism $r\ra p^r$ of $\Q$ in $\tio{\C_p}$, we could write any element of $\tio{\C_p}$ in a unique manner under the form $p^{v_p(x)}\omega u$, where $\omega$ is a root of unity of order prime to $p$ and $u\in D(1,0^+)$; we must thus have ${\log_\scr{L}}x=\log u+\scr{L}v_p(x)$, this proves the uniqueness. The existence follows from the fact that $x\mapsto u$ and $x\mapsto v_p(x)$ are the morphisms of groups.

Remark 4.8 The choice of $\scr{L}$ amounts to a fixed branch of the Logarithm. From the point of view of arithmetic, imposing $\log p=0$ is natural, we obtain then the Logarithm of Iwasawa.
4.2.2-The exponential function
Denote $[x]$ as the integral part of $x$ if $x\in \R$ (not to confuse with the Teichmuller representative)
Proposition 4.9 If $n\in\N$ then $$v_p(n!)=\sum_{k\geq 1}\left[\dfrac{n}{p^k}\right]=\dfrac{n-S_p(n)}{p-1}$$ where $S_p(n)$ denotes the sum of numbers of writing $n$ in base $p$.
Proof Let $a_k$ (respectively $b_k$) be the cardinality of the set of integers $i$ verifying $1\leq i\leq n$ and $v_p(i)=k$ (respectively $v_p(i)\geq k$). We have $b_k=\sum_{k\geq\ell}a_\ell$ and $$v_p(n!)=\sum_{k\geq 1}ka_k =\sum_{k\geq 1}b_k=\sum_{k\geq 1}\left[\dfrac{n}{p^k}\right]$$ which gives us the first equality, the second is left as an exercise.

Proposition 4.10 The series $\exp(x)=\sum_{n\geq 0}x^n/n!$ converges if and only if $v_p(x)>1/(p-1)$, and induces an isomorphism of groups of the open ball $D\left(0,\left(\dfrac{1}{p-1}\right)^+\right)$ equipped with addition to the open ball $D\left(1,\left(\dfrac{1}{p-1}\right)^+\right)$ equipped with multiplication, inverse is the $\log$ map.

Proof The determination of radius of convergence comes from the fact that there is an infinite number of $n$ such that $S_p(n)=1$ (namely, the powers of $p$). The rest of the proposition is an exercise.
  Let $\pi$ be the solution of the equation $\pi^{p-1}=-p$ (it is the $\pi$ of Dwork).
Proposition 4.11
(i) The formal series $E_\pi(X)=\exp(\pi(X-X^p))$ belongs to $1+\pi X+\pi^2X^2\oo_{\C_p}[[X]]$, and there exists $r<0$ such that the formal series $E_\pi(X)=\exp(\pi(X-X^p))$ converges on $D(0,r)$.
(ii) If $\omega:\F_p\ra\Z_p$ designates the map representing Teichmuller, then $x\mapsto E_\pi(\omega(x))$ is an isomorphism of groups of $\F_p$ on $\mu_p$.
Remark The result is a bit surprising, because, $\omega(x)-\omega(x)^p$ being null, we have the impression that we should have $E_\pi(\omega(x))=1$ for any $x\in\F_p$. The point is that the series $\exp(\pi x)$ does not converge if $v_p(x)=0$.




A substantial part is a translation of LES NOMBRES p-ADIQUES, NOTES DU COURS DE M2 by Pierre Colmez.