p-adic Analysis

Introduction

The field $\mathbb{Q}_p$ is the standard example of everything in this category: a complete field with a discrete valuation, a compact ring of integers, a finite residue field, and an analysis in which the balls are compact and split into finitely many smaller balls to all depths. This article develops that analysis concretely. It begins with the analytic geometry of $\mathbb{Q}_p$ and of $\mathbb{C}_p$ — the compactness of $\mathbb{Z}_p$, the partition of a ball into $p$ balls of the next level, the failure of local compactness for $\mathbb{C}_p$ — and then treats three bodies of results that have no analogue for a general valued field: Mahler's theorem, which expands every continuous function on $\mathbb{Z}_p$ in the binomial polynomials with coefficients tending to $0$ and thereby identifies the Banach space of continuous functions with the space $c_0$; the Newton polygon, which reads the valuations of the zeros of a power series from the valuations of its coefficients; and the Weierstrass preparation theorem on the unit disc, which factors a power series with a unit coefficient into a polynomial and a series invertible on the disc.

The general theory is taken from Analytic Functions and Power Series and Non-Archimedean Analysis: the radius of convergence, the ultrametric convergence criterion, the exponential and the logarithm with their discs, and the maximum principle are used here without rederivation, and this article supplies the arithmetic that the concrete field permits. The topological and arithmetic background is The $p$-adic Numbers and Local Fields for the construction of $\mathbb{Q}_p$, Hensel's lemma and the structure of the unit group. The other articles of this category build on this one: one supplies the measure, one the theory of linear equations, one the higher-dimensional and affinoid theory, and one the normed theory. The arithmetic applications of the analytic machinery — the Kubota–Leopoldt zeta function and the $p$-adic $L$-functions — belong to the arithmetic part of this Part.

Throughout, $p$ is a prime, $\mathbb{Q}_p$ is the field of $p$-adic numbers with the absolute value normalised by $\lvert p \rvert_p = p^{-1}$, $\mathbb{Z}_p$ is its valuation ring, $v = v_p$ the valuation, $\mathbb{F}_p$ the residue field, $\mathbb{C}_p$ the completion of an algebraic closure of $\mathbb{Q}_p$, and $\mathcal{O}_p = \mathbb{Z}_p$. The disc notation is that of Analytic Functions and Power Series: $B(a,r)$ and $\overline{B}(a,r)$ for the open and closed discs, and $\rho_{\exp} = \lvert p \rvert^{1/(p-1)}$ for the radius of the exponential. The binomial polynomial is $\binom{x}{n} = x(x-1)\cdots(x-n+1)/n!$, a polynomial with rational coefficients that is integer-valued on $\mathbb{Z}$.

The $p$-adic Fields and Their Analytic Geometry

$\mathbb{Q}_p$ and $\mathbb{C}_p$

Theorem (properties of $\mathbb{Q}_p$). The field $\mathbb{Q}_p$ is complete, locally compact, totally disconnected and not discrete; its valuation ring $\mathbb{Z}_p$ is a compact open subring whose maximal ideal is $p\mathbb{Z}_p$ and whose residue field is $\mathbb{F}_p$; the value group is $p^{\mathbb{Z}}$, so the valuation is discrete and $p$ is a uniformiser. The absolute value is non-Archimedean and $\mathbb{Z}_p$ is the closed unit ball.

Proof. These are the standard properties of the $p$-adic completion, established in The $p$-adic Numbers and Local Fields: compactness of $\mathbb{Z}_p$ from its expression as the inverse limit of the finite rings $\mathbb{Z}/p^n\mathbb{Z}$, total disconnectedness from the clopen partition of $\mathbb{Z}_p$ into residue classes modulo $p^n$, and local compactness because $\mathbb{Z}_p$ is a compact neighbourhood of $0$.

Theorem (properties of $\mathbb{C}_p$). The field $\mathbb{C}_p$ is complete, algebraically closed, non-Archimedean, with residue field the algebraic closure of $\mathbb{F}_p$ and value group $p^{\mathbb{Q}}$. It is not locally compact, so its closed unit disc is not compact and cannot be covered by finitely many translates of a smaller disc.

Proof. Completeness and algebraic closure are standard. The value group is $p^{\mathbb{Q}}$ because the absolute value of $\mathbb{Q}_p$ extends uniquely to the algebraic closure and the completion has dense value group. A locally compact non-Archimedean field has finite residue field and discrete value group, by the classification in Local Fields; $\mathbb{C}_p$ has neither, so it is not locally compact.

The contrast is decisive for the analysis. Over $\mathbb{Q}_p$ the unit ball is compact, so continuous functions on it are uniformly continuous and the finiteness results of Non-Archimedean Analysis apply. Over $\mathbb{C}_p$ there is no compact unit ball, the maximum principle of Non-Archimedean Analysis must be applied to discs whose radius is small enough that compactness is irrelevant, and the global arguments are replaced by the affinoid ones.

The Partition of a Ball

Proposition. For every $a \in \mathbb{Q}_p$ and every integer $n \geq 0$, $$ a + p^n\mathbb{Z}_p = \bigsqcup_{j=0}^{p-1} \bigl(a + jp^n + p^{n+1}\mathbb{Z}_p\bigr), $$ the union being disjoint, and each of the $p$ pieces is an open and closed ball of radius $p^{-(n+1)}$ contained in the ball of radius $p^{-n}$.

Proof. The quotient $p^n\mathbb{Z}_p/p^{n+1}\mathbb{Z}_p$ has $p$ elements, represented by $0, p^n, \dots, (p-1)p^n$, so the cosets displayed are the residue classes of the subgroup $p^{n+1}\mathbb{Z}_p$ inside $p^n\mathbb{Z}_p$; a set of coset representatives is exactly a partition. That each coset is clopen is the ultrametric nesting property: if $\lvert x - y \rvert < p^{-(n+1)}$ then $x + p^{n+1}\mathbb{Z}_p = y + p^{n+1}\mathbb{Z}_p$, so each ball is a union of balls and its complement is a union of balls.

The partition is the algebraic expression of total disconnectedness: a ball of $\mathbb{Q}_p$ is a disjoint union of finitely many clopen sub-balls, and the decomposition can be iterated to all depths. It is also the mechanism behind the fractal-like behaviour of continuous functions on $\mathbb{Z}_p$ and behind the recursive construction of the functions of Non-Archimedean Analysis with vanishing derivatives.

Continuous Functions and Mahler's Theorem

Definition. Let $C(\mathbb{Z}_p, \mathbb{Q}_p)$ be the $\mathbb{Q}_p$-vector space of continuous functions $f : \mathbb{Z}_p \to \mathbb{Q}_p$, with the supremum norm $\lVert f \rVert_\infty = \sup_{x \in \mathbb{Z}_p} \lvert f(x) \rvert$. The difference operator is $$ (\Delta f)(x) = f(x+1) - f(x), $$ and $\Delta^n f$ is defined by iteration.

Theorem (uniform continuity). Every $f \in C(\mathbb{Z}_p, \mathbb{Q}_p)$ is uniformly continuous, and $C(\mathbb{Z}_p,\mathbb{Q}_p)$ with the supremum norm is a Banach space over $\mathbb{Q}_p$.

Proof. $\mathbb{Z}_p$ is compact, and a continuous function on a compact uniform space is uniformly continuous. The supremum norm is non-Archimedean, the space is complete because a uniform limit of continuous functions on a compact space is continuous and bounded functions with values in a complete field are complete for the supremum norm, as in Modes of Convergence.

Theorem (Mahler). Every $f \in C(\mathbb{Z}_p,\mathbb{Q}_p)$ has a unique expansion $$ f(x) = \sum_{n \geq 0} a_n \binom{x}{n}, \qquad a_n = (\Delta^n f)(0) \in \mathbb{Q}_p , $$ the coefficients satisfying $a_n \to 0$; the series converges uniformly on $\mathbb{Z}_p$, and the map $f \mapsto (a_n)$ is an isometry $$ C(\mathbb{Z}_p,\mathbb{Q}_p) \cong c_0 = \bigl\{ (a_n) \in \mathbb{Q}_p^{\mathbb{N}} : a_n \to 0 \bigr\} $$ onto the space $c_0$ with the supremum norm. Moreover $f$ takes values in $\mathbb{Z}_p$ for every $x \in \mathbb{Z}_p$ if and only if $a_n \in \mathbb{Z}_p$ for every $n$.

Proof sketch. For the uniqueness, evaluate the expansion at $x = 0, 1, \dots, N$: since $\binom{m}{n} = 0$ for $n > m$, the values on $\mathbb{Z}_{\geq 0}$ determine the $a_n$ recursively by finite triangular inversion, and the formula $a_n = \Delta^n f(0)$ is the inverted system. For the existence, define $a_n = \Delta^n f(0)$; the estimate $\lvert a_n \rvert \leq \sup_{\mathbb{Z}_p} \lvert f \rvert$ shows the coefficients are bounded, and a compactness argument with the uniform continuity of $f$ shows $a_n \to 0$ and that the sum converges to $f$ uniformly. The isometry and the integrality statement follow from $\lVert \binom{x}{n} \rVert_\infty = 1$ on $\mathbb{Z}_p$ and from the corresponding integrality of the binomial polynomials.

Example. The polynomial $f(x) = x = \binom{x}{1}$ has expansion with $a_1 = 1$ and all other coefficients zero; the polynomial $f(x) = x^2 = \binom{x}{1} + 2\binom{x}{2}$ has $a_1 = 1$, $a_2 = 2$, the rest zero. The function $$ f(x) = \sum_{n \geq 0} p^n \binom{x}{n} $$ is continuous, has coefficients tending to $0$ but not integral, and is not the restriction of a polynomial; it is the standard example of a continuous non-analytic function on $\mathbb{Z}_p$.

Mahler's theorem is the concrete form of the identification of the continuous functions on the unit ball with a space of sequences. It makes the Banach space $C(\mathbb{Z}_p,\mathbb{Q}_p)$ computable, and it is the basis of the theory of $p$-adic distributions: a $p$-adic distribution is a continuous linear functional on the space of continuous functions, hence a sequence of bounded coefficients dual to the Mahler basis.

Locally Analytic Functions and the Newton Polygon

Locally Analytic Functions

Definition. A function $f : \mathbb{Z}_p \to \mathbb{Q}_p$ is locally analytic if for every $a \in \mathbb{Z}_p$ there is an integer $n \geq 0$ and a power series $\sum_k c_k (x-a)^k$ converging on $a + p^n\mathbb{Z}_p$ whose sum agrees with $f$ there. The locally analytic functions form a $\mathbb{Q}_p$-algebra containing the analytic functions and the polynomials, and the inclusion is strict.

The definition is the affine local form of the analytic functions of Analytic Functions and Power Series; the same notion with a uniform radius of convergence over the ball gives the more restricted class of globally analytic functions, and the difference is genuine. The structure of the ring of locally analytic functions, its norms and its completions belong to the theory, where the affinoid algebra of a closed disc is defined, where the Banach and Fréchet spaces of such functions are treated.

The Newton Polygon

Definition. Let $f = \sum_{n \geq 0} a_n x^n$ be a power series over $\mathbb{Q}_p$ converging on the closed unit disc, so that $a_n \to 0$. The Newton polygon of $f$ is the lower convex hull in the plane of the set of points $(n, v(a_n))$ for those $n$ with $a_n \neq 0$: it is the boundary of the intersection of the closed half-planes lying above the lines through pairs of the points, and its edges are the segments joining the vertices of that hull. The slope of an edge is its vertical displacement divided by its horizontal displacement, and the horizontal length of an edge is the difference of the first coordinates of its endpoints.

Theorem (Newton polygon theorem). Let $f = \sum_{n \geq 0} a_n x^n$ converge on the closed unit disc with $a_n \to 0$ and $f \neq 0$, and let an edge of its Newton polygon have slope $s$ and horizontal length $\ell$, with no vertex of the polygon strictly between its endpoints. Then $f$ has exactly $\ell$ zeros $x$ with $v(x) = -s$, counted with multiplicity.

Proof sketch. Write the zero set as a multiset of $K$-points after passing to $\mathbb{C}_p$; for each zero $\alpha$ the linear factor $x - \alpha$ has Newton polygon the segment from $(0, v(\alpha))$ to $(1,0)$, of slope $-v(\alpha)$ and horizontal length $1$, and the polygon of a product is the sum of the polygons with slopes added in the same order. Hence a root of valuation $m$ contributes to an edge of slope $-m$, and the total horizontal length of the edge is the number of zeros of that valuation contributing to it. The content of the theorem is that this formal bookkeeping is valid for a convergent series and not merely for a product of linear factors, which is proved by the Weierstrass preparation theorem below, by induction on the edges. The degenerate cases in which the polygon has a vertical edge or an infinite sequence of edges of vanishing slope are handled by taking limits along the edges.

Corollary (counting zeros disc by disc). With $f$ as in the theorem, the number of zeros of $f$ in the closed unit disc, counted with multiplicity, is the sum of the horizontal lengths of the edges of slope $\leq 0$; the number of zeros on the unit circle is the sum of the horizontal lengths of the edges of slope $0$; and the number of zeros in the open unit disc is the sum of the horizontal lengths of the edges of slope $< 0$.

Example (the primitive $p$-th cyclotomic polynomial). Let $\Phi_p(x) = (x^p-1)/(x-1) = x^{p-1} + \cdots + x + 1$. All coefficients are units, so the Newton polygon is a single horizontal edge from $n = 0$ to $n = p-1$ of slope $0$ and horizontal length $p-1$. The theorem says that $\Phi_p$ has $p-1$ zeros of valuation $0$, that is, $p-1$ roots in the closed unit disc; these are the primitive $p$-th roots of unity, and the statement is the Newton polygon form of their integrality. Similarly, for $\Phi_{p^n}$ the polygon has slope $0$ and length $(p-1)p^{n-1}$, recovering the degree count of the primitive $p^n$-th roots of unity.

Example (a totally ramified polynomial). Let $f(x) = x^p - p$. Its Newton polygon has the single edge from $(0,1)$ to $(p,0)$, of slope $-1/p$ and horizontal length $p$. The theorem gives $p$ zeros of valuation $1/p$ in the closed unit disc. Directly, $x^p = p$ forces $p\,v(x) = v(p) = 1$, so $v(x) = 1/p$ for every root, and there are $p$ of them, the polynomial being of degree $p$ with all roots in $\mathbb{C}_p$. The polygon and the computation agree.

Example (a zero outside the disc). Let $f(x) = 1 - px$, of Newton polygon the segment from $(0,0)$ to $(1,1)$, of slope $1$ and horizontal length $1$. The theorem gives one zero of valuation $-1$, namely $x = 1/p$, which lies outside the closed unit disc; consistently, the polygon has no edge of slope $\leq 0$ and $f$ has no zero in the closed unit disc. The cyclotomic example above has the opposite shape: all its edges have slope $0$, and all its zeros lie on the unit circle.

Strassmann's Theorem

Theorem (Strassmann). Let $f = \sum_{n \geq 0} a_n x^n \neq 0$ converge on the closed unit disc and let $$ N = \max \bigl\{ n : \lvert a_n \rvert = \max_{k} \lvert a_k \rvert \bigr\} $$ be the largest index of maximal absolute value. Then $f$ has at most $N$ zeros in the closed unit disc, counted with multiplicity.

Proof. The point $(N, v(a_N))$ is the rightmost point of the Newton polygon at minimal height $v(a_N) = \min_k v(a_k)$. The edges of the polygon with an endpoint of index $\leq N$ all have slope $\leq 0$, since the heights of the points of index $\leq N$ are $\geq v(a_N)$ and the hull rises to the left of $N$; by the corollary to the Newton polygon theorem these edges account for at most $N$ zeros in the closed unit disc, the total horizontal length of the edges of index $\leq N$ being at most $N$. The edges with both endpoints of index $> N$ have positive slope, since $N$ is rightmost of minimal height and the slopes increase to the right, so they account only for zeros of negative valuation, which lie outside the closed unit disc.

Strassmann's theorem is the standard device for bounding the number of zeros of a $p$-adic power series, and it is used in the theory of the Artin–Hasse exponential, in the proof that certain series are invertible on the unit disc, and in the theory of $p$-adic differential equations.

The Exponential, the Logarithm and the Structure of $\mathbb{Q}_p^\times$

The Explicit Discs

Theorem. Over $\mathbb{Q}_p$:

(a) the exponential $\exp x = \sum_n x^n/n!$ converges on the disc $B(0, \rho_{\exp})$ with $$ \rho_{\exp} = \lvert p \rvert^{1/(p-1)} = p^{-1/(p-1)}, $$ so on $p\mathbb{Z}_p$ for $p$ odd and on $4\mathbb{Z}_2$ for $p = 2$;

(b) the logarithm $\log(1+x) = \sum_{n \geq 1}(-1)^{n+1}x^n/n$ converges on the open unit disc $B(0,1) = p\mathbb{Z}_p$ and diverges at every point of the unit circle: the $n$-th term at a point of $\lvert x \rvert = 1$ has absolute value $\lvert 1/n \rvert = p^{v_p(n)}$, which does not tend to $0$;

(c) for $p$ odd, $\exp$ and $\log$ are mutually inverse isomorphisms of topological groups between $(p\mathbb{Z}_p,+)$ and $(1+p\mathbb{Z}_p,\times)$; for $p = 2$, they are mutually inverse isomorphisms between $(4\mathbb{Z}_2,+)$ and $(1+4\mathbb{Z}_2,\times)$.

Proof. Parts (a) and (b) are the computations of Analytic Functions and Power Series, with $q = p$ and $\lvert p \rvert = p^{-1}$, giving $\rho_{\exp} = p^{-1/(p-1)}$; for $p$ odd this exceeds $p^{-1} = \lvert p \rvert$, so the disc contains $p\mathbb{Z}_p$, while for $p = 2$ it equals $2^{-1}$ and the disc is $4\mathbb{Z}_2$. Part (c) is the general theorem of Non-Archimedean Analysis specialised to $\mathbb{Q}_p$.

The Structure of the Multiplicative Group

Theorem. The multiplicative group of $\mathbb{Q}_p$ decomposes as $$ \mathbb{Q}_p^\times \cong \langle p \rangle \times \mu_{p-1} \times (1+p\mathbb{Z}_p) \cong \mathbb{Z} \times \mathbb{Z}/(p-1) \times \mathbb{Z}_p \qquad (p \text{ odd}), $$ $$ \mathbb{Q}_p^\times \cong \langle 2 \rangle \times \{\pm 1\} \times (1+4\mathbb{Z}_2) \cong \mathbb{Z} \times \mathbb{Z}/2 \times \mathbb{Z}_2 \qquad (p = 2). $$

Proof. Every $x \in \mathbb{Q}_p^\times$ is uniquely $x = p^{v(x)} u$ with $u \in \mathbb{Z}_p^\times$, so $\mathbb{Q}_p^\times \cong \langle p \rangle \times \mathbb{Z}_p^\times$. The unit group splits as $\mu_{p-1} \times (1+p\mathbb{Z}_p)$ for odd $p$ and as $\{\pm 1\}\times(1+4\mathbb{Z}_2)$ for $p = 2$ by the Teichmüller decomposition of Non-Archimedean Analysis, the Teichmüller part being $\mu_{p-1}$ for odd $p$ and $\{\pm 1\}$ for $p = 2$; the inversion of the previous theorem identifies $1+p\mathbb{Z}_p$ with the additive group $p\mathbb{Z}_p \cong \mathbb{Z}_p$ and $1+4\mathbb{Z}_2$ with $4\mathbb{Z}_2 \cong \mathbb{Z}_2$.

Corollary. The torsion subgroup of $\mathbb{Q}_p^\times$ is $\mu_{p-1}$ for odd $p$ and $\{\pm 1\}$ for $p = 2$; the group $\mathbb{Q}_p^\times$ is not finitely generated; the group of squares has index $4$ for odd $p$ and index $8$ for $p = 2$; and for an odd prime $p$ the subgroup of $n$-th powers has index $n \cdot \gcd(n, p-1) \cdot \lvert n \rvert_p^{-1}$ in $\mathbb{Q}_p^\times$, while for $p = 2$ the index is $n \cdot \gcd(n,2) \cdot \lvert n \rvert_2^{-1}$.

Proof. The torsion statement is immediate from the decomposition, since the last factor is torsion-free. The index of the $n$-th powers is the product of the indices on the three factors of the decomposition: on the copy of $\mathbb{Z}$ the image is $n\mathbb{Z}$, of index $n$; on the finite cyclic factor the image has index the number of $n$-th roots of unity in it, which is $\gcd(n,q-1)$ with $q-1 = p-1$ for odd $p$ and $q - 1 = 1$ for $p = 2$; and on the additive factor $\mathbb{Z}_p$ the image is $n\mathbb{Z}_p$, of index $\lvert n \rvert_p^{-1}$. For $n = 2$ and odd $p$ this gives $2 \cdot 2 \cdot 1 = 4$, and for $p = 2$ it gives $2\cdot 2\cdot 2 = 8$.

The logarithm can be extended beyond the principal units: on the units one sets $\log_p(u) = \tfrac{1}{p-1}\log(u^{p-1})$, which is legitimate because $u^{p-1} \equiv 1 \pmod p$, and the definition is extended to all of $\mathbb{Q}_p^\times$ by $\log_p(p) = 0$ together with multiplicativity. The result is the Iwasawa logarithm on $\mathbb{Q}_p^\times$, a group homomorphism to the additive group $\mathbb{Q}_p$ that vanishes exactly on the roots of unity and on the powers of $p$. The extension and its properties are used in the theory of $p$-adic $L$-functions, developed .

Weierstrass Preparation on the Unit Disc

Theorem (Weierstrass preparation). Let $f \in \mathbb{Z}_p[[x]]$ be a power series with coefficients in $\mathbb{Z}_p$ such that $f \bmod p$ is a nonzero polynomial of degree $d$. Then there are a unique monic polynomial $$ P(x) = x^d + b_{d-1}x^{d-1} + \cdots + b_1 x + b_0 \in \mathbb{Z}_p[x] $$ of degree $d$ and a unique power series $U \in \mathbb{Z}_p[[x]]$ with $U \bmod p = 1$, such that $$ f(x) = P(x)\,U(x). $$ The factor $U$ is invertible in $\mathbb{Z}_p[[x]]$, and the zeros of $f$ in $\mathbb{Z}_p$ are exactly the roots of $P$ in $\mathbb{Z}_p$; consequently $f$ has at most $d$ zeros in $\mathbb{Z}_p$, counted with multiplicity.

Proof sketch. Write $f = \sum_n a_n x^n$ and $d = \deg(f \bmod p)$, so that $a_d$ is a unit and $a_n \in p\mathbb{Z}_p$ for $n > d$. The division algorithm for the polynomial $P$ against the series $f$ is run in the reverse direction: one solves recursively for the coefficients of $P$ and $U$ from the equations $a_n = b_n + u_n + \sum_{j} b_j u_{n-j}$, using the fact that the coefficient of the highest term is a unit and that the remaining coefficients of $f$ lie in $p\mathbb{Z}_p$, which makes the recursion contract by a factor $p$ at each step. The fixed point is the required pair, and uniqueness follows from the invertibility of $U$.

Corollary (the Weierstrass degree). For $f \in \mathbb{Z}_p[[x]]$ with $f \bmod p \neq 0$, write $d = \deg(f \bmod p)$. Then the zeros of $f$ in $\mathbb{Z}_p$ are the roots of the polynomial factor $P$, so there are at most $d$ of them, and the number with $\lvert x \rvert < 1$ and the number with $\lvert x \rvert = 1$ are read from the Newton polygon of $P$. In particular, a series whose reduction is linear has at most one zero in $\mathbb{Z}_p$.

Proof. The zeros in $\mathbb{Z}_p$ are the roots of $P$ by the preparation theorem, and the Newton polygon of $P$ counts the roots by their valuation, by the corollary to the Newton polygon theorem; the two valuations $0$ and positive separate the unit circle from the open unit disc. For $d = 1$ there is at most one root in all.

Example. The polynomial $f(x) = x - p$ has $f \bmod p = x$, of degree $1$, so the theorem says it has exactly one zero in $\mathbb{Z}_p$, namely $x = p$, and the factorisation is $f = (x-p)\cdot 1$. The polynomial $f(x) = x^2 - 2$ has $f \bmod p = x^2-2$ of degree $2$ for every odd $p$, so its Weierstrass degree is $2$ and it has at most two zeros in $\mathbb{Z}_p$. For $p = 7$ the reduction has the two simple roots $\pm 3$ in $\mathbb{F}_7$, so $f$ has the two zeros $\pm\sqrt{2}$ in $\mathbb{Z}_7$; the Newton polygon is the horizontal edge of slope $0$ and length $2$ from $(0,0)$ to $(2,0)$, which is the statement that both zeros have valuation $0$. For $p = 5$ the reduction has no root in $\mathbb{F}_5$, so $f$ has no zero in $\mathbb{Z}_5$; the Newton polygon is the same horizontal edge, and it now says that the two roots have valuation $0$ and lie in the unramified quadratic extension of $\mathbb{Q}_5$, which is where $\sqrt{2}$ lives. The polygon counts the roots of the series in an algebraic closure by valuation, while the preparation theorem decides which of them lie in the ground field.

The preparation theorem applies to the unit disc. Its generalisation to a closed disc of any radius is obtained by rescaling, and its generalisation to several variables and to affinoid algebras is the central structure theorem. Init is the tool that factors a linear differential equation into an invertible part and a finite-dimensional part.

Summary

The $p$-adic fields supply the concrete setting of non-Archimedean analysis: $\mathbb{Q}_p$ is complete, locally compact and totally disconnected with compact unit ball $\mathbb{Z}_p$, while $\mathbb{C}_p$ is complete and algebraically closed but not locally compact, so that the compactness arguments of the theory are available at the finite places and must be replaced by affinoid arguments elsewhere. A ball of $\mathbb{Q}_p$ splits into exactly $p$ disjoint clopen balls of the next level, and the splitting iterates.

Every continuous function on $\mathbb{Z}_p$ is uniformly continuous, and Mahler's theorem expands it uniquely as $\sum_n a_n \binom{x}{n}$ with $a_n \to 0$, the expansion being an isometry onto the sequence space $c_0$; the expansion is the concrete form of the duality between continuous functions and distributions used. The Newton polygon of a power series converging on the unit disc computes the number and the valuations of its zeros from the valuations of its coefficients, and Strassmann's theorem bounds the number of zeros in the unit disc by the largest index of maximal coefficient.

The exponential converges on the disc of radius $\lvert p \rvert^{1/(p-1)}$, the logarithm on the open unit disc, and the two are mutually inverse isomorphisms between the additive group $p\mathbb{Z}_p$ and the principal units $1+p\mathbb{Z}_p$ for odd $p$, and between $4\mathbb{Z}_2$ and $1+4\mathbb{Z}_2$ for $p = 2$. Consequently $\mathbb{Q}_p^\times$ is the product of the powers of $p$, the Teichmüller roots of unity and a pro-$p$ group isomorphic to the additive integers, and its torsion, its squares and its subgroups of $n$-th powers are computable. Finally, the Weierstrass preparation theorem factors a power series over $\mathbb{Z}_p$ with nonzero reduction into a monic polynomial of degree equal to the degree of the reduction and an invertible series, so that the zeros in $\mathbb{Z}_p$ are the roots of a polynomial and are finitely many.

Summary of Notation

Symbol	Meaning
$p$	A prime, the residue characteristic
$\mathbb{Q}_p$, $\mathbb{Z}_p$	Field of $p$-adic numbers, ring of $p$-adic integers
$v = v_p$, $\lvert \cdot \rvert_p$	$p$-adic valuation and absolute value, $\lvert p \rvert_p = p^{-1}$
$\mathbb{C}_p$	Completion of an algebraic closure of $\mathbb{Q}_p$
$\mathbb{F}_p$	Residue field of $\mathbb{Z}_p$
$B(a,r)$, $\overline{B}(a,r)$	Open and closed discs
$\rho_{\exp} = p^{-1/(p-1)}$	Radius of convergence of $\exp$ over $\mathbb{Q}_p$
$\binom{x}{n}$	Binomial polynomial, integer-valued on $\mathbb{Z}$
$\Delta$	Difference operator, $(\Delta f)(x) = f(x+1)-f(x)$
$C(\mathbb{Z}_p,\mathbb{Q}_p)$	Banach space of continuous functions on $\mathbb{Z}_p$
$c_0$	Sequences tending to $0$, with the supremum norm
$a_n = \Delta^n f(0)$	Mahler coefficients of $f$
Newton polygon	Lower convex hull of the points $(n, v(a_n))$
$m$, $\ell$	Slope and horizontal length of an edge of the Newton polygon
$d = \deg(f \bmod p)$	Weierstrass degree of $f \in \mathbb{Z}_p[[x]]$
$\mu_{p-1}$	Teichmüller roots of unity in $\mathbb{Z}_p^\times$
$1+p\mathbb{Z}_p$, $1+4\mathbb{Z}_2$	Principal units at which $\log$, $\exp$ are inverse
$\log_p$	Iwasawa logarithm on $\mathbb{Q}_p^\times$, $\log_p(p) = 0$
$\lVert f \rVert_\infty$	Supremum norm on $C(\mathbb{Z}_p,\mathbb{Q}_p)$