Currently, I'm reading a chapter about p-adic equations in A Course in Arithmetic by Jean-Pierre Serre and I have a hard time understanding it.
My questions/thoughts are in textboxes like these. I tried to anwser most of them myself, but I'm quite unsure, if the anwsers are right.
Lemma: Let $\ldots \to D_n \to D_{n-1} \to \ldots \to D_1$ be a projective system and let $D = \varprojlim D_n$ be its projective limit. If the $D_n$ are finite and nonempty, then $D$ is nonempty
Proof: The fact that $D \neq \varnothing$ is clear if the $D_n \to D_{n-1}$ are surjective.
Question 1: Is this correct?: Let $p_n: D_n \to D_{n-1}$ be surjective. Therefore for all $x_{n-1} \in D_{n-1}$ there exist a $x_n \in D_n$, such that $p(x_n) = x_{n-1}$ for all $n$. Hence, $(\ldots,x_n, x_{n-1}, \ldots, x_1) \in D$ and hence $D \neq \varnothing$.
We are going to reduce the lemma to this special case. Denote by $D_{n,p}$ the image of $D_{n+p}$ in $D_n$. For fixed $n$, the $D_{n,p}$ form a decreasing family of finite nonempty subsets, hence this familiy is stationary, i.e. $D_{n,p}$ is independent of $p$ for $p$ large enough.
Question 2: Are the $D_{n,p}$ nonempty, because of the very definition of a mapping/function?
Let $E_n$ be this limit value of the $D_{n,p}$.
Question 3: That means, for all $p$ large enough it is $E_n = D_{n,p}$?
One checks immediately that $D_n \to D_{n-1}$ carries $E_n$ onto $E_{n-1}$.
Question 4: Why is the mapping $E_n \to E_{n-1}$ surjective?
Since the $E_n$ are nonempty, we have $\varprojlim E_n \neq \varnothing$ by the remark made at the beginning.
Question 5: Similar to question 3: For a $p$ large enough, it is $D_{n,p} = E_n$. Since $D_{n,p}$ is nonempty, $E_n$ is nonempty, too?
Hence, a fortiori $\varprojlim D_n \neq \varnothing$.
Notation If $f \in \mathbb{Z}_p [ X_1, \ldots, X_m]$ is a polynomial with coefficients in $\mathbb{Z}_p$ and if $n$ is an integer $\geq 1$, we denote by $f_n$ the polynomial with coefficients in $A_n = \mathbb{Z} / p^n \mathbb{Z}$ deduced from $f$ by reduction $(\mod p^n)$.
Question 6: Let $f \in \mathbb{Z}_p[X]$. Then $f(X) = \sum_{i=0}^m a_iX^i$, where $a_i = (\ldots, a_i^{(n)}, a_i^{(n-1)}, \ldots, a_i^{(1)}) \in \mathbb{Z}_p$. By reduction, does it mean, that I have a map $\mathbb{Z}_p \to A_n$, where $a_i \to a_i^{(n)}$, that is $f_n(X) = \sum_{i=0}^{m} a_i^{(n)}X^i$?
Proposition: Let $f^{(i)} \in \mathbb{Z}_p [X_1, \ldots, X_m]$ be polynomials with p-adic integer coefficients. The following are equivalent:
i) The $f^{(i)}$ have a common zero in $(\mathbb{Z}_p)^m$.
ii) For all $n>1$ the polynomials $f_n^{(i)}$ have a common zero in $(A_n)^{m}$.
Proof: Let $D$ (resp. $D_n$) be the set of common zeros of the $f^{(i)}$ (resp. $f_n^{(i)}$). The $D_n$ are finite and we have $D = \varprojlim D_n$. By the above lemma, $D$ is nonempty if and only if the $D_n$ are nonempty, hence the propostion.
Question 7: Here we need the $D_n$ to form a projective system. Can I choose $\phi_n: D_n \to D_{n-1}$, where $(x_1 \mod p^n, \ldots, x_m \mod p^{n}) \mapsto (x_1 \mod p^{n-1}, \ldots, x_m \mod p^{n-1})$. Then if $(x_1 \mod p^n, \ldots, x_m \mod p^{n})\in D_n$ is a zero, then it is a zero in $D_{n-1}$, right?
Question 8: If we take $\phi_n$ from question 7, we have $\varprojlim D_n \subset D$. If $x \in D$ then after plugging in $f$ each component has to be $0$. After rearranging we get that $x \equiv 0 \mod p^n$ for all $n$. Is this correct?
Question 9: The $D_n \subset (A_n)^m$ are finite, because $(A_n)^m$ is finite for all $n$? Then since the $D_n$ are finite and nonempty we can apply the above lemma to obtain that $D$ is finite and nonempty, too.
Question 10: If $D$ is nonempty, then $\varprojlim D_n$ is nonempty, too, therefore the $D_n$ are nonempty?
Definition/Remark: A point $x = (x_1, \ldots, x_m)$ of $(\mathbb{Z}_p)^m$ is called primitive if one of the $x_i$ is invertible, that is, if the $x_i$ are not all divisible by $p$.
Question 11: If $x_i$ is not divisible $p$, does it mean, that each component of $x_i$ is not divisible by $p$?
One definies in a similiar way the primitive elements of $(A_n)^m$.
Question 12: That is, an element $x$ of $(A_n)^m$ is primitve if not all of $x_i$ is not divisible by $p$?
Notation: Let $x$ be a nonzero element of $\mathbb{Z}_p$. Write $x$ in the form $p^nu$ with $u \in U$, where $U$ denotes the group of invertible elements of $\mathbb{Z}_p$. The integer $n$ is called the p-adic valuation of $x$ and denoted by $\nu_p(x)$. Proposition: Let $f^{(i)} \in \mathbb{Z}_p[X_1, \ldots, X_m]$ be homogenous polynomials with p-adic integer coefficients. The following are equivalent:
a) The $f^{(i)}$ have a non trivial common zero in $(\mathbb{Q}_p)^m$. b) The $f^{(i)}$ have a common primitive zero in $(\mathbb{Z}_p)^m$. c) For all $n>1$, the $f_n^{(i)}$ have a common primitive zero in $(A_n)^m$.
Proof: Them implication $b) \implies a)$ is trivial.
Question 13: If $x$ is a primitive zero in $(\mathbb{Z}_p)^m$ , then one of the components is divisible by $p$, that is this specific component is not zero, hence it is not a trivial zero in $(\mathbb{Z}_p)^m$ and therefore not a trivial zero in $(\mathbb{Q}_p)^m$?
Conversely, if $x = (x_1, \ldots, x_m)$ is a nontrivial common zero of the $f^{(i)}$ put $ h = \inf (\nu_p(x_1), \ldots, \nu_p(x_m))$ and $y = p^{-h}x$. It is clear that $y$ is a primitive element $(\mathbb{Z}_p)^m$ and that it is a common zero of the $f^{(i)}$. Hence $b) \implies a)$
Question 14: $\nu_p(y) = \nu_p(p^{-h}x) = \nu_p(x) - h \geq 0$. Since $\nu_p(z) \geq 0$ iff $z \in \mathbb{Z}_p$, it is $y \in \mathbb{Z}_p$?
Question 15: The infimum is probably a minimum. That means there exists $x_i$, such that $\nu_p(y_i) = \nu_(x_i) - h = 0$. Therefore $y_i$ is nor divisible by $p$ and hence $y$ is primitve?
Question 16: However, I do not understand, why after modifiying $x$ to $y$ it is still a zero.
Question 17: Can $b) \implies c)$ and $c) \implies b)$ be proven exactly like in the previous proposition, where $D_n$ is the set of common primitive zeros of the $f^{(i)}$?
I think this is enough for now...